Investigating Simulation-Based Metrics for Characterizing Linear Iterative Reconstruction in Digital Breast Tomosynthesis
Sean D. Rose, Adrian A. Sanchez, Emil Y. Sidky, Xiaochuan Pan
IInvestigating Simulation-Based Metrics for Characterizing Linear IterativeReconstruction in Digital Breast Tomosynthesis
Sean D. Rose, Adrian A. Sanchez, Emil Y. Sidky, and Xiaochuan Pan
University of Chicago, Dept. of Radiology MC-2026, 5841 S. Maryland Avenue, Chicago IL, 60637 (Dated: 25 September 2018)
Purpose : Simulation-based image quality metrics are adapted and investigated for charac-terizing the parameter dependences of linear iterative image reconstruction for DBT.
Methods : Three metrics based on a 2D DBT simulation are investigated: (1) aroot-mean-square-error (RMSE) between the test phantom and reconstructed image,(2) a gradient RMSE where the comparison is made after taking a spatial gradi-ent of both image and phantom, and (3) a region-of-interest (ROI) Hotelling observer(HO) for signal-known-exactly/background-known-exactly (SKE/BKE) and signal-known-exactly/background-known-statistically (SKE/BKS) detection tasks. Two simulation studiesare performed using the aforementioned metrics, varying voxel aspect ratio and regulariza-tion strength for two types of Tikhonov regularized least-squares optimization. The RMSEmetrics are applied to a 2D test phantom with resolution bar patterns at varying angles,and the ROI-HO metric is applied to two tasks relevant to DBT: lesion detection, modeledby use of a large, low-contrast signal, and microcalcification detection, modeled by use of asmall, high-contrast signal. The RMSE metric trends are compared with visual assessment ofthe reconstructed bar-pattern phantom. The ROI-HO metric trends are compared with 3Dreconstructed images from ACR phantom data acquired with a Hologic Selenia DimensionsDBT system.
Results : Sensitivity of the image RMSE to mean pixel value is found to limit its applicabilityto the assessment of DBT image reconstruction. The image gradient RMSE is insensitive tomean pixel value and appears to track better with subjective visualization of the reconstructedbar-pattern phantom. The ROI-HO metric shows an increasing trend with regularizationstrength for both forms of Tikhonov-regularized least-squares; however, this metric saturatesat intermediate regularization strength indicating a point of diminishing returns for signaldetection. Visualization with the reconstructed ACR phantom images appear to show asimilar dependence with regularization strength.
Conclusions : From the limited studies presented it appears that image gradient RMSEtrends correspond with visual assessment better than image RMSE for DBT image recon-struction. The ROI-HO metric for both detection tasks also appears to reflect visual trendsin the ACR phantom reconstructions as a function of regularization strength. We point out,however, that the true utility of these metrics can only be assessed after amassing more data.
I. INTRODUCTION
Image reconstruction for digital breast tomosynthesis(DBT), whether direct or iterative, invariably involvesa variety of parameters and implementation choices,such as voxel size, voxel aspect ratio, and regulariza-tion strength . Experience from computed tomography(CT) provides some insight into the parameter depen-dence of these algorithms, but this experience alone isinsufficient for the tailoring of algorithms to DBT. As inCT, exhaustive exploration of implementations and pa-rameter settings is warranted for every algorithm, task,and DBT system design under consideration. Efficientlycomputable objective image quality metrics are necessaryto facilitate this task.Numerous authors have investigated metrics address-ing this issue, some quantifying image artifacts spe-cific to DBT, some using DBT tailored versions oftraditional CT metrics, and others calculating task- specific detectability indices. A number of theseworks focus on optimization of system acquisitionparameters, or assessment of reconstructionalgorithms.
