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Characterization of preliminary breast tomosynthesis data
h [electronic resource] :
noise and power spectra analysis /
by Madhusmita Behera.
[Tampa, Fla.] :
University of South Florida,
Thesis (M.S.B.E.)--University of South Florida, 2004.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
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Title from PDF of title page.
Document formatted into pages; contains 105 pages.
ABSTRACT: Early detection, diagnosis, and suitable treatment are known to significantly improve the chance of survival for breast cancer (BC) patients. To date, the most cost effective method for screening and early detection is screen-film mammography, which is also the only tool that has demonstrated its ability to reduce BC mortality. Full-field digital mammography (FFDM) is an extension of screen-film mammography that eliminates the need for film-processing because the images are detected electronically from their inception. Tomosynthesis is an emerging technology in digital mammography built on the FFDM framework, which offers an alternative to conventional two-dimensional mammography. Tomosynthesis produces three-dimensional (volumetric) images of the breast that may be superior to planar imaging due to improved visualization. In this work preliminary tomosynthesis data derived from cadaver breasts are analyzed, which includes volume data acquired from various reconstruction techniques as well as the planar projection data. The noise and power spectra characteristics analyses are the focus of this study. Understanding the noise characteristics is significant in the study of radiological images and in the evaluation of the imaging system, so that its degrading effect on the image can be minimized, if possible and lead to better diagnosis and optimal computer aided diagnosis schemes. Likewise, the power spectra behavior of the data are analyzed, so that statistical methods developed for digitized film images or FFDM images may be applied directly or modified accordingly for tomosynthesis applications. The work shows that, in general, the power spectra for three of the reconstruction techniques are very similar to the spectra of planar FFDM data as well as digitized film; projection data analysis follows the same trend. To a good approximation the Fourier power spectra obey an inverse power law, which indicates a degree of self-similarity. The noise analysis indicates that the noise and signal are dependent and the dependency is a function of the reconstruction technique. New approaches for the analysis of signal dependent noise were developed specifically for this work based on both the linear wavelet expansion and on nonlinear order statistics. These methods were tested on simulated data that closely follow the statistics of mammograms prior to the real-data applications. The noise analysis methods are general and have applications beyond mammography.
Adviser: John J. Heine.
x Biomedical Engineering
t USF Electronic Theses and Dissertations.
Characterization of Preliminar y Breast Tomosynthesis Data: Noise and Power Spectra Analysis by Madhusmita Behera A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Biomedical Engineering Department of Chemical Engineering College of Engineering University of South Florida Major Professor: John J. Heine, Ph.D. Maria Kallergi, Ph.D. Marla Hersh, M.D. William E. Lee III, Ph.D. Date of Approval: July 06, 2004 Keywords: mammography, filtering, signal-depe ndent noise, wavelet-expansion, Fourier transform Copyright 2004, Madhusmita Behera
ACKNOWLEDGEMENTS In my two years at USF and the Imaging Scie nce Research Division (ISRD), I have been fortunate to meet some wonderf ul people who have directly or indirectly influenced this work and my achievements. I wish to take th is opportunity to cove y my appreciation for them. I would first like to thank Dr Maria Kallergi for providing me the opportunity to work at ISRD, which helped me define my career pa th. She has been a constant source of support and inspiration, never letting me falter in my path at any moment. She has been a wonderful person to work with and I just cannot thank her enough. I would like to thank my advisor and mentor Dr. John Heine, without whose help I would have never been able to achieve what have today. He has been guiding me for more than two years now and I have learned more from him than I had in 16 years of formal education. His guidance has been invaluable Working with him has provided me with the self confidence that I had been completely lacking in before. I would like to thank Dr. William Lee for gi ving me an opportunity in the Biomedical Engineering program and for his guidance in the last two years. I woul d like to thank Dr. Marla Hersh for being on my co mmittee and for giving me her valuable time to review my work.
Among my ISRD friends and coll eagues, I would like to than k Joe and Angela for their support during this work. Mugdha, Anand and Anu have been very helpful and their moral support has been invaluable. Lastly, I would like to thank my family and friends. Without their support and encouragement I could not have achieved this.
i TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES v LIST OF ABBREVIATIONS viii ABSTRACT x CHAPTER 1 INTRODUCTION 1 1.1 BC Imaging and Mammography 3 1.1.1 Magnetic Resonance Imaging 3 1.1.2 Breast Ultrasound 3 1.1.3 Positron Emission Tomography 3 1.1.4 Mammography 4 1.2 Screen-Film Mammography 6 1.3 Full Field Digital Mammography 7 1.4 Tomosynthesis 9 1.5 Mammography Data 12 1.6 Image Noise 22 1.7 Spectral Analysis 23
ii CHAPTER 2 SIGNAL DEPENDENT NOISE 25 2.1 Noise Models 25 2.1.1 Additive Noise 26 2.1.2 Multiplicative Noise 26 2.1.3 Signal-dependent Noise 27 2.2 SDN and Mammography 28 2.3 Imaging Parameters 30 2.4 Related Work on Noise Analysis 31 CHAPTER 3 NOISE CHARACTER IZATION AND MODELING 37 3.1 Introduction 37 3.2 Simulated Data 38 3.2.1 Simulation One: 1/ f Field 38 3.2.2 Simulation Two: Mammographic SDN Simulation 40 3.3 Noise Estimation 42 3.3.1 Method I: The Filtering Method 44 3.3.2 Method II: The Order-Statistics Method 51 3.4 Noise Analysis of Tomosynthesis Data 57 3.4.1 Projection Data 58 3.4.2 Volume Data 60 3.4.3 FFDM Data 67 3.5 Discussion 68
iii CHAPTER 4 SPECTRAL ANALYSIS OF TOMOSYNTHESIS DATA 70 4.1 Power Spectrum Estimation 71 4.2 Constant Ring Model 72 4.3 Spectral Modeling of the Tomosynthesis Data 73 4.3.1 Projection Data 74 4.3.2 Volume Data 77 4.3.3 Spectral Comparison Between Volume Sets 81 4.4 SDN and Power Spectra Behavior 82 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 86 5.1 Conclusions 86 5.2 Recommendations 87 REFERENCES 88
iv LIST OF TABLES Table 3.1 The Average Slope of the Li ne and the Average Linear Correlation Across the 44 Slices of Each Volume Set 66 Table 3.2 The Correlation Between the Slopes of the Line (Shown in Fig.3.24 and Fig.3.25) Fitted Over the Volume Slices of the Four Reconstruction Sets 66
v LIST OF FIGURES Figure 1.1 Anatomy of the Female Breast 2 Figure 1.2 Schematic Representation of Screen-film Mammography System 5 Figure 1.3 The Working of a Digital Mammography System 8 Figure 1.4 The X-ray Beam Creates 2D Projection View of the 3D Breast 10 Figure 1.5 Comparison of the Visibility of a Lesion in 2D and 3D Mammograms 11 Figure 1.6 Filtered Back Projection Showing Projection (left) and Backpr ojection (right) 16 Figure 1.7 The 2-D Projection Image from the Cadaver 18 Figure 1.8 Volume Slice Example of a Cadaver from FBP and ART 19 Figure 1.9 Volume Slice from GF BP (left) and OSBP (right) 20 Figure 1.10 ROI Inscribed Within the Breas t (top) and Excised ROI (bottom) 21 Figure 3.1 The Simulated 1/ f Field 39 Figure 3.2 Simulated Image Gene rated as a True Mammogram 41 Figure 3.3 A Zero-mean Gaussian Noise Field 43 Figure 3.4 The Noisy Image f which is Contaminated with SDN with = 2 44 Figure 3.5 d1 Image 46 Figure 3.6 d2 Image 46 Figure 3.7 The f1 Version by the Wavelet Expans ion of Image Shown in Fig.3.4 47
vi Figure 3.8 The Signal Average versus Noise Variance 49 Figure 3.9 The Theoretical Curve (jagged) against the Empirical Curve (solid) Shown in Fig.3.8 49 Figure 3.10 This Shows the Non-signal Dependent Case 50 Figure 3.11 The Empirical Curve (jagged lin e) is Linearly Fit (solid line) 51 Figure 3.12 The Jagged Curve (empirical ) Shows the Noise as a Quadratic Function of Signal 52 Figure 3.13 Theoretical Curve agai nst Empirical Curve (jagged) 53 Figure 3.14 The Empirical cu rve (jagged line) is Linear ly Fitted (solid line) 54 Figure 3.15 The Noise Variance is Plotte d as a Function of the Mean Signal 55 Figure 3.16 The Noise Variance is Plotte d as a Function of the Mean Signal 55 Figure 3.17 The Noise Variance (jagged) is Shown When the Noise Term Contains Mean Value Equal to Five Times the Standard Deviation 56 Figure 3.18 The Noise Variance when Mean of Noise is One 57 Figure 3.19 The Noise Variance (jagged) as a Function of the Mean Signal 58 Figure 3.20 The Slope Across 11 Projection Images 59 Figure 3.21 The Noise Power for Each of the11 Projection Images (asterisks) 60 Figure 3.22 The Noise Estimated From GFBP (top) and OSBP (bottom) 62 Figure 3.23 The Noise Estimated From FBP (top) and ART (bottom) 63 Figure 3.24 The Slope of the Line fo r GFBP (top) and OSBP (bottom) 64 Figure 3.25 The Slope of the Line for FBP (top) and ART (bottom) 65 Figure 3.26 The Average Noise Power (diamonds) 67 Figure 3.27 The Noise Variance from FFDM Data (jagged) Shown as a Function of the Mean Signal 68 Figure 4.1 Raw Image (left) and the Im age in Frequency Domain (right) 71
vii Figure 4.2 The PS of the Projection Data (top) and Log Display (bottom) 75 Figure 4.3 The Normalized Ense mble PS of Projection Data 76 Figure 4.4 The Absolute Ensemble PS (solid) of Projection Data 77 Figure 4.5 The PS of a Volume Slic e from the ART Method (diamonds) 78 Figure 4.6 The PS of a Volume Slice from the GFBP Method (diamonds) 78 Figure 4.7 The PS of a Volume Slice from the OSBP Method (diamonds) 79 Figure 4.8 The PS of a Volume Slice from the FBP Method 79 Figure 4.9 The Normalized Ensemble PS from Volume Slices 80 Figure 4.10 The Absolute Ensemble PS from Volume Slices 80 Figure 4.11 The Comparison of Tw o Sets of Volume Data 81 Figure 4.12 PS of an Image when the Noise Modulating it is Zero-mean 83 Figure 4.13 The PS of z ( x y ) 84 Figure 4.14 The PS of s ( x y ) 84
viii LIST OF ABBREVIATIONS BC Breast Cancer ACS American Cancer Society ART Algebraic Reconstruction Technique DQE Detective Quantum Efficiency FFDM Full-field Digital Mammography FBP Filtered Back-Projection GE General Electrics GFBP Generalized Filtered Back-Projection MAS Milli-Ampere Seconds MRI Magnetic Resonance Imaging MTF Modulation Transfer Function NMNV Non-stationary Mean Non-stationary Variance OSBP Order Statistics Based Back-Projection PET Positron Emission Tomography PS Power Spectra ROI Region of Interest RV Random Variable SDN Signaldependent Noise
ix SNR Signal-to-Noise Ratio TD Tomosynthesis Data
x CHARACTERIZATION OF PRELIMINARY BREAST TOMOSYNTHESIS DATA: NOISE AND POWER SPECTRA ANALYSIS Madhusmita Behera ABSTRACT Early detection, diagnosis, and suitable treat ment are known to significantly improve the chance of survival for breast cancer (BC) pa tients. To date, the most cost effective method for screening and early detection is screen-film mammography, which is also the only tool that has demonstrated its ability to reduce BC mortality. Full-field digital mammography (FFDM) is an extension of sc reen-film mammography that eliminates the need for film-processing because the images are detected electronically from their inception. Tomosynthesis is an emerging technology in digital mammography built on the FFDM framework, which offers an alternative to conventional two-dimensional mammography. Tomosynthesis produces three-dimens ional (volumetric) images of the breast that may be superior to planar imaging due to improved visualization.