Metrics for DBT imaging fall into two categories: thosethat are applied to actual DBT scans, and those that areapplied to DBT simulations. Metrics for real data areclearly more direct and are useful for DBT systems char-acterization and optimization, but there is the potentialfor error due to the fact that the truth is unknown. Inparticular, for subtle features such as small spiculationsor calcifications, error in assessing their detectability ormorphology can be large. It is also precisely for thesesubtle DBT features where image reconstruction param-eters can have the greatest impact. For this reason, it isalso desirable to have metrics that summarize qualitiesof DBT image reconstruction from simulated data.In this work, we follow the latter approach and seek a r X i v : . [ phy s i c s . m e d - ph ] J un to establish additional metrics for DBT image recon-struction parameter characterization based on simula-tion, where the underlying true object is known. To es-tablish relevance of the proposed metrics, we compare themetric trends with trends in the visualization of recon-structed DBT images from simulation and actual scannerdata.Due to the limited scanning angular range, DBT can-not provide an accurate 3D reconstructed volume. Thisbasic feature of DBT imaging can limit the usefulnessof image quality metrics based on tomographic imagefidelity. We illustrate this problem with studies basedon image RMSE. While absolute image fidelity is notachievable for image reconstruction in DBT even underideal simulation with no noise, there are features of thescanned object which are recovered. We modify imageRMSE in such a way that the new RMSE-based measureis sensitive only to features of the scanned object thatare recoverable in a DBT scan. We also investigate animage quality metric based on a signal detection task.Such metrics reflect an important imaging task for DBTand their applicability is not hindered by the inabilityto obtain accurate tomographic image reconstruction inDBT.In this work, we investigate three image quality met-rics for DBT image reconstruction, two based on imagefidelity and one based on a detection task. The task-based metric extends on our previous work in the contextof analytic reconstruction algorithms in breast CT andDBT and constitutes a region of interest (ROI)based implementation of the Hotelling observer (HO).For brevity, we refer to it as an ROI-HO metric. Investi-gation of the image fidelity metrics extends our previouswork on parameter selection in breast CT to a limitedangular scanning geometry. Our studies focus on thebehavior of these metrics with respect to voxel aspect ra-tio and regularization strength for Tikhonov-regularizedleast-squares optimization.The paper is organized as follows. In section II, the re-construction optimization problems, image quality met-rics, and study designs are described. In section III wepresent the results of our investigation. The results areseparated into three studies: (1) a simulation study in-vestigating the behavior of the two image fidelity met-rics with varying voxel aspect ratio and regularizationstrength (2) a simulation study investigating the task-based metric with reconstructions of varying regulariza-tion strength, and (3) a real data study using reconstruc-tions of the ACR mammography phantom for visual as-sessment of realistic reconstructions using parameters inthe ranges investigated in the simulation studies. In sec-tion IV we discuss our results in greater depth. Lastly,we conclude in section V. !" %&’&(’)* !+ ( , -.*/012(/334351/*( - %&’&(’)* :4;&174&8 -.*/012(/334351/*( < * &/ ’ Fig.
Fig. recon used forcalculating reconstruction operator. B) ROI HO used for computingsignal detection signal-to-noise ratio. II. METHODS
II.A. Image Reconstruction Problems
LSQI - The first of two optimization reconstruc-tion problems investigated in this manuscript is a leastsquares problem with identity Tikhonov regularization,referred to as LSQI. The optimization problem for thisreconstruction formulation can be written asargmin x (cid:8) (cid:107) Ax − b (cid:107) + ( λ (cid:107) A (cid:107) ) (cid:107) x (cid:107) (cid:9) (P1)where x ∈ (cid:60) n is a pixelized image, b ∈ (cid:60) m is a sinogramdata vector, A ∈ (cid:60) m × n is a matrix modeling forwardprojection, and λ ∈ (cid:60) is a scalar regularization param-eter. The notation (cid:107) · (cid:107) will be used to denote the (cid:96) (Euclidean) norm throughout. When the argument of (cid:107) · (cid:107) is a matrix, it is defined to return the maximumsingular value of its argument.Identity Tikhonov regularization can be interpreted interms of the singular value decomposition (SVD) of thesystem matrix A . Let A = U Σ V T be an SVD of A , whereΣ ∈ (cid:60) m × n is a diagonal matrix with elements σ i . Thesolution to P1 can be written x ∗ = V Γ U T b, Fig. − . where the diagonal matrix Γ ∈ (cid:60) n × m has elementsΓ ii = σ i σ i + λ σ As λ →
0, the matrix V Γ U T limits to the pseudo-inverseof A . The parameter λ can be interpreted as the fractionof the maximum singular value of A below which the ef-fects of the singular value spectrum on the reconstructionare significantly diminished.A second important observation is thatlim λ →∞ x ∗ (cid:107) x ∗ (cid:107) = A T b (cid:107) A T b (cid:107) meaning that in the limit of infinite regularizationthe LSQI reconstruction is visually equivalent to back-projection. LSQD - The second problem considered is a leastsquares optimization with derivative Tikhonov regular-ization, referred to as LSQD. The optimization problemcan be writtenargmin x (cid:8) (cid:107) Ax − b (cid:107) + ( λ (cid:107) A (cid:107) / (cid:107) D (cid:107) ) (cid:107) Dx (cid:107) (cid:9) (P2)where D ∈ (cid:60) dn × n is a finite difference matrix approx-imation of the gradient operator, and d is the spatialdimensionality of the image array (2-D or 3-D). The ra-tio ( (cid:107) A (cid:107) / (cid:107) D (cid:107) ) is employed in the regularization term tonormalize the strengths of the data fidelity and regular-ization terms.lim λ →∞ x ∗ (cid:107) x ∗ (cid:107) = ( D T D ) − A T b (cid:107) ( D T D ) − A T b (cid:107) meaning the reconstruction is visually equivalent to theinverse Laplacian of the back-projection image. Thislimit holds if the image boundary conditions are suchthat boundary pixels are set to zero. In our implementa-tion of LSQD, we enforce this boundary condition. Linearity - The solutions of problems P1 and P2 can bewritten as linear functions of the data vector b . Specifi-cally, let the matrix C represent the scaled linear operatorinvolved in regularization C = (cid:40) λ (cid:107) A (cid:107) I if LSQI( λ (cid:107) A (cid:107) / (cid:107) D (cid:107) ) D if LSQDFormulate the matrix ˜ A and the vector ˜ b as˜ A := (cid:18) AC (cid:19) ; ˜ b := (cid:18) b (cid:19) (1)Then the solution to each reconstruction problem can bewritten x ∗ = ˜ A † ˜ b where ˜ A † is the pseudo-inverse of ˜ A . Since only the first m elements of ˜ b are non-zero, we can simplify this ex-pression by writing ˜ A † in block-form˜ A † = ( R H )where R ∈ (cid:60) n × m and H ∈ (cid:60) n × n for LSQI or H ∈ (cid:60) n × dn for LSQD, so that x ∗ = Rb Implementation - The conjugate gradient least squares(CGLS) (ref. 19, p.289) algorithm was used for recon-struction for both LSQI and LSQD. The algorithm wasrun to numerical convergence in all cases. Direct inver-sion was employed in implementing the ROI-HO metric.Further details will be provided later in the manuscriptwhen the ROI-HO metric is discussed in greater detail.The matrix A was defined using a distance driven forwardprojection model. FBP - In previous work we investigated parameter se-lection for filtered back-projection (FBP) reconstructionwith a Hanning apodizing window in DBT. Here weinclude FBP results as a reference for comparison in sim-ulation studies. The cutoff frequency parameter for theHanning window is specified as a fraction of the Nyquistfrequency f max = 12∆ u where ∆ u is the detector bin size. Note that decreas-ing the cutoff frequency in FBP reconstruction is analo-gous, though not equivalent, to increasing regularizationstrength in LSQI and LSQD reconstruction in the sensethat it reduces sensitivity of the reconstruction to high-frequency components in the data. For this reason, wewill refer to decreasing the Hanning window cutoff fre-quency as increasing the regularization strength.In the limit of infinite regularization, the Hanning win-dow becomes an impulse at zero frequency. In our im-plementation of FBP, as is common, the zero frequencyvalue of the ramp filter is actually a small nonzero num-ber (ref. 21, p. 74). In the limit of infinite regularization,each projection is therefore filtered by a scaled impulse atzero frequency in the Fourier domain. This is equivalentto convolving with a constant function in the spatial do-main. Each filtered projection is a constant as a functionof detector bin, and the reconstruction is the backprojec-tion of these “flat” projections.
32 1
Fig.
II.B. Simulation-based Image Quality Metrics
Three image quality metrics were investigated in ourstudies: (1) image RMSE, (2) gradient image RMSE,and (3) an ROI-based implementation of the HO. Foreach of the three image quality metrics we restrict theDBT system model to a 2D plane perpendicular to thedetector containing the X-ray source trajectory whichwe refer to throughout as the scanning-arc plane, illus-trated in Figure 1. The idea of using the scanning-arcplane for computing the simulation-based image qualitymetrics is that the principle imaging properties of DBTcan be captured in this plane. Use of this 2D-modelalso significantly decreases computation time facilitat-ing parameter-dependence studies for accurate solutionof LSQI and LSQD.
Image RMSE - Given an image vector ˜ x ∈ (cid:60) n and areconstruction x ∈ (cid:60) n , the image RMSE is defined asRMSE = (cid:107) x − ˜ x (cid:107)√ n This global image fidelity metric is a normalized versionof the Euclidean distance between two image vectors. Itis sensitive to the mean pixel values of the image, and inDBT, since the reconstruction is only quasi-3D, the meanpixel values of the reconstruction can be quite differentthan those of the true image.