xi In this work preliminary tomosynthesis data derived from cadaver breasts are analyzed, which includes volume data acquired from va rious reconstruction techniques as well as the planar projection data. The noise and power spectra characteristics analyses are the focus of this study. Understanding the noise characteristics is signi ficant in the study of radiological images and in the evaluation of the imaging system, so that its degrading effect on the image can be minimized, if possible and lead to be tter diagnosis and optimal computer aided diagnosis schemes. Likewise, the power spectra behavior of the data are analyzed, so that statistical methods developed for digitized f ilm images or FFDM images may be applied directly or modified accordingly for tomosynthesis applications. The work shows that, in general, the power spectra for three of the reconstruction techniques are very similar to the spectra of planar FFDM data as well as digitized film; projection data analysis follows the same trend. To a good approximation the Fourier power spectra obey an inverse power law, wh ich indicates a degree of self-similarity. The noise analysis indicates that the noise and signal are dependent and the dependency is a function of the reconstruction technique. New approaches for the analysis of signal dependent noise were developed specifically for this work based on both the linear wavelet expansion and on nonlinear order st atistics. These methods were tested on simulated data that closely follow the statis tics of mammograms prior to the real-data applications. The noise analysis methods are general and have applications beyond mammography.
1 CHAPTER 1 INTRODUCTION Breast cancer is the most commonly diagnosed cancer and the second leading cause of cancer death among women in United States [ ACS 2003 ], following only lung cancer. In 2004, 40,580 people (40,110 women and 470 men) are projected to die of BC [Jemal et al 2004 ]. Statistics show that the lifetime risk of BC in the United States has almost tripled in the past 50 years. In the 1940Â’s, a womanÂ’s lifetime risk of BC was 1 in 22 that increased to 1 in 8 in the year 2002 [ MBCC 2002 ]. Although it is primarily a disease of wo men, about 1% of BCs occur in men [ Jemal et al 2003; Anderson et al 2004 ]. Breast cancer is caused by th e uncontrolled growth of cells in the breast. The female breast as illustrate d in Fig.1.1 is primarily composed of lobules (milk-producing glands), ducts (milk channels that link the lobules to the nipple), and the stroma. Approximately 90% of BCs begin in th e milk ducts, and 10% begin in the lobules of the breast [ Wellings et al 1975 ]. When the cancer cells re main within the ducts, the cancer is referred as in situ and the probability of cure is high. Once the cells have broken through the wall of a duct or lobule, the cancer is called invasive. The most common types of invasive BCs are Ductal Carcinoma and Lobular Carcinoma.
2 Figure 1.1 Anatomy of the Female Breast [Image from: http://www.loradmedical.com/ly320.html] Before the 1990's, BC mortality rates were constant for nearly four decades. During 1989-1995 the BC mortality declined by 1.6% and by 3.5% from 1995-1999 [ MBCC 2002 ]. Most medical experts agree that this dec line in the mortality rate can be attributed to the increasing awareness in the public that leads to the early detection of BC followed by proper treatment and regular follow-up. This is in agreement with previous studies that have shown that early detection, diagnosis and suitable treatment can significantly improve the chance of survival for patients with BC [ Chan et al 1995; Lester 1984 ]. This can be successfully accomplished by effective screening methods of BC [ Yaffe 2000 ].
3 1.1 BC Imaging and Mammography It is established that early detection of BC can reduce BC mortality. This is most commonly accomplished with regular screen ing and mammographic imaging. The only definite method of determining the ma lignancy of the breast tissue is by a biopsy The breast biopsy involves removing the tissue samp le surgically, or w ith a less-invasive needle core sampling procedure, to determ ine whether it is can cerous or benign. Most biopsy methods rely on image guidance to help the radiologist or brea st surgeon precisely locate the lesion or abnormality within the breast. Imaging techniques of the breast are therefore vital for early detec tion of cancer, and localization of the suspicious lesion in the breast for a biopsy procedure. Some of the imaging modalities ava ilable today for breast imaging are: 1.1.1 Magnetic Resonance Imaging (MRI) This is a diagnostic procedure that uses magnetic fields and computers to create images of areas inside the breast. With MRI, the contrast between the soft tissues in the breas t is 10 to 100 times greater than that obtained with X-rays [ Azar 2000 ]. This is an expensive procedure. 1.1.2 Breast Ultrasound This technique uses sound wave s to create an image of the breast tissue and project it onto a computer screen. 1.1.3 Positron Emission Tomography (PET) This diagnostic procedure involves injecting the patient with a ra dioactive compound that is ta ken up by suspicious cells. The
4 positron radioactivity emitted by the compound is recorded by a PET camera and processed by computer. The areas of greates t metabolic activity light up on a computer generated image which assists th e radiologist in identifying su spicious tissue or lesion. 1.1.4 Mammography It is an X-ray screening technique that is used to create detailed images of the breast. Among the imaging syst ems discussed here, the most widely used modality for BC detection and diagnosis is mammography for cost effectiveness and its ability to reduce BC mortality. A mammogram is an X-ray projection of the breast. These projections are usually taken from twoviews: 1) Cranio-caudal, where X-rays are passed through the breast from top to bottom, and 2) Medio-lateral, where the X-rays are passed through the breast from the side The schematic representation of the mammographic imaging system is illustrated in Fig.1.2. When a mammogram is performed, a beam of X-ray is incident on a compressed breast. The energy spectrum of the beam is characteristic of the X-ray target filter and tube voltage The intensity of the beam is correlated with the breast size and composition to some degree due to the experience of the X-ray technician involved with the process and the automated exposure control of modern systems; basically larger and more dense the breast implies longer exposure times. The photon interaction with the breast involves both absorption and scattering. The absorption characteristic of a given tissue is depende nt upon the spectral character of the incoming beam. The X-ray photons exiting the breast pass through an anti-scatter grid before reach ing a phosphorous intensifying sc reen. If an X-ray photon is absorbed by the screen, light photons are emitte d that expose a film or more generally interact with some form of detector. In areas of the breas t where large absorption takes
5 place the signal is weakest and an area wh ere less absorption occurs, the signal is strongest. Thus, the resulting image represen ts a crude abstraction of the average attenuation properties of the breast above th e detector. As the X-rays pass through the breast, they are attenuated by the differe nt tissue densities within the breast. Figure 1.2 Schematic Representation of Screen-film Mammography System [Image from: http://detserv1.dl.ac.uk/Her ald/images/MammoSet.gif ]
6 The appearance of a female breast on a mamm ogram varies due to the differences in Xray attenuation in the relative am ounts of fat, connective and epithelial tissue. Fat appears radiolucent or dark on a mammogram, epithelial and connective tissues are radiographically dense and appear lighter or white in the deve loped image. Some relevant findings or abnormalities in a mammogram include [ Kaul et al 2002 ]: soft-tissue lesions These are recognized as a mass or an architectural distortion. A mass is often defined as a region of increased dens ity usually with a distinct edge, making it distinguishable from the surrounding breast tissue. Architectural distortions are irregular breast patterns caused by abnormal tissue. microcalcifications These are seen as small calcium deposits in the breast tissue. They can typically build up in clusters. Depending on their number in a cluster and the overall shape of the cluster they indi cate a possible risk of BC. 1.2 Screen-Film Mammography To date, the most cost effective method for screening and early detection is screen-film mammography, which is also the only tool that has demonstrat ed its ability to reduce BC mortality [ Yaffe 2000 ]. Conventional screen film mammography uses low energy X-rays that pass through a compressed breast duri ng a mammographic examination. The exiting X-rays are absorbed by film (screen film co mbination), which is then developed into a two dimensional or planar mammographic image, that is interpreted by the radiologist for diagnostic purposes, with the use of a lig ht box normally. Although there are benefits
7 associated with film-based screening, there ar e considerable interpretation errors that are in part caused by the image quality, restricted intensity latitude, and close similarities between normal tissue and suspicious tissue. In order to process these images with computer techniques, they must be digitize d. Since this work involves the analysis of film-less acquired mammograms, film detected images will not be discussed further. 1.3 Full Field Digital Mammography FFDM is an advancement of sc reen-film mammography that eliminates the need for filmprocessing because the images are detected electronically for their inception. From a subjectÂ’s perspective, the examination is the same as in film mammography, where the breast is positioned between two flat plates and lightly compressed. In FFDM system, the screen-film image receptor is replaced with a flat detector which pr ovides an electronic signal that is proportional to the X-ray exposure [ Yaffe 2000 ].This system is intended to replace conventional film based imaging in the future. FFDM acquired images are ideal for digital or computer processing without further manipulation and are viewed on softcopy display when viewed by the mammographer FFDM offers numerous advantages such as digital image management, digital da ta transfer, digital image processing. With the capability to process the digital images with a computer, new medical applications will emerge, for example, real-time Computer-Aided Diagnosis (CAD), contrast medium imaging etc. Fig.1.3 illustrates the working of a mammography system [ Moore 2001 ]. All mammography begins with an X-ray source, whic h projects photons th rough a patient. In