Gradient Image RMSE - In an attempt to obtain ametric with less sensitivity to the mean pixel values ofreconstructions, we study the gradient image RMSE, de-fined asgradient RMSE = (cid:107) D ( x − ˜ x ) (cid:107)√ n where, as before, D ∈ (cid:60) d × n is a finite differencing op-erator. This metric can be interpreted as the summedRMSE of the d gradient images. ROI-HO Metric - The last metric we investigate fo-cuses on how effectively a given task can be performedusing the information within a specified ROI of a recon-struction. The figure of merit for a detection task per-formed by the ROI-HO is a signal-to-noise ratio (SNR), which quantifies the ability of the ROI-HO to classifya given image into one of two cases: signal-present, i.e.signal plus background, or signal-absent, i.e. backgroundonly. We apply the ROI-HO metric to two different typesof detection tasks: SKE/BKE and SKE/BKS.For the SKE/BKE detection task, the confoundingphysical factor is quantum noise, and the noise modelis taken to be additive, zero-mean Gaussian with vari-ance proportional to the noise-free projection of the back-ground object. The signal is assumed small, and we em-ploy the same noise model for signal-present and signal-absent cases.For the SKE/BKS detection task, in addition to quan-tum noise, the presence of random nonuniform back-ground structure is considered, further complicating sig-nal detection. A statistical distribution is used to modelthis background variability. The model used here as-sumes the background structure can be described bya nonuniform component, modeled as a zero-mean sta-tionary random process, added to a uniform backgroundcomponent of known attenuation. A radially symmetricnoise power spectrum of the form h bg ( ν ) = α | ν | − β (2)is considered for the nonuniform component, with β = 1 . The value of α is chosen so that the fullwidth at half maximum of each pixel’s marginal distribu-tion is equal to the difference between attenuation coeffi-cients of fibroglandular and fatty breast tissue at 20keV.The SNR for signal detection in the image is given bySNR x = w Tx s x where w x is the image domain Hotelling template, and s x is reconstruction of the signal of interest. The Hotellingtemplate is defined implicitly through the equation K x w x = s x (3)where K x is the reconstructed image covariance matrix.The ROI-HO is a form of the Channelized HotellingObserver where the channels are the pixels within anROI. In defining the ROI-HO, we perform a decimationoperation M ∈ (cid:60) p × n which removes pixels outside ofa specified ROI. The ROI is referred to as ROI HO andis illustrated schematically (ROI sizes in the Figure arenot reflective of what was used in experiments) in Fig-ure 2. This restricts the observer to a 0.42cm region (30pixel width) of a single row of the reconstructed imagecontaining the true volume of the physical signal. Theresulting task is analogous to the case in 3D DBT imagereconstruction of determining whether or not a signal ispresent from an ROI within a single axial slice of thereconstruction.Denoting the linear reconstruction operator by R ∈(cid:60) n × m , the image covariance matrix after decimation canbe expressed K x = M RK b R T M T (4)where K b ∈ (cid:60) m × m is the data covariance matrix. Dueto the low dimensionality of K x resulting from the dec-imation operation, direct inversion of the matrix can beperformed for calculation of the Hotelling template.While decimation conveniently allows for efficient di-rect inversion of K x , calculation of the reconstructionmatrix R still poses a problem. Doing so directly requirespseudo-inversion of the concatenated matrix ˜ A , definedin equation 1. The large dimensionality of ˜ A makes thisa time consuming step. The idea of exploiting local shiftinvariance may help with this issue. For this work,however, we calculate an approximation of R using a sub-set of image pixels and detector bins. In particular,in calculating the forward projection operator A we con-sider only the middle l > p columns of the image and,correspondingly, only the middle l detector bins in eachprojection. This is illustrated by ROI recon in figure 2.The evaluation of the regularization matrix C is simi-larly confined in the image space. This defines a matrix A ( l ) ∈ (cid:60) m l × n l and a corresponding matrix ˜ A ( l ) of lowenough dimensionality for efficient direct inversion, yield-ing a reconstruction operator R ( l ) ∈ (cid:60) n l × m l . The scalars m l and n l refer to the number of measurements and im-age pixels when considering only the middle l detectorbins and image columns, respectively.Referring to equation 4, the question of how well thisapproximation holds is dependent on how well the ma-trix R ( l ) K ( l ) b R ( l ) T approximates the elements of RK b R T within the middle l columns of the image. As the recon-struction operator R ( l ) can vary significantly with thesize of l , it is important to check that l is large enoughfor the approximation to be reasonable. This was doneby calculating the ROI-HO metric for a range of valuesof l .Instead of reporting SNR x directly, we report the ef-ficiency (cid:15) which relates SNR x to the HO SNR for signaldetection in the projection data domainSNR b = s Tb ( K − b s b ) , where s b is the projection of the signal of interest. Theexpression in the parenthesis is the data domain Hotellingtemplate.The efficiency is then written as (cid:15) = SNR x SNR b . The efficiency is less than or equal to one for linear imagereconstruction and it is a measure of how well the relevantinformation for performing a given task was preservedthrough the reconstruction and decimation procedure.In order to compare detection performance forSKE/BKE and SKE/BKS tasks directly, we will reportthe relative efficiency for SKE/BKS tasks, which we de-fine as (cid:15) r = SNR x,bg SNR b . where SNR x,bg is the image domain ROI-HO SNR foran SKE/BKS task and SNR b is the data domain HO SNR for an SKE/BKE task. The relative efficiency (cid:15) r for a SKE/BKS task will always be less than the ef-ficiency (cid:15) for the corresponding SKE/BKE task. Dueto the common normalization factor, comparison of theSKE/BKE efficiency (cid:15) with the SKE/BKS relative effi-ciency (cid:15) r is equivalent to comparing their image domainsquared SNRs. Subjective visualization - Along with plots of the im-age quality metrics, we show reconstructed images cor-responding to some investigated parameter settings toallow for comparison of metrics with subjective visual-ization of the 2D-model DBT reconstructions. In addi-tion, we also show 3D DBT images reconstructed fromphantom data acquired with a DBT scanner.
II.C. DBT simulation studies
We present two specific DBT simulation studies thatcharacterize parameter dependences of image reconstruc-tion by LSQI and LSQD. The simulation studies havetwo purposes: (1) to demonstrate applicability of thepresented image quality metrics, and in the process, (2)to reveal important characteristics of linear iterative im-age reconstruction for DBT. The simulation studies arenot meant to be comprehensive as the main purpose isthe development and demonstration of the image qualitymetrics.
Simulation System Geometry - For the real data stud-ies, projection data was acquired using a Hologic SeleniaDimensions unit. For simulation studies, the 2D systemgeometry was defined to mimic the geometry of the mea-surements taken by a single detector row of the Hologicsystem. Specifically, the radius of the source trajectoryand source-to-detector distance were both taken to be70 cm and 15 simulated projections were acquired in 1degree increments. The detector bin size was taken to be0 .
14 mm.
II.C.1. Simulation study 1: Pixel anisotropy andregularization strength
The first study aims at characterizing image recon-struction by LSQI and LSQD as a function of pixelanisotropy and regularization strength for a pseudo-continuous model of mean DBT data. The dominantfeature of the DBT scanning system is its limited scan-ning angular range. This causes the inherent resolutionof the DBT system to be highly anisotropic. Accordingly,this first study focuses on setting the pixel aspect ratioand the impact of regularization, isolating the limitedscanning angular range aspect of the DBT system. Theidea is to characterize what features of the test objectcan be recovered from the DBT sampling conditions andhow well the RMSE-based metrics reflect the recovery ofvisual features.For this first study, a discrete phantom is defined ona high-resolution grid with pixels much smaller thanthe detector bins so that its projection yields pseudo-continuous data. The pixel grid is 4096 × x × z dimensions 20 . × . µ m × µ m. An image of the phantom is shown inFigure 3 along with the axis directions. The x axis liesparallel to the detector plane. The bars on the left pro-vide visual assessment of resolution at small angles fromthe horizontal axis. The disks on the right are meant forboth evaluating detection of large, low contrast signalsand assessing resolution in the vertical direction. Thisphantom is defined in the scanning-arc plane and its usefor visualization is non-standard, because it is orientedperpendicular to the usual 3D DBT slice images. Visu-alization of reconstructions in this non-standard orienta-tion is simplified by the phantom’s uncomplicated design.Data was generated by projecting the phantom ontothe detector line at 100 times the detector bin resolutionusing the line-intersection method. This high-resolutionprojection was subsequently converted to X-ray trans-mission employing Beer’s law, assuming monochromaticX-rays at 20 keV. The transmission factors were then av-eraged within each detector bin, then converted back toprojection data by taking the negative logarithm. Re-construction was performed onto grids covering the sameregion as the high resolution grid on which the phantomwas defined. The x pixel dimension of reconstructionswas held constant at 0.14mm — matched to the detectorbin size — while the z pixel dimension was varied.The image RMSE was calculated by first resampling,via nearest-neighbor interpolation, the reconstructed im-age to the high-resolution grid, then calculating theRMSE. The gradient RMSE was evaluated by calculat-ing a finite difference approximation of the gradient atthe reconstructed resolution then resampling the x and z direction gradient images to the high resolution grid.Resampled gradient images were compared to the gra-dient images of the phantom to calculate the gradientimage RMSE. Both metrics were plotted as a functionof the pixel aspect ratio, defined as the ratio of z to x pixel dimension. This process was repeated to obtainRMSE curves at different regularization strengths. Forreference, FBP reconstructions were performed at an as-pect ratio of 1.0. This aspect ratio for FBP was pickedbecause it yielded the minimum gradient image RMSE. II.C.2. Simulation study 2: ROI-HO detection efficiency asa function of regularization strength
The second study improves on the data model real-ism of the first study by including quantum noise andcharacterizes image reconstruction as a function of reg-ularization strength for LSQI and LSQD. The impact ofthe regularization strength is quantified by the ROI-HOdetection efficiency (cid:15) .We investigate two types of detection tasks as de-fined in section II.B: SKE/BKE and SKE/BKS. ForSKE/BKE tasks, observer performance is purely quan-tum limited, and the task does not challenge the re- construction to resolve the signal of interest from othernearby structures which can differ between data sets.An SKE/BKE task only challenges the reconstruction tokeep the signal distinguishable from the quantum noise.An SKE/BKS task with nonuniform background furtherchallenges the reconstruction to resolve the signal fromvariable background structure. In DBT, backgroundstructure is of particular importance due to blur in thedepth direction.Two signals were considered for investigation in bothSKE/BKE and SKE/BKS detection tasks. The first sig-nal was a 0 .