8 conventional mammography, the development of a film-screen cassette generates the mammogram. There are two principal methods of detection in the digital version of mammography: indirect, in which a scintillator converts X-rays into visible light that is collected by a solid-state dete ctor; and direct, which invol ves employing a coating of a material such as amorphous selenium to conve rt the X-rays into electron-hole pairs for sensing by a transistor array. Figure 1.3 The Working of a Digital Mammography System [Image from: http://www.spectrum.ieee.or g/pubs/spectrum/0501/cancerf1.html ] Standard planar mammography techniques, film or FFDM, suffer from the limitation that three-dimensional anatomical information is projected onto a two-dimensional detector. Thus, the spatial arrangement of tissues cannot be pr eserved, causing the loss of
9 morphological information. Cancers may be masked by radiographically dense fibroglandular breast tissue which may encompass th e cancer. Likewise, the true character of breast tissue is somewhat lost or obscured. 1.4 Tomosynthesis Tomosynthesis is a forthc oming technology in digital mammography built on the FFDM framework, which offers an alternative to conventional two-dimensional mammography. Tomosynthesis is a technique that allows the radiologist to vi ew individual planes of the breast, potentially reducing th e problem of superimposed structures that may limit conventional mammography techniques. Sin ce this technology is new and very few systems are available today, most of the resear ch on the subject has been limited to using phantom images. The tomosynthesis data (TD) used for this study are images generated from cadaver breasts. How Tomosynthesis works Tomosynthesis combines a conve ntional tube with a digital Xray detector and sophisticated computer algorithms. In this method, multiple projection images from different angles are acquired as the X-ray tube is moved in an arc above the stationary breast and digital detector. A r econstruction technique is then applied to capture a volume of three-dimensional inform ation from the series of projection images. The image obtained at each angle is of low radiation dose, with th e total radiation dose required for imaging the entire breast being somewhat less than the dose used for a twoview film-screen mammogram. Since the resulting volumetric information is in digital format, it can be reconstructed in any plane. A key aspect of tomosynthesis is that in the
10 reconstructed volume, the in -plane resolution is 100m and the inter-plane resolution is 1000m which is an artifact of limited number of projections. The reconstructed volume will have slices available at various depths or planes within the breast, as illustrated in Fig.1.4. It is anticipated that by stepping through the slices, one can eliminate the superimposed tissues that might be hiding the tumor or malignancy. Figure 1.4 The X-ray Beam Creates 2D Projec tion View of the 3D breast. A 3D reconstruction of the breast can be viewed as a sequence of 2D planes (shown as colored). Thus, the tomosythesis image data represents two distinct representations (1) the 12 projections that appear similar to regular planar X-ray image but with more variation due to the reduced X-ray strength, and (2) the r econstructed volumetric data Prospective benefits of tomosynthesis Tomosynthesis allows three-dimensional (volumetric) imaging of the breast and poten tially allows BC to be better visualized
11 without the superimposed dense breast ti ssue that may obscure BC in conventional mammography. The potential benefit will be gr eatest in women with radiographically dense breasts. In addition, improved visibi lity of a lesion, lesion extent and lesion margins may improve specificity and treatm ent. Tomographic imaging provides three basic advantages over conven tional projection mammography as discussed by Dobbins et al [Dobbins et al 2003], which are significant for BC re search. First, it allows depth localization of a lesion. Sec ond, it improved conspicuity of structures by removing the visual clutter associated with overlying anatomy. Third, it im proves the contrast of local structures by restricting the overall image dynami c range to that of a single slice. These advantages provide an exciti ng opportunity for researchers to study BC in depth and the risk factors associated with it. This ha s been illustrated by an example in Fig.1.5. (a) (b) (c) (d) Figure 1.5 Comparison of The Visibility of a Le sion in 2D and 3D Mammograms.(a) 2D planar mammogram with 2 lesions barely vi sible. (b)Shows where lesions should be (c, d)Re-sliced planes of 3D rec onstruction clearly show each lesion
12 (e) (f) Figure 1.5 -continued (e, f) For comparison, shows how lesions would appear in the resliced phantom [Figures (1.4-1.5) from: http://clio.r ad.sunysb.edu/mipl/projects/tomosyn.html] The focus of this study is on the tomosynt hesis breast images that includes planar projection images and volumetric slices acquired from different reconstruction techniques. The methods developed in this work has also been demonstrated on 2dimensional FFDM data and simulated images for comparison. This study primarily has two aims: (1) to analyze the noise characteri stics of the TD and develop general signaldependent noise models and (2) analyze th e power spectra behavior of the TD. A description of the data that are used in this study is pr ovided in the following section. 1.5 Mammography Data The planar FFDM data analyzed in this work is of 100 m /pixel spatial resolution and of 14bit pixel dynamic range and was acqui red with the General Electrics (GE) Senographe 2000D. The tomosynthesis data basica lly has two represen tations (1) the 12 projections that are the same as the plan ar FFDM images acquired from the planar FFDM system and (2) the reconstructed volume consisting of 44 slices.
13 As described in section 1.4, digital tomos ynthesis is a technique for producing slice images (volume) using modified conventiona l X-ray systems with a limited number of projections. By shifting and adding these pr ojection images, specific planes may be reconstructed. Different types of reconstr uction algorithms are used to produce the tomosynthesis volume images. Images used in this study are from cadave r breasts. This is a case-study of one cadaver breast. There is a set of 2-D projections and also 4 different reconstructed datasets. The projection dataset consists of 12 projection images. Usually the 12th frame has only very low dose, and nominally the tube position doe sn't change between the 11th and the 12th shot. So, for all practical purposes, only the fi rst 11 images are considered in this study. The X-ray tube moves in an arc from -25 to +25 degrees with 5 degree increments and multiple projections are acquired. However, th e pivot point is located 22.4 cm above the detector, and the tube arm is 44cm long. Therefor e, the effective angle with respect to the center of the detector is smaller; th e angles (in degrees) are approximately, -16.6249 -13.2829 -9.95229 -6.63021 -3.31371 0.00000 3.31371 6.63021 9.95229 13.2829 16.6249 The image obtained at each angle is of low radiation dose, with th e total radiation dose required for imaging the entire breast being somewhat less than the dose used to acquire the two views of a film-screen mammogram.
14 Digital tomosynthesis is unive rsally practiced with the sh ift-and-add or backprojection techniques [Dobbins et al 2003]. However, there are seve ral iterative reconstruction techniques described in the literature for th e reconstruction of a th ree-dimensional object from two-dimensional projection images [Colsher 1977]. Colsher discusses three basic approaches for reconstruction, namely, 1) su mmation approach, 2) Fourier techniques, and 3) algebraic methods. Some iterative a pproaches for reconstruction are algebraic reconstruction techniques (ART), simultane ous iterative reconstr uction technique and iterative least squares technique. Only one of these techniques has been used by GE for generating the volume data along with several backprojection techniques. The reconstruction algorithms used to genera te the volume data for the cadaver breasts are: Algebraic Reconstruction Technique Standard Filtered Back projection Order Statistics-Based Back projection Generalized Filtered Backprojection A brief description of the first three techniques is provided here. Algebraic Reconstruction technique (ART) This is an iterative reconstruction algorithm in which computed projections or ray sums of an estimated image are compared with the original projection measurements and the resul ting errors are applied to correct the image estimate. In ART, the corrections are computed and applied on a ray-by-ray or view by view basis. The manner in which the image converges depends on the order in which the ray-sums are considered. In most ART appli cations, the reconstruc ted image is assumed
15 to consist of an array of square pixels which are of uniform density. The computed projections are obtained by summ ing the values of the pixels whose centers lie within a path of finite width. The average error is then computed and added to the pixels included in the ray sum. Filtered Back projection (FBP) Filtered backprojection as a concept is relatively easy to understand and is one of the popular reconstructi on techniques, which is illustrated by an example in Figs. 1.6. Let's assume that we ha ve a finite number of projections of an object which contains radioactive sources (F ig. 1.6(left)). The projections of these sources at 45 degree intervals are represen ted on the sides of an octagon. Fig.1.6(right) illustrates the basic idea behind back projecti on, which is to simply run the projections back through the image (hence the name ``back projection'') to obtain a rough approximation to the original. The projections w ill interact contstructively in regions that correspond to the emittive sources in the origin al image. A problem that is immediately apparent is the blurring (star-like artifacts) that occur in other parts of the reconstructed image. The optimal way to eliminate these patterns in the noisel ess case is through a ramp filter. The combination of back projection and ramp filtering is known as filtered back projection.
16 Figure 1.6 Filtered Back Projection Showing Projection (left) and Backprojection (right) [Image from: http://www.owlnet.rice.ed u/~elec539/Projects97/cult/node2.html] Order-Statistics Based Back Projection This reconstruction method is based on simple backprojection, which generates high contrast reconstructions with minimized artifacts at a relatively low computational complexity [Claus et al 2002]. The first step in this method is a simple backprojection with an or der statistics-based operator (e.g., minimum) used for merging the backprojected images into a reconstructed slice. Accordingly, a given pixel value does not gene rally contribute to all slices. The percentage of slices where a given pixel value does not contribute, as well as th e associated reconstructed values, are collected. Using a form of re-p rojection consistency constraint, projection images are then updated, and the order statis tics backprojection r econstruction step is repeated, but now it uses the "enhanced" proj ection images calculated in the first step.