25 cm disk lying parallel to the detector planewith X-ray attenuation coefficient equal to that of fibrog-landular breast tissue at 20keV. As the study was per-formed in 2D with a slice perpendicular to the detectorplane, a cross section of the disk — i.e. a rectangle —was used as the signal. The second signal was a 0 .
32 mmmicro-calcification modeled as a Gaussian with full widthat half maximum (FWHM) equal to the calcification’swidth.For the SKE/BKE tasks, a uniform 4 . × . HO was held at p = 30 throughout (seeFigure 2), while the size of ROI recon was varied from 40to 70 or 70 to 100 for LSQI and LSQD reconstruction,respectively. The x pixel dimension of reconstructionswas 0 .
14 mm and the aspect ratio was 9.2.The quantum noise model is based on a Gaussian ap-proximation to Poisson noise. The variance of the sino-gram data for the SKE/BKE tasks is (ref. 31, p. 542)Var( b i ) = 1¯ N i + 1¯ N where ¯ N denotes the average number of incident X-raysand ¯ N i denotes the mean number of detected photons inthe i th ray after passing through the object. The meanX-ray transmission is¯ N i = ¯ N exp( − ¯ b i ) , where the mean projection is computed in the same waydescribed in Sec. II.C.1. The data was assumed to bea vector of identically independently distributed (i.i.d.)Gaussian random variables, and accordingly, off-diagonalelements of the covariance are zero. Explicitly, the datadomain covariance for the SKE/BKE task is( K b, BKE ) ij = (cid:40) N i + N if i = j i (cid:54) = j . (5)To calculate K b for the SKE/BKS task, the image do-main background noise model given in equation 2 is prop-agated through the projection model and added to thequantum noise via the equation K b, BKS = AK bg A T + K b, BKE
Fig. λ . where K bg is the covariance matrix of the variable back-ground. The fact that the contributions of the quantumnoise and background variability to the data covariancematrix are additive has been demonstrated in ref. 22.The image covariance matrix K bg is calculated from thevariable background noise power spectrum via the equa-tion K bg = W ∗ diag( u ) W where W is the matrix representation of the discreteFourier transform (DFT), u is a vector with elementsequal to discrete samples of h bg , and diag( u ) is a diago-nal matrix with diagonal elements u . II.D. Real data study: ACR phantom data
In order to show the potential relevance of the 2D ROI-HO metric to the actual 3D DBT system we present3D DBT reconstructed images by LSQI and LSQD forphysical phantom data. Projection data were acquiredfor the mammography accreditation phantom listed bythe American College of Radiology (ACR) for mea-suring the physical standards baseline in mammogra-phy since the beginning of the Mammographic QualityStandards Act (MQSA). Reconstruction was performedonto a grid of pixels with x × y × z pixel dimensions Fig. − . FBP display window: [-0.05,0.06] cm − Fig. λ = 10 − . and three aspect ratios.Aspect ratios are displayed above each image. Display window:[0.15, 0.55] cm − . .
14 mm × .
14 mm × . z is the directionperpendicular to the detector plane. Note the same x pixel dimension was used in the simulation studies. Thethree ROIs indicated in Figure 4 were chosen for visualevaluation of reconstructions.To obtain reconstructions more representative of the2D SKE/BKS detection tasks, a realization of nonuni-form background structure was simulated, projected, andadded to the measured data prior to reconstruction. Therealization of nonuniform background was simulated byfiltering white noise with the square root of h bg in equa-tion 2 as described in detail in ref. 32. III. RESULTS
III.A. Simulation study 1: Pixel anisotropy andregularization strength
LSQI - Image and gradient image RMSE curves fromLSQI reconstructions are shown in Figure 5. The image
Fig. λ . Fig. − . FBP display window: [-0.05,0.06] cm − RMSE of the FBP reconstruction is not shown in thetop panel because it lies at value of 0.289, far above theRMSE of the LSQI reconstructions. This results from theFBP reconstruction having almost no DC component.Image RMSE is seen to be relatively insensitive to as-
Fig.