17 In the following pages, examples of the to mosynthesis images from the cadavers are illustrated. First, a 2-D projection image of a breast is demonstrated in Fig. 1.7 followed by a volumetric slice from the four recons tructions techniques in Fig.1.8 and Fig.1.9. In this study, the analysis has been performed on the 11 projection images and 44 volume slices. From each breast image (projection a nd volume), a large rectangular section was excised, which is termed as the Region of Interest (ROI). The experiments and analysis are restricted to this section which is ex tracted by user-interaction methods from the breast region. Fig.1.10 (top) illustrates the in scribed region or section on the breast, which is to be excised. The associat ed ROI is shown in Fig.1.10 (bottom)
18 Figure 1.7 The 2-D Projection Image from the Cadaver Breast
19 Figure 1.8 Volume Slice Example of a Cadaver from FBP and ART. The left image is obtained from the FBP and the right from the ART. Note that the appearance is not the same. This difference is brought out in the spectral analysis
20 Figure 1.9 Volume Slice from GFBP (left) and OSBP (right). Note that they are somewhat similar and are also similar to Fi g. 1.8 (right), these likenesses are indicated by the spectral analysis
21 Figure 1.10 ROI Inscribed Within the Breast (top) Excised ROI (bottom)
22 1.6 Image Noise All radiological images are corrupted with unwanted fluctuations or uncertainties that arise from several sources in the imaging system. These undesirable variations are referred to as noise or mottle by physicists and radiologists which is any component in the signal that interferes with the true signa l. From a general point of view, a noise may be described as the set of all obstructive si gnals superimposed on th e useful signal at a given location in an image [Bochud 1999]. In the context of medical imaging, the useful signal contains valuable dia gnostic information, whereas the noise represents a hindrance in understanding the information relayed by th e useful signal. Nois e is detrimental to radiological images because it im pairs the reliable detection of subtle or low contrast structures. In mammograms, the dominant cause of this problem arises from statistics of X-ray quanta, which can be compounded by noise fr om other sources, like structure of fluorescent screen, granularity of film emulsion etc. Understanding the noise characteristics is therefore significant in th e study of radiological images and in the evaluation of the imaging system, so that its degrading effect on the image can be minimized, if possible and l eads to better diagnosis. One aim of this work is to analyze the noise in TD, which is approached from the broad framework of signal dependent noise (SDN) analysis. In some cases it may be appropriate to study this type of noise through direct expe rimentation of the imaging system based on phantom image analyses and in other cases it may be that the images
23 are analyzed directly, if that is all that is available (if there is no access to imaging equipment). In this work, we follow that latter avenue since the imager is not local. Thus, in this work the general questions are addres sed: given an image, are the signal and noise related and if so, what is the functional c onnection? The methods developed to address these questions are therefore applicable in th e wider sense to all t ypes of data that may have signal dependent noise. This study has aimed to analyze the characteri stics of the noise in the images acquired by digital tomosynthesis. The results have been compared with the findings from 2D FFDM images. Two methods for estimating the noise are developed in this work. Prior to applying on mammographic data, these methods are first experimented and studied on simulated images in the blind, which forms a part of this work. Understanding the noise characteristics is significant in the study of radiological imag es and in the evaluation of the imaging system, so that its degrading effect on the image can be minimized, if possible and lead to better diagnosis and optimal computer aided diagnosis schemes. 1.7 Spectral Analysis The other aim on this work is to investig ate the spectral charact er of TD. Since the tomosynthesis is a newer technology, very little is discu ssed about the image characteristics in literature. In this wo rk, the frequency domain characteristics of tomosynthesis breast data are analyzed. Previ ous work by Heine et al has shown that the power spectra of FFDM images obey an i nverse power law, to a good approximation [Heine and Velthuizen 2002]. This also coincides with the spectral analysis of digitized
24 film data [Heine et al 1999]. In this work, the power spectra of the tomosynthesis projection and volume images have been studie d and compared with that of planar FFDM images. This analysis may be helpful in unde rstanding the image co rrelation and texture properties. The purpose of this work is to develop an understanding of the power spectra behavior so that statistical methods devel oped for digitized film images or FFDM images, may be applied to the tomosynthesis data. This work has been conducted as a two-part study: 1) Noise measurements from the simulated and the mammograhic (tomosynthesi s and FFDM) images, and 2) The spectral analysis of the tomosynthesis data and co mparison with regular FFDM data. From this point, the thesis has been organized as fo llows: Chapter 2 discusses the X-ray imaging system and the different sources of noise in detail. The different types of noise and the related work by others are discussed in th is chapter. The two methods for estimating noise are discussed in chapter 3 followed by the results from tomosynthesis images. The spectral analysis of the TD is discussed in chapter 4. Comparisons of the planar and volume slices are made. Conclusions from the work are discussed in chapter 5 including recommendations for future work.
25 CHAPTER 2 SIGNAL-DEPENDENT NOISE The predominant noise component in mammogram s is best defined as signal dependent due to the photon statistics responsible fo r the image creation, when considering two dimensional planar X-ray images. Although the focal point of this work is the analysis of tomosynthesis breast images, it is first necessa ry to consider the broader general idea of signal dependent noise without regard to mammography, which incl udes a summary of related work. In this chapter, general signal dependent noise models are discussed as well as the specific phenomena related to pl anar mammography images. Naturally, more emphasis is placed on the latter, which includ es a brief exposition on the imaging system as well as the specific noise mechanism. Th e more general idea of SDN should be taken in the context that the volume images are constructed from planar mammographic projections with various rec onstruction techniques. Thus, th e ideas that apply to the projections may not correspond exactly to the resulting noise qualities in the volume data. 2.1 Noise Models Like any physical measurement, the radiogra phic imaging system consists of errors and uncertainties which may be broadly distinguished as systemic and random errors
26 [Barrett 1981]. Systemic errors remain unchanged w ith every repetition of the process, such as geometric distortion, miscalibration of the detector, computation errors when the image is reconstructed etc. Random errors, on the other hand vary with every repetition of the same measurement. Certain examples of random noise are film grain noise, photon noise, electronic noise, interferen ce due to scattered radiation et c. This study concentrates exclusively on random noise, sp ecifically on the noise due to X-ray quanta, which is the most dominant form of noise in radiographic images [Barrett 1981]. In the most general terms, noi se associated with the resul ting image may have both signal dependent and signal independe nt components. Throughout the course of this study the term signal refers to the 2-D image signal unde r consideration. The following noise models are useful for a wide variety of situations: 2.1.1 Additive Noise This noise is independent of the signal. The noise model is of the form, n s r (2.1) where r is the noisy signal, s is the pure signal and n is signal independent random noise, that may have a spectral form other than flat (other than white noise). Example of this type of noise is thermal noise from the detector. 2.1.2 Multiplicative Noise This noise is modeled as s n r (2.2)
27 where r, s and n are the same as described above. Exam ple of this type is Â“film-grainÂ” or Â“speckleÂ” noise. 2.1.3 Signal-dependent Noise The above expression may be extended to a more general model 2 1) ( n n s f s r (2.3) where r is the noisy image, s is the pure signal, f is a general function and n1 and n2 are signal independent random noise processes [Froelich 1981]. The middle term in Eq. (2.3) gives the SDN component of r and the additive component is given by the last term. SDN is commonly encountered in many si gnal and image processing applications [Cunningham 1975]. Multiplicative noise is a limiting case of SDN, where the amplitude of the noise term is proportional to the value of the pure or noise-free signal [Aiazzi et al 1997]. A key aspect of SDN is that a certain amount of signal information is embedded in the noise [Kasturi 1983]. In the event in which the noise term dominates the signal term, the signal information presen t in the noise term may be greater than the signal term. Hence recovering the signal from the noise te rm in order to have it in a usable form becomes a significant task. Several studies have investigat ed the various forms of SDN. Various methods have been proposed in these studies to estimate the noise and to recover the sign al. A brief review of the work performed by others is presented in section 2.3.
28 2.2 SDN and Mammography The mammographic image formation process is initiated with the photon output of the Xray tube. If we assume that a uniform parallel beam interacts with an object (the beam is orthogonal to its face and we assume the object is a cube) that is characterized by a linear attenuation coefficient and thickness t, there is simple expression that relates the output beam (emerging through the other side of the object) with the incident beam ) exp( t I Ii o (2.4) This is known as BeerÂ’s law and is really a probabilistic expression in that the probability of the transmission is given by ) exp( t I Ii o The linear attenuation coefficient is a function of the spectral char acter of the incoming beam. Putting this in context of mammography, if we consider a photon incide nt on the breast above the detector at spatial location ( x y ) the probability of transmission through the breast of constant thickness t at ( x y ) is given by ) ( ) ) ( exp( ) ) , ( exp( y x P t y x dt z y xT z (2.5) Here, z may be considered as the average attenuation along the path above the image plane at ( x y ). The number of photons incident upon the breast is a statistical quantity that follows a Poisson distribution ) exp( ) ( ) ( m n m n pn n (2.6) where the units may be changed to include phot on density or total fluence, whichever is needed. Assume the above equation holds in the vicinity of the spatial location ( x, y ) and is similar for all ( x, y ), which is a generalization of a uniform beam from a statistical
29 point. Note that with the Poisson distribution the expected value < n > = m which is also equal to its variance. In this situation, th e expected signal and associated noise are proportional. Hence, the noise is signal dependent. Holding n constant for the moment, the probability of k photons transmitting through the breast at ( x, y ) with transmission probability p = PT ( x,y ) given n incident photons may be ex pressed as a conditional probability that follows a binomial distribution k n kp p k k n n n k p ) 1 ( )! ( ) ( (2.7) Including the statistical nature of n gives the total probability of transmitting k photons through the breast when there is an expected number of m incident photons traveling through ( x, y ). Substitution of Eq.(2.5) and Eq.(2.6) and letting m = pm gives ) exp( ) ( ) ( ) ( k pm pm n p n k pk k n n (2.8) The sum initiates at k because k cannot be less than n Thus, the transmitted photons also follow a Poisson distribution with the mean and variance attenuated by the transmission probability. Likewise, if the detector at ( x, y ) has a detection efficiency, (which is just the probability of detection), the detected signa l statistics also follows a Poisson law by the same arguments as above, whic h leads to what is termed as cascaded random process [ Barrett 1981 ]. These arguments have ignored sc attering, beam hardening and heel effects that are associated with the actual imaging process. Photon noise is the most dominant source of noi se in a radiographic image. This results in the final image consisting of a relatively small number of detected quanta since the
30 radiation dose delivered to the patient is li mited. As the dose increases, the number of photons per unit area of the receptor increases, an d the relative noise level decreases. The relative noise level is inversely proportional to the square root of the dose, i.e. reduction of the noise level to half will require that the dose is increased four times [ Yaffe 2000 ]; this follows by considering the relative signal to noise ratio m m /. The visibility of low-contrast objects is thus heavily de pendent on the relative noise level. One of the important considerations for the evaluation of any imaging system is the quality of the image produced [ Dobbins 1995 ]. The quality of the image acquired by any type of imaging system can be measured by certain imaging parameters like signal-tonoise ratio, modulation transfer function, de tective quantum efficiency etc, which are discussed in the next section. 2.3 Imaging Parameters The quantitative evaluation of an imaging system, digital or screen-film, can be performed by taking certain imaging concepts into consideration. Although they are not used for the analyses in this work sin ce this work focuses on SDN due to the photon counting process, a brief review of some of these concepts is provided here. Modulation transfer function (MTF) The sharpness of the imaging system is characterized by the MTF, which is basically the Fourier response due to a delta function spatial input.
31 Signal-To-Noise Ratio (SNR) This is the ratio of the useful information to the random fluctuations or noise that can obscure the us eful information in the image. High SNR is thus desirable in an imaging system for superior image quality. Noise power spectrum This is a measure of the no ise power per unit frequency, sometimes called the power spectral density Contrast Resolution Represents the number of shades of gray that a detector can capture. Flat-panel digital detectors typical ly offer resolution of 12-14 bits. Detective Quantum Efficiency (DQE) DQE is the measure of combined effect of noise, efficiency and contrast resolution performance of an imaging system. It is expressed as a function of the object detail or spatial freque ncy. High DQE is a widely accepted measure of improved digital image quality and object detectability. 2.4 Related Work on Noise Analysis A review of related SDN analyses is provided in this section. Cunningham et al have studied the problem of detecting a kn own 2-D signal or object in an image corrupted by SDN by approaching it from the classical sta tistical technique of hypothesis testing [ Cunningham et al 1976 ]. A general solution is formulated by the derivation of a decision rule using a likelihood ratio test for a signal corrupted by an unknown noise which may include SDN. Using the decision rule, the probability of
32 detection is evaluated from a prior knowledge of the noise and imaging system. The use of this technique is limited by the necessity of accurate prior knowle dge of the signal and the imaging system. But it may be useful in evaluating the relative performance of various imaging systems. Froehlich et al have proposed some estimator s for SDN related to film grain. The noise model used for this study is the general model shown in Eq. (2.3) [ Froehlich 1981 ]. The noise terms n1 and n2 in the model are assumed to be zero-mean and normal (Gaussian distributed) random variables (rv) and the pr obability density func tion of the signal is also assumed. One of the estimators proposed is minimum mean square error estimate (MMSE), which exhibits greater sensitivity to SDN term rather than the additive one. The performance of this estimator suffers if the assumed probability distribution function of the signal is not accurate. Another method is derived by weighted spatial averaging for estimating the sample mean. For example, the estimation procedure in an image will be to replace each pixel with the average of that pixel and its eight neighboring pixels. Kasturi et al have proposed a simple techni que for signal recovery from Poisson noise [ Kasturi et al 1983 a ]. In this work, the noisy image, R is modeled as ] [ 1 S P R (2.9) where ] [ S P represents a Poisson process with as the parameter and S is the noisy pure signal. An estimate of the signal was obtained by computing the mean, which was then subtracted from the noisy signal. A s econd estimate of the signal was then obtained
33 from the difference image which still containe d some signal information. This estimate was obtained by using the relationship 2Âˆ ÂˆdS (2.10) where 2Âˆd is the estimate of the local variance of the difference image. Another study by Kasturi et al has developed some methods for restoring the image by transformation of SDN to additi ve signal independent noise [Kasturi et al 1983 b]. One method involves transforming the SD process to an additive process using the general noise model given by Eq. (2.3). Another tec hnique is based on a contrast manipulation transformation. However, both the methods described here are bound by the assumption that the noise processes in the general model are normally distributed zero-mean processes. In a study by Kuan et al, non-stationary 2-D recursive filters were developed for image restoration based on a non-st ationary mean, non-stationa ry variance (NMNV) image model [Kuan et al 1984]. The recursive filters developed here are for restoration of images corrupted with multiplicative and Po isson noise. In the NMNV model, an image is decomposed into a nonstationary mean component and an uncorrelated residual random process that can be characterized by it s non-stationary varian ce. In this model, the non-stationary mean contains the gr oss structure of the image while the nonstationary variance contains the edge information.
34 Another study develops a method for imag e restoration in SDN using a Markovian covariance matrix [Kasturi et al 1984]. The image distribution is modeled as a spatially non-stationary process having a Mar kovian covariance model given by 2 1)] ( [ N N s f S R (2.11) where R and S are vectors representing the noisy observation and signal respectively. 1N and 2N are noise processes. Note that Eq.(2. 11) is similar to Eq.(2.3).The signal dependence is modeled as a memoryless spatia lly stationary process characterized by the matrix [f(S)]. The vector S is estimated based on the observation R, a knowledge of the matrix [f(S)], and the statistical parameters of the vectors S, 1N and 2N. Karssemeijer has developed a method for estim ation of high frequency noise level as a function of grey value [Karssemeijer 1993] in mammograms. An adaptive approach was used in this study, in which the high frequency noise for each image was determined as a function of the grey level, a nd this information was used for rescaling the images to equalize image noise. Aiazzi et al have presente d a form of SDN given by [Aiazzi et al 1997] ) ( ) ( ) ( ) (y x n y x s y x s y x r (2.12) where r(x, y) represents the noisy imag e value at pixel position (x, y), s(x, y) represents the noise-free image value and n(x, y) is a stationary random process with zero-mean and variance 2n. The second term in the equation re presents the SDN term, which is a special case of Eq. (2.3). The value of is estimated to be between 0 and 1. For = 0,
35 the model reduces to additive noise model. From this model the parameter is calculated. This is achieved by measuring the mean and variance from several homogenous patches (regions of interest) of the noisy image, and the for a pair of patches is given by j i j i log log log log (2.13) where i and j denote the measurements from two different patche s. A consistent estimate of is then calculated by averag ing over all possible pairs. An algorithm was developed by Rank et al to estimate the noise variance of an image [Rank et al 1999]. In this work, the algorithm is basically organized in three steps. The image is first preprocessed by a difference operator to minimize the influence of the original image. A histogram of local standard deviations is then computed. In the final step the histogram is evaluated in order to receive the desired estimate of the noise variance. This algorithm was designed for image models c onsisting of only the additive noise component. In order to put the noise in context with the image structur e, work by Bochud et al is discussed here briefly [Bochud et al 1999]. The effect of system noise may be negligible compared to anatomical fluctuations in some situations. The effect of variations in anatomical background on detectio n tasks has been quantified. Experiments in this work show that the human observerÂ’s behavior wa s highly dependent on both system noise and the anatomical background. The anatomy has b een identified as part ly acting as signal and partly acting as pure noise th at disturbs the detection process. This dual nature of the
36 anatomy has been quantified and it was shown its effect varies according to its amplitude and the profile of the object being detected. Salmeri et al have presented an algorithm to obtain estimations of a Gaussian additive noise [Salmeri et al 2001].This method uses a fuzzy system that processed certain parameters which can be easily extracted from the image. This method can be applied to distributions other than Gaussian but is limited to additive noise models. Veldkamp and Karssemeijer have developed a method for detection of microcalcifications in digital mammograms by noise equalization [Veldkamp and Karssemeijer 2000]. In this work, an adaptive approach is optimized by investigating a number of alternative approaches to es timate image noise. The estimation of high frequency noise as a function of gray scale is improved by a new technique for dividing the gray scale in sample intervals and by using a model for additive high frequency noise. Several other studies [Aghdasi et al 1994] have been conducted on the subject of radiographic noise which is cited in literatu re. This further emphasizes the importance of analyzing and characterizing noi se in radiographic image whic h can be beneficial for the detection and diagnosis of disease.
37CHAPTER 3 NOISE CHARACTERIZ ATION AND MODELING 3.1 Introduction In the previous chapte r a review of several studies involving noise in an imaging system was provided with an emphasis on SDN from a general point of view In most of the studies performed on this subject, the noise fi eld in the image models was assumed to be zero-mean noise. That is the si gnal dependent noise model is s n where n, the noise factor, is zero mean and s is the pure signal raised to a power. While, several methods have been proposed for estimation and detection of the zero-mean type SDN, very little is found in the literature that addr esses the other case, where the expected value of the noise factor is not zero mean. As demonstrated belo w, this is an important consideration when assuming noise models and developing estima tion methods dependent on the noise model assumptions. This idea must be taken in the context that in some instances the imaging physics may lead to the proper model, as in the Poisson case for photon counting, and in other cases the data prior to acquisition a nd manipulation methods may not be known or are inaccessible. Since the work presente d here follows the la tter avenue, it may be considered as a signal dependent noise detecti on algorithm. That is, given a signal, is it possible to determine if the noise is signal dependent and if so, what is the functional relationship between the two?
38 In the first part of this chapter a description of the simulated data that are used for testing is provided The simulations are significant because they represent the ideal cases that will show whether the ideas ar e worth pursuing further. The two methods for estimating the form of SDN are also presented in this chapter and the methods are applied to the simulated data for validation purposes. 3.2 Simulated Data The noise estimation methods described in this chapter are first tested and validated with 2-D simulations. In this section, discussion on the generation of the simulated fields is provided. Two related mammographic simulations are described below. 3.2.1 Simulation One: 1/ f Field Simulation-one is a 2-D random field that ha s statistically simila r PS as a mammogram, which is referred to as a 1/f process. This terminology arises from the fact that the 2-D noise power spectra of the fiel d drops off as approximately 2/ 1f, where f represents the 2-D frequency coordinates and is a positive parameter in the neighborhood of 1.5, which to a good approximation describes the mammographic spectra. This simulation was developed as part of a previous study [Heine et al 1999]. The 1/f simulation is generated by Fourier domain filtering ) ( ) ( ) (0f S f H f S (3.1) with f = (fx, fy), where (fx, fy) are 2-D Cartesian coordinates in the frequency domain. S (f) is the Fourier transform (FT) of the simulation input field, s(x, y), which is a noise
39 process that is distribute d proportional to a modified hyperbolic Bessel function. H (f) is the frequency domain transfer function given by 2 / 2 2) ( ) ( y x y xf f f f H (3.2) where = 1.5. S0 (f) is the FT of the output or simulation process, S0 (x, y), which can be obtained by taking the inverse FT. Fig.3.1 illustrates the simulated field generated by this process. The image shown below appears similar to a mammogram, but does not have similar signal dependent noise properties. In the ne xt section, it is shown how this may be achieved. Figure 3.1 The Simulated 1/f Field
403.2.2 Simulation Two: Mammographic SDN Simulation In this method, a 1/f image is generated as the first step, which is used as the raw image or input image for this simulation. In orde r to simulate a mammogr am, the thickness of a compressed breast is assumed to be 5 cm. In a human breast, the attenuation of glandular tissue is estimated to be 0.9/ cm and 0.5/cm for fat tissue [Highnam 1999]. Thus if the Xray traversed through a 5cm breas t, the natural log of the tr ansmission probabilities would range over the interval ( 2.5, 4.5).Using these values, the 1/f image was mapped between the attenuation values of fat a nd glandular tissue (which are 2. 5 and 4.5 for a breast of 5 cm thickness). For a typical X-ray tube, at the tube voltage of 28 kVp, the photon flux rate is given as 0.366 106 photons/mAs/mm2 [Highnam 1999]. If the exposure value measured in mAs (milli-Ampere seconds) is assumed to be 25 and the pixel area as 1 mm2, then the average number of photons over one pixel area of an image at 30m is given by (photon flux rate) (mAs) (30 10-3)2 10,000 (3.3) The probability of transmission of a photon through each pixel (shown in Eq.(2.5)) is given by ) exp( z pT (3.4) where, z is the resulting image obtained by mapping the 1/f image into the attenuation values of 2.5 and 4.5 of a breast of 5cm thickness. For each pixel position a rv, w, is picked at random from a Poisson distri bution with =10000. This gives the maximum number of X-ray photons that may pass through the breast above the detect or at the position (x, y). For each photon a uniform rv, u,
41 distributed over the interval (1 ,100) is picked at random. If the transmission probability is less than u/100, the photon is transmitted through to the detector, otherwise it is absorbed. This is repeated for each photon for the given location above the detector. The process is then repeated for each pixel. In this case we have the prefect lin ear detector with an efficacy of 100 %, implying that every photon in teracting with the detector is detected. This resulted in the formation of an image which is considered to be a simulated mammogram of the same size as the input image z, which is illustrated in Fig.3.2. Figure 3.2 Simulated Image Generate d as a True Mammogram
423.3 Noise Estimation Two methods of noise estimati on and SDN modeling are presen ted in this section. The methods are first tested on simulated data of known noise characte ristics and validated with blind simulated data. A 512 512 size 1/f image is generated (shown in Fig.3.1). A 512 512 size, zero-mean Gaussian noise field is generated using the random noise generator function in IDL [RSI systems], which is illustrated in Fig.3.3. The signal-dependent noise is given by s n r (3.5) where, n is a zero-mean Gaussian noise field, and s is the pure signal or the image created as a 1/f field with = 2 in this case and = 1.5 from Eq. (3.2). In this case, note that s is not a conventional deterministic signal but is a statistical entity itself that follows from the simulation-one method. The variance of r is given by 2 2 2s n r (3.6) The importance Eq. (3.6) will become clear when modeling the signal dependent noise behavior. The SDN in Eq.(3.5) is then added to the signal term giving the noisy image r s f (3.7) Fig.3.4 illustrates the noisy image f.