10 LSQD reconstructions at λ = 10 − . and three aspectratios. Aspect ratios are displayed above each image. Displaywindow: [0.15, 0.65] cm − pect ratio while increasing with increasing regularizationstrength. The gradient image RMSE shows a decreasingtrend with decreasing aspect ratio for each level of regu-larization, noting that the curves for λ ≤ − . exhibitmore of a plateau for aspect ratios ≤ .
0. The rank orderof the curves appears to switch rapidly for aspect ratiosin the range of 8 . .
0, and the sensitivity of thismetric with regularization strength appears to increasewith aspect ratio during and beyond this critical range.We note that, for all aspect ratios, image RMSE favorsthe smallest shown regularization strength of λ = 10 − . ,while gradient image RMSE favors λ = 10 − . .The image dependence of the regularization strengthis seen in Figure 6 in which reconstructions at a fixedaspect ratio of 10.7 are shown. This aspect ratio is cho-sen for visualization because both RMSE measures showsensitivity to regularization strength at 10.7. At the low-est shown regularization strength λ = 10 − . , there areclear high frequency artifacts. These artifacts decreasein conspicuity with increasing regularization. The recon-structed images also show increased blur and decreasedmean value as regularization strength increases. We alsonote that the FBP image appear to be of comparableimage quality to the LSQI images. Image RMSE ranksthese images in the order of best to worst as λ = 10 − . ,10 − . , 10 − . , and FBP with FBP being “off-the-charts”bad. Gradient image RMSE ranks the images in the or-der λ = 10 − . , FBP, λ = 10 − . , and 10 − . . The latterordering of gradient image RMSE appears to be more inline with subjective visualization of the images in Figure6. In Figure 7, reconstructions at three aspect ratios areshown at a constant level of regularization for λ = 10 − . .This value is chosen because image RMSE dependenceof aspect ratio is nearly constant, while gradient imageRMSE shows much variation particularly for large aspectratios. Again, subjective visualization agrees more withthe gradient image RMSE trend. LSQD - Image and gradient image RMSE curves forLSQD reconstruction are shown in Figure 8. The trends
LSQI: Disk DetectionLSQI: Microcalcification Detection
Fig.
11 Hotelling observer efficiency (cid:15) for LC disk (top) and HCcalcification (bottom) SKE/BKE detection tasks as a function ofregularization strength for LSQI reconstruction. Legends indicatesize of ROI (value of l ) used for estimating the reconstruction op-erator. of these curves are quite similar to those of LSQI shownin Figure 5. The main difference is that the image RMSEcurve for LSQD does actually show some non-trivial de-pendence on aspect ratio as opposed to the nearly con-stant dependence shown for LSQI. Again, the FBP imageRMSE value is not indicated in Fig. 8, because it is muchlarger than the shown values for LSQD.For the same fixed values of pixel aspect ratio and reg-ularization strengths used for the LSQI, Figures 9 and10 show reconstructed image dependences on λ and pixelaspect ratio, respectively. As before with LSQI, the vi-sual subjective image quality trend in both figures alignsmore with image gradient RMSE than it does with imageRMSE. III.B. Simulation study 2: ROI-HO detection efficiency asa function of regularization strengthIII.B.1. SKE/BKE Tasks
LSQI - Figure 11 shows the ROI-HO detection effi-ciency (cid:15) in LSQI reconstruction as a function of regular-
LSQD: Disk DetectionLSQD: Microcalcification Detection
Fig.
12 Hotelling observer efficiency (cid:15) for LC disk (top) andHC calcification (bottom) SKE/BKE detection tasks with variablebackground as a function of regularization strength for LSQD re-construction.
Fig.
13 Hotelling observer efficiency (cid:15) for LC disk and HC calci-fication SKE/BKE detection tasks as a function of spectral filtercutoff frequency for FBP reconstruction. Note the x -axis is invertedso regularization strength increases from left to right. ization strength for the two SKE/BKE detection tasksand 4 values of the ROI recon size parameter l . Recallfrom section II.C.2 that ROI recon is used to estimate thereconstruction operation R , and does not change the dec-0 LSQI: Disk Detection with Background VariabilityLSQI: Microcalcification Detection with Background Variability
Fig.