43 Figure 3.3 A Zero-mean Gaussian Noise Field
44 Figure 3.4 The Noisy Image f which is Contaminated with SDN with = 2 3.3.1 Method I: The Filtering Method One approach to estimate the noise-signal rela tion is to apply some method that separates the two components. For the moment, two assumptions are be made (1) the signal dependent noise field is not co rrelated with the signal term, and (2) the spectral character of the noise term is flat, implying that is white noise. The merits of these assumptions will be discussed later in the work. These tw o assumptions relate to the signal and noise
45 in following way: the signal varies slow ly (long range correlation) due to the 1/f behavior and has more power located near the zero fr equency region, whereas the noise term has its power distributed evenly. Thus, a filtering operation may separate the signal from the noise term if the assumptions hold true. The wa velet transform has been used here for this application. Wavelet Expansion The noisy image f is expanded into a sum of uncorrelated images by a applying wavelet expansion method th at was used in a previous study [Heine et al 1997] j jf d d d f 2 1 (3.8) where, dj = f j1 fj The expansion may be te rminated for any value of j. The important relation here is that dj = f j1 fj The d images are difference images that contain the detail information as the image fj -1 is blurred to the next coarse resolution fj. The dj images get coarser with increasing value of j, implying that the d1 image contains finer detail than the d2 image and the fj is a half resolution and smoothed version of fj1. This latter network may be considered as the out put of a filter bank, where the outputs are linearly independent. Figs.(3.5 Â– 3.7) illustrate the d1 d2 and f1 versions of the image shown in Fig.3.4. For the purpose of this work, only d1 and f1 images are considered for noise modeling. If a noise field is white, of the power will be contained in the d1 component. Thus under the assu mptions described above, the d1 is a good approximation for the noise while the f1 image approximates the true signal.
46 Figure 3.5 d1 Image Figure 3.6 d2 Image
47 Figure 3.7 The f1 Version by the Wavelet Expansi on of Image Shown in Fig.3.4 Signal and Noise Measurements An 8 8 size window or box is shifted across the f1 image. For every location of the box, the mean signal, m is calculated and stored. A similar procedure is then applied to the d1 image. But in this case the variance2 is tabulated. The stored average values are then sorted in ascending order. The spatial location of each average value in the final so rted arrangement is retained. The estimated variance is then aligned such that the resulting combination forms ordered pairs of the ascending averages (the indepe ndent variable) and the associated noise variance estimates from the same spatial location. Since the data is integer, there will be repeated values along the independent axis (signal axis). When this case exists, the a ssociated noise terms
48 are averaged to genera te the resultant ordered pair. A box car sliding average of length 10 is applied to the noise si gnal prior to the modeling. The result as applied to the Fig.3.4 is shown in Fig.3.8 where the noise variance (jagged) is plotted as a function of the mean, m. A polynomial fit is applied to this function which is shown as the solid curve. Polynomials from degree 0 to 5 are fitted to this curve and the fit with the minimum error is retained as the best fit. The polynomial fitted is of the form 2x b a y (3.9) which follows from Eq 3.6 with an added degree of freedom indi cated by the additive constant a. In Fig.3.9, the theoretical curve is pl otted against the empirical curve showing the linearity between th e two curves. Since was given a value of 2 in this particular case, the function is estimated as a fourth degree polynomial (t he independent variable is the local sorted mean signal) gave the best fit. As a counter example, the value of was set to 0 in Eq.(3.5). In this case plotting the noise as a function of signal as shown in Fig.3.10 produces a constant function; clearly indicating no functional dependency of the noise on the signal. The noise is termed Â“additiveÂ” in this case with approximately zero slope. Hence, we see that given a known dependency of the noise on the signal, the functional form of the noise can be estimated with this method. This is validated in the following section by using blind data, in the case wher e the functional dependency of the noise is assumed unknown.
49 Figure 3.8 The Signal Average Versus Noise Varian ce. Empirical curve (jagged line) and theoretical curve (solid line) resulting from the curve fitting analys is. The form of the noise is given by 4 910 5 2 84 550x y Figure 3.9 The Theoretical Curve (jagged) Agains t the Empirical Curve (solid) Shown in Fig.3.8. This has the effect of making the re lation linear when the polynomial relation is in agreement. A linear fit (solid) is applied with a slope, m= 0.98 0.001 and correlation, r = 0.99 Me anSignal Noise Variance TheoreticalCurve Em p irical Curve
50 Figure 3.10 This Shows the Non-signal Dependent Case. When the noise is not related to the signal, the outcome is a straight line with approximately zero slope. This is encountered with addi tive random white noise Validation of Method 1 It was demonstrated in the prev ious section that the method is useful for finding the noise-s ignal relation. Now a blind image is generated using simulation method 2 (Poisson pr ocess), which is a different phenomenon than the signal dependent simulation developed in Eqs. 3.5 and 3.7. This image is decomposed into d1 and f1 images by wavelet expansion (discussed above and shown in Fig.3.7) and the signal noise measurements follow the methodI prescription. Plot ting the noise as a function of signal is shown in Fig. 3.11. The noi se variance in this case is found to be a linear function of the signal which is to be expected due to the Poisson statistics where the expected value is equal to the variance. Mean Signal Noise Variance
51 Figure 3.11 The Empirical Curve (jagged line) is Linearly Fit (solid line).This shows noise variance is a linear functi on of the signal with slope 04 0 54 11 x mand r = 0.98 3.3.2 Method II: The Order-Statistics Method In this subsection another method is develope d for estimating the si gnal-noise relation. The necessity for this will become clear in th e following sections. The image generated in Eqs.(3.5-3.7) is used as the raw image. As shown in the previous section, a known value of is set for testing purposes. Signal and Noise Measurements As in the previous method, an 8 8 size window or box is shifted across the raw image in both dire ctions (vertical and horizontal). At each location of the box, the pixel va lues were sorted in ascending order and the median value, q, approximates the average (analogous to me thod one). From each location of the box, q of the sorted values and the minimum pixe l value (min) and the maximum pixel value Noise Variance Mean Signal
52 (max) were calculated. The noise standard de viation is estimated at each box site by the following operation 2 min) ( max) ( q q k (3.7) The above expression is an a pproximation and the constant k must be determined. This represents a non-linear operat ion that is sensitive to the image correlation and distribution. This was studied with simulations that indicate k = 6 1 approximately for mean-symmetric type distributions, which require further investigation. The analysis continues in exactly the same fashion as in method I from this point. The results of the technique are displayed below in Fig.3.12 on th e same simulations as displayed in Fig 3.8 above with similar results. The linearity be tween the theoretical a nd the empirical curve is shown in Fig.3.13. Figure 3.12 The Jagged Curve (empirical) Shows th e Noise as a Quadratic Function of Signal. It is fitted with a 4th or der polynomial (solid curve) given by4 910 3 2 12 1467x y Noise Variance M ean Si g na l
53 Figure 3.13 Theoretical Curve against the Empiri cal Curve (jagged). This makes the relation linear when the fit is a reasonable ap proximation. A linear fit (solid) is applied with a slope, m = 1.02 0.06 and correlation r = 0.97, which shows the technique produces the expected behavior Validation The method is validated with the same Poisson process as in the previous case. The relation is linear as previously a nd is displayed in Fig. 3.14 with the slope m = 21.7 0.05 and the linear correlation coefficient, r = 0.91. The two methods predict the same form when considering the simulation presented in Fig. 3.1 when considering the scaling constant s. However, in the Poisson case (Fig.3.2), the scaling constants are of the same magnitude but differ by roughly a factor of two. Theoretical Curve Em p irical Curve
54 Figure 3.14 The Empirical Curve (jagged line) is Lin early Fitted (solid line).It shows that the noise is a linear function of the signal given by6 17102 7 21 x y The two methods discussed above have been tested with a mu ltiplicative model zeromean case and validated with a Poisson pr ocess. The two methods predict the same functional relations for these simulations under the original assumptions, which may not always be the case. In the multiplicative case, changing the DC bias of the multiplicative noise field to something other than zero mean alters the original assumption of the noise character. These ideas are illu strated in Figs.(3.15-3.18). The same simulation displayed above is used for this demons tration with the noise term in Eq.(3.5) having a mean value that is ten times its variance. The results obtained by applying the two methods are illustrated below. Mean Signal Noise Variance
55 Figure 3.15 The Noise Variance is Plotted as a Fu nction of the Mean Signal. The noise shown here is estimated using method I. It is evident that this me thod does not predict the true character of the noise in the model Figure 3.16 The Noise Variance (jagged) is Plotte d as a Function of the Mean Signal It is fitted with 4th degree polynomial of the form, 4 1010 7 1 59 18209x y The noise Mean Signal Mean Signal Noise Variance Noise Variance