14 Hotelling observer relative efficiency (cid:15) r for LC disk (top)and HC calcification (bottom) SKE/BKS detection tasks as a func-tion of regularization strength for LSQI reconstruction. Dashedlines show HO efficiency (cid:15) for the corresponding SKE/BKE task. imation operator M , which is held constant throughout.The efficiency curves appear to approach a limiting curveas the value of l is increased, suggesting that the largestvalue employed ( l = 70) provides a good approximationof the behavior of the ROI-HO. We therefore limit ourdiscussion to this value of l for the LSQI results.The efficiency curves are monotonically increasing withincreasing regularization strength for both tasks, startingfrom a low-efficiency plateau, then rising and saturatingat a large efficiency which is close to the theoretical maxi-mum of (cid:15) = 1 .
0. The location and width of the transitionas a function of λ clearly depends on the characteristicsof the signal. The saturation of the efficiency curves sug-gests that there is a point beyond which increasing reg-ularization does not further improve preservation of taskrelevant information through the reconstruction. Also,there is no penalty in terms of HO SNR for detectionfor increasing the regularization strength arbitrarily. Re-call from section II.A that the LSQI reconstructed imageapproaches the back-projection image as λ → ∞ . LSQD - Figure 12 shows the efficiency curves for LSQDreconstruction. The efficiency curves again approach a
LSQD: Disk Detection with Background VariabilityLSQD: Microcalcification Detection with Background Variability
Fig.
15 Hotelling observer relative efficiency (cid:15) r for LC disk (top)and HC calcification (bottom) SKE/BKS detection tasks as a func-tion of regularization strength for LSQD reconstruction. Dashedlines show HO efficiency (cid:15) for the corresponding SKE/BKE task. limit with increasing values of l , in this case suggest-ing the value of l = 100 yields a reasonable approxima-tion of the ROI-HO. Similar to LSQI, the efficiency satu-rates with increasing regularization strength in a signal-dependent manner. FBP - Figure 13 shows the efficiency for FBP recon-structions of varying cutoff frequency for the Hanningapodizing window. The x -axis of the plot has been in-verted so that regularization increases from left to right,consistent with the LSQI and LSQD efficiency curves.By contrast to the LSQI and LSQD efficiency curves,the FBP efficiency curves do not display saturating be-havior with increasing regularization strength, instead in-dicating that an SNR penalty is incurred for apodizingtoo heavily for a quantum noise limited task. Recall thatin the limit of infinite regularization, the FBP algorithmconvolves each projection with a constant function priorto back-projection.In the limit of decreasing regularization strength, aplateau in efficiency is observed. In this limit the recon-struction algorithm approaches FBP with a pure rampfilter, so the level of the plateau is representative of the1 Fig.
16 ROIs of LSQI reconstructions containing 0.25, 0.5, and0.75cm disks of ACR phantom. Regularization increases from leftto right. Back projection image is shown for reference. efficiency of FBP with no apodization. Moving further tothe right, a monotonic increase in efficiency is observedprior to a peak. The location of the peak appears to besignal dependent, paralleling the signal-dependent loca-tion of the transition region in the efficiency curves forLSQI and LSQD. Note that in both cases, the locationthe of the peak/transition region for calcification detec-tion occurs at lower regularization strength than for diskdetection.
III.B.2. SKE/BKS Tasks
LSQI - Figure 14 shows the ROI-HO relative detectionefficiency (cid:15) in LSQI reconstruction as a function of regu-larization strength for the two SKE/BKS detection tasksalongside the ROI-HO detection efficiencies for the cor-responding SKE/BKE tasks. Based on the limiting be-havior observed in the SKE/BKE tasks, only the l = 70ROI recon results are shown.The relative efficiencies show a clear decrease in taskperformance relative to the quantum limited SKE/BKEcase for both tasks, though the magnitude of the dropin SNR due to the presence of background variability isgreater for the disk detection task than for the calcifi-cation detection task. Aside from this decrease in taskperformance, the overall trends in the efficiency curvesare quite similar to their SKE/BKE counterparts. Thatthe curves plateau at large regularization strengths indi-cates that the SNR is not penalized for increasing regu-larization strength arbitrarily, even in the presence of avariable, nonuniform background.At low regularization strengths, the relative efficiencycurves of the SKE/BKS tasks track well with the effi-ciency curves of the SKE/BKE tasks. The curves sep-arate at intermediate regularization strengths suggest-ing that the main contributing factor to SKE/BKS task Fig.
17 ROIs of LSQI reconstructions containing 0 .
54, 0 . , and0 .
32 mm specks of ACR phantom. Regularization increases fromleft