56 shown here is estimated using method II. The result is consistent with the results shown in previous sections for zero-mean noise Thus, it is seen that method I fails to estimate the true char acter of the noise when the noise term contains a large mean value. As the mean in the noise term is small or approaches zero, this method pr ovides results close to Eq.( 3.9) but it is not exactly accurate. This is further illust rated in Figs.(3.17-3.18). In the first figure the noise is shown when the mean in the noise is five times the variance and the second figure shows the case when the mean is equal to the variance. Figure 3.17 The Noise Variance (jagged) is Sh own When the Noise Term Contains Mean Value Equal to Five Times the Standard Deviation. It is f itted with a polynomial (solid)of the form 2023 0 82 97840x y Mean Signal Noise Variance
57 Figure 3.18 The Noise Variance when Mean of Nois e is One. The noise variance when the mean in the noise is equal to the sta ndard deviation and is fitted with a polynomial (solid) of the form3 510 7 2 98 2387x y Thus, it is seen that as the mean in the noise gets closer to zero, the estimated noise gets closer to its true form 3.4 Noise Analysis of Tomosynthesis Data Two methods for estimating SDN have been disc ussed in section 3.3. Both methods were applied to the TD and the methods produce consistent results; th e results that are presented here follow method-one. The noise characteristics as estimated from the projection data and the volume data are an alyzed and compared. Likewise, comparisons are referenced to the sta ndard planar FFDM images. Mean Signal Noise Variance
583.4.1 Projection Data The results for one projection example are sh own in Fig.3.19. In the figure, the noise variance is plotted as a function of the mean signal. A linear fit has been applied to the curve showing that the noise is a linear function of the signal, which is of the form b s m s n ) ( (3.8) where, n(s) refers to the noise variance, s is the mean signal m is the slope of the line fitted and b is the constant term. The empirical cu rve (jagged line) is shown fitted with the theoretical curve (solid line). This result is found to be consistent across the 11 projection images, with a varying slope of th e fitted line. The average slope across the 11 projections, = 0.24 0.02, and average correlation coefficient, = 0.86 0.03. The slope of the line for each projection is plotted in Fig.3.20. Figure 3.19 The Noise Variance (jagged) as a Functi on of the Mean Signal. A linear fit is applied to it (solid) with the slope m = 0.20 0.004, and linear correlation coefficient, r = 0.87 Noise Variance MeanSignal
59 Figure 3.20 The Slope Across the 11 Projection Images The average noise power across the 11 projec tion images was calculated from the first detailed image obtained by the wavelet expansio n of the raw image. This image contains three-fourth of the noise power of the raw image if we consider the noise as wide-band white noise. This image captures the effective average interference of the SDN and electronic thermal noise. The plot in Fig.3.21 shows the average or effective noise power for each of the projection image. Projection Index Slo p e ( m )
60 Figure 3.21 The Noise Power for Each of the 11 Pr ojection Images (asterisks). The solid line represents the general trend 3.4.2 Volume Data The four sets of reconstructed volume data ar e analyzed and presented in this section. Figs. (3.22-3.23) illustrates th e functional form of the noise as estimated from the volume data. In these figures, the noise variance is pl otted as a function of the mean signal. The relationship was modeled by cons idering polynomial from zer o through fifth degree and applying error analysis. A linear fit showed th e least error indicating that the noise is a linear function of the signal a nd is of the form given by Eq .(3.8). The empirical curve (jagged line) is shown fitted with the theoretical curve (solid line). The linear form of the Noise Powe r Projection Index
61 noise estimated from the two methods is co nsistent across the 44 volume slices for each volume set. The average slope and the average li near correlation coefficients of 44 slices for each volume set are tabulated in table 3.1. The slope of the line for each volume set as a f unction of the slice is plotted in Figs.(3.243.25). The correlation between the slopes of the f our sets has been tabulated in table 3.2. As in the previous case the average power was calculated from the first detailed image from the wavelet expansion of each volume slice for the 4 sets. This is plotted as a function of the volume slice, shown in Fig.3.26, which illustrates the GFBP data. A linear fit is applied to it which has a slope, m = 89.69, with linear correlation coefficient r = 0.88. The average power measure from each vo lume set is found to be following this trend.
62 Figure 3.22 The Noise Estimated From GFBP (top) and OSBP (bottom). It is shown as function of the mean signal, fitted with the theoretical line (solid). The form of the noise is found to be linear in both cases shown here with the slope, m = -15.49 0.051 (top) with correlation coefficient, r = -0.93 and m = 1.4 0.012 (bottom), with the correlation coefficient, r = -0.91 Mean Signal Mean Signal Noise Variance Noise Variance
63 Figure 3.23 The Noise Estimated From FBP (top) and ART (bottom). It is shown as function of the mean signal, fitted with the theoretical line (solid). The form of the noise is found to be linear in both cases shown here. For FBP, the slope m = -3.78 0.03 and r = -0.90, and for ART, m = -0.12 0.01, and r = -0.94 Mean Signal Noise Variance Mean Signal Noise Variance
64 Figure 3.24 The Slope of the Line for GFBP (top) and OSBP (bottom). This is the slope of the line fitted with the noise estima ted across the 44 volume slices for each reconstruction set SliceIndex Slo p e o f the line Slo p e o f the line SliceIndex
65 Figure 3.25 The Slope of the Line for FBP (top) and ART (bottom).This is the slope of the line fitted with the noise estimated acros s the 44 volume slices for each reconstruction set Slo p e o f the line SliceIndex SliceIndex Slo p e o f the line
66Table 3.1 The Average Slope of the Line and the Average Linear Correlation Across the 44 Slices of Each Volume Set RECONSTRUCTION SETAVERAGE SLOPE, ERROR AVERAGE CORRELATION ERROR GFBP -15.64 1.35 -0.92 0.01 OSBP -1.37 0.057 -0.91 0.03 FBP -3.86 0.75 -0.85 0.07 ART -0.12 0.009 -0.93 0.02 Table 3.2 The Correlation Between the Slopes of the Line (Shown in Fig.3.24 and Fig.3.25) Fitted Over the Volume Sli ces of the Four Reconstruction Sets RECONSTRUCTION SET I RECONSTRUCTION SET II CORRELATION BETWEEN THE SLOPES OF I AND II GFBP OSBP 0.25 GFBP ART -0.62 GFBP FBP -0.05 OSBP ART 0.14 OSBP FBP 0.016 ART FBP 0.19
67 Figure 3.26 The Average Noise Power (diamonds). This is calculated from the first detailed image of the 44 volume slices for one volume set shown with a line (solid) fitted over it with slope m = 89.69 3.58, with linear correlation coefficient r =0.88. The average power for all four sets of volume im ages follow similar trend as shown here 3.4.3 FFDM Data The noise is estimated from regular FFDM data using the methods described in this work. Fig.3.27 illustrates the noise variance (jagged) as a function of the mean signal and fitted with a line (solid). The noise is found to be a linear func tion of the signal. Previous studies have shown that the noise from th e digital mammography system, estimated as a function of the X-ray exposure, exhibits a linear behavior [Cooper 2003]. This is expected, since the FFDM detector has a linear response with the X-ray exposure [Vedantham et al 2000]. Slice Inde x Average Noise Power
68 Figure 3.27 The Noise Variance from FFDM Data (jagged) Shown as a Function of the Mean Signal. It is fitted with a line (solid). The noise is found to a linear function of the signal with the slope, m = 0.075 0.03 and r = 0.96 3.5 Discussion From the noise analysis, the slopes of the li near fit give the sca ling between incremental changes in the noise relative to the signal. The work shows that the noise-signal dependency is strongest for the GFBP met hod and weakest for the ART method. Also note that the relationship is stronger with th e volume data sets in comparison to the projection sets. It is also in teresting to note that within a reconstruction method, the slopes are similar although not correlated acro ss the reconstruction sets ; this holds except for the FBP method. In as much that th e wavelet image capture s or represents the Mean Signal Noise Variance
69 effective noise, the work shows, that the projection angle has l ittle influence on the randomness as is evident from Fig. 3.21, whic h is consistent with the ensemble slope average and variations of the projections. The detector response in the FFDM system is linear with respect to the X-ray exposure. Thus, the resulting pixel value, PV is rela ted to the exposure, E, by the relation, PV = m E + c [Vedantham et al 2000], where m and c are constant parameters. Since c is small, the PV and PV va riance relation should follow the Poisson relation.
70CHAPTER 4 SPECTRAL ANALYSIS OF TOMOSYNTHESIS DATA To date, very little analysis has been publis hed relating to the spectral characteristics of the TD, since itÂ’s a newer tec hnology and not many systems are in clinical use. In this work, the frequency domain characteristics of the TD are studied and analyzed. Previous work by Heine et al shows that the power sp ectra of FFDM planar da ta obey an inverse power law, to a good approximation [Heine et al 2002]. An effort has been made to understand the power spectra behavior of the tomosynthesis (project ion and volume) data so that statistical methods developed for di gitized film images and FFDM images may be adapted to this data. In addition to mode ling the tomosynthesis spectra, the PS of the FFDM data are also discussed for comparison purposes. The PS behavi or in relation to SDN has also been discussed with examples in this chapter, which is relevant in understanding and analyzing the nois e characteristics in images. Fourier methods are applied to estimate a nd analyze the PS of the TD. The analysis technique applied here referred to as the constant ring model, was developed previously for planar FFDM analysis [Heine and Velthuizen 2002].
714.1 Power Spectrum Estimation The power spectrum (PS), or more aptly termed the spectral density, describes the characteristics of the data in the frequency domain. Figs.4.1 illust rates a typical example of an image in the Fourier domain as obtai ned in Cartesian coor dinate systems. Figure 4.1 Raw Image (left) and the Image in Frequency Domain (right)
72 The FT of an image f(x, y) of size MN is given by )) ( 2 exp( ) ( 1 ) (00N vy M ux j y x f MN v u FM x N y (4.1) where, u and v denote the frequency domain coordinates. The PS of f(x, y) is approximated from this operation by 2) ( ) (v u F v u s (4.2) 4.2 Constant Ring Model The constant ring method provides a summary an alysis of the spectral behavior of the data. In this method, the power is analyzed by integrating over constant ring widths in a radial coordinate system. For this analysis, th e spectrum is divided into 51 sections. This provides a ring width of 5/51 or about 0.098 cycl es/mm per ring. The ring that covers the zero frequency range [0-0.098] is not included in the analysis to a void any dc bias. The other 50 equally spaced rings are analyzed. Sin ce the ROIs are rectangul ar in size and not square, the rings appear elli ptical instead of circular. Integration of the rings Approximating the spectral char acter of a mammogram as a 1/f process and integrating the power spectral de nsity over equally spaced rings in radial coordinates gives ) 1 ( 2 2 2 2 2] ) 1 ( [ 2 ) (n d dn in n k f df f c n P (4.3) where k denotes all constant factors, n is an integer that refers to the ring position, and d is the constant ring width ( 0.098). Here, n represents the frequency index; increasing n
73 indicates increasing frequency. Here, f represents the two-dime nsional frequency domain coordinates and is a positive parameter. The subscript i is the image index. Pi (n) provides the total power in the radi al frequency range that covers [nd, (n+1) d]. Eq. (4.3) provides a means for parametric spectral m odeling. However, the operation described by Eq. (4.3) may also be presented empirically by integrating the power spectra of the image without regard to parametric assumptions. Normalization A measure of the fractional power per ring is provided by normalizing Eq.(4.3), which is given by 50 1) ( ) ( ) (n i i in P n P n p (4.4) where, the lower case p denotes the fractional power per ring. The normalization provides the relative power per ring, with respect to the total power in the 50 rings. This ensures that all images are treated with the sa me weight in the analysis. The index i indicates the data form or representation. 4.3 Spectral Modeling of the Tomosynthesis Data The above method is applied to the various forms of TD in this secti on. The intent of the work here is to examine if the TD shows si milar behavior as that of other mammography data, particularly the FFDM data. This work involves modeling the projection data and the four sets of volume data as a 1/ f process. Note in the figures that the projection image and the first three sets of the volume images have been theoretically modeled, but Fig.4.6 showing the PS of the volume slice from FBP has not been modeled.
744.3.1 Projection Data The normalized power spectra (solid curve) as a function of the frequency index is plotted and theoretically modeled (diamonds) w ith Eq.(4.3), as illustrated in the following figures Figs.4.2. The top figure shows the PS of the projections image followed by PS displayed on a log scale (bottom) that s hows the minute differences. The divergence of the curve from the theoretical model at the tail as seen in the log display is due to the noise in the data in th e high frequency region. Ensemble Power Spectra The ensemble PS was obtained by averaging over the 11 projection images. Fig.4.3 shows the ensemble behavior of the PS with the standard deviation curves, displaying the error margin s. This is displayed in an un-normalized scale in Fig.4.4. The projections show the 1/ f behavior with the average < > = 1.36 0.04
75 Figure 4.2 The PS of the Projection Da ta (top) and Log Display (bottom).The integrated PS (solid curve) of the projection data is displayed over the theoretical 1/ f model (diamonds) with = 1.30. The figure is displayed in a log scale to show minute differences Powe r Lo g Powe r Frequency Index Frequency Index
76 Figure 4.3 The Normalized Ensemble PS of Proj ection Data. The normalized ensemble power (solid) of the 11 projection images is shown with the standard deviation curves given by04 0 36 1 .The (average + deviation) is shown in diamonds and the (averagedeviation) is shown as (+). This is displayed in a log scale Frequency Index Log Power ( Ensemble)
77 Figure 4.4 The Absolute Ensemble PS (solid) of Projection Data It is shown with the standard deviation curves given by given by043 0 36 1 The (average + deviation) is shown in diamonds and the (avera gedeviation) is show n as (+). This is displayed in a log scale 4.3.2 Volume Data The spectral modeling of the volume data from each reconstructions set are displayed in Figs.(4.5-4.8) Ensemble Behavior The ensemble PS was obtained by averaging over the 44 volume slices. Fig.4.9 shows the normalized ensemble behavior of the PS with the standard deviation curves and Fig 4.10 shows the sa me on an absolute scale. The plots are displayed on a log scale to show minute differences. Frequency Index Lo g Power ( ensemble )
78 Frequency Index Figure 4.5 The PS of a Volume Slice from the ART Method (diamonds) It is displayed over the theoretical 1/ f model (solid) with = 1.56 Figure 4.6 The PS of a Volume from the GFBP Method (diamonds) It is displayed over the theoretical 1/ f model (solid) with = 1.49 Lo g Powe r Lo g Powe r Frequency Index
79 Frequency Index Figure 4.7 The PS of a Volume Slice from the OS BP Method (diamonds). It is displayed over the theoretical 1/ f model (solid) with = 1.46 Frequency Index Figure 4.8 The PS of a Volume Slice from the FBP Method. It clearly indicates a different behavior than that of other volume images. This is consistent with the appearance of the reconstructed data (see the figures in chapter 1) Lo g Powe r Powe r
80 Figure 4.9 The Normalized Ensemble PS from Volume Slices (solid). It is shown with the standard deviation curves given by069 0 426 1 The (average + deviation) is shown in diamonds and the (average deviation) is shown as (+) Figure 4.10 The Absolute Ensemble PS from Volume Slices (solid). It is shown with the standard deviation curves given by069 0 426 1 The (average + deviation) is shown in diamonds and the (averagedeviation) is shown as (+) Fre q uenc y Index Lo g Power ( ensemble ) Frequency Index Lo g Power ( ensemble )
81 The above analysis indicates that there is little variation across the volume slices. The volume slices show the 1/ f behavior with the average069 0 426 1 4.3.3 Spectral Comparison betw een the three volume sets The spectral characteristics of the three volume sets showing 1/ f behavior were compared by performing a paired t test. The 44 PS measurements obtained from each volume set were compared with the same from every other set. This results in 50 pair wise comparisons, which are the p-values that cover each frequency division. Fig.4.11 shows the p-values from the t-statistic comparing two volume sets. Figure 4.11 The Comparison of Two Sets of Volume Data. The plot shows the p-values for each frequency index obtained from t-statistic The actual points are displayed as diamonds Frequency Index P -Value
82 This result was consistent between every two volume sets. From the t-statistic it was evident that the spectral characteristics of the volume images obtained from different reconstruction techniques are similar at a si gnificance level of 0.01 in the low frequency region. In the high frequenc y region (above 12 cycles/mm), they are much different. The spectral analysis shows that (1) the 1/ f behavior is not dependent of the projection angle, and (2) three of the reconstructi on techniques as well as the projections approximately follow the 1/ f model and are similar to previous work in modeling mammograms. 4.4 SDN and Power Spectra Behavior In this section, certain aspects of SDN (signa l-dependent noise) are di scussed in relation to Fourier spectrum, by means of an example. A 1/ f image, s is generated (discussed in section 3.2) and a Gaussian noise field n of the same size as s is generated. Modulating the image s with n gives ) ( ) ( ) ( y x s y x n y x z (4.6) where, x and y represent the pixel position in th e image domain. Taking the FT of Eq.(4.6) gives ) ( ) ( ) ( v u S v u N v u Z (4.7) where the capitals denote the Fourier domain. Eq.(4.7) represents a standard convolution integral. From this, example, two cases are insp ected that relate to the expected behavior of n ( x y ).
83 Case 1 This is termed as the zero-mean case. In this case noise field n is zero-mean and stationary. The resulting PS of z is flat, indicating that it is a white noise, as illustrated in Fig.4.12. Figure 4.12 The PS of an Image when the Noise Modulating it is Zero-mean The Â“flatÂ” argument follows this reasoning: the FT of the noise field will also be a random noise field that meanders about the ze ro axes in the frequency domain. Thus the convolution gives a meandering process with no structure. Case 2 This is termed as the non-zero mean case. In this case the n ( x y ) has an appreciable mean value relative to it variance and is stationary. In this case the PS of
84 z ( x y ) is similar to s ( x y ). The power spectra for both fields are shown in Figs.4.13 and Fig.4.14, respectively. Figure 4.13 The PS of z ( x y ) Figure 4.14 The PS of s ( x y )
85 This result shows than when the noise modul ating a signal has a nonzero mean, it is not possible to separate the PS of the noisy imag e from the pure image. The noise term acts as a reproducing kernel in this case and returns the PS of the pure image [ Heine et al 2004 ]. The correlation between the two images shown in Figs.(4.13-4.14) was found to be 1.0. In this case the FT of the noise field follows the previous example everywhere except at zero frequency region. If the mean is appreciable, the FT of n will appear as a spike at the zero frequency. Hence, Eq. (4.7) is acting like a delta function convol ved with the FT of s which just returns the spectrum of s (in an average sense). This result is relevant for the estimation of SDN in images. This characteristic of the power spectra behavior in images in presen ce of noise with a large mean has shown a significant impact in the results of the two noi se estimation methods that are discussed in chapter 3.
86CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions In this work, the noise and spectral characteristics of the preliminary tomosynthesis data are analyzed. Two methods are derived for the estimation of signal-dependent noise. Both methods have proven to be successful in estimat ing the functional form of the noise in the event of the pure noise term modulating the signal bei ng zero-mean. Method I (the filtering method) was discovered to be unsuccessf ul in estimating the form of the noise in the event of the pure noise term having a large-mean. On the other hand, method II (order-statistics method) was proven to be successful on both events. The failure of the filtering technique arises from the fact that in the event of the noise having a large mean, the spectrum of the SDN and the pure signa l cannot be differentiated (discussed in chapter 4). The noise from the tomosynthesis projection da ta and four sets of reconstructed volume is estimated as a linear function of the signal, as illustrated in chapter 3. The functional form of noise estimated from the FFDM data is found to be linear as well. Since the tomosynthesis technique is built on the FFDM framework, these results are consistent.
87 The spectral characteristics of the projections follow a 1/ f behavior, which shows a similar nature as mammograms. The images from reconstruction sets followed the 1/ f behavior except the FBP, which deviated fr om this behavior. Comparing the spectral characteristics of the other three volume sets, the images show similarity in the low frequency region. 5.2 Recommendations Characterization of the images provided a be tter understanding of the data acquired from the tomosynthesis system. The TD can be investigated in more detail with images from living women, when available. Since the projection and volume images show similar characteristics as regular mammograms, ex isting statistical methods for mammography data may be modified for tomosynthesis appl ications for further research. Multiresolution analysis of the data could be an extension of the work presented in this study. Further research could be conducted to compare the volume images from different reconstruction techniques to determine the technique that produces the best results for clinical applications. Texture analysis of the projec tion and volume images may be an interesting area of research and an extension of this study. Since tomosynthesis is a new technique, there is much scope for further study.
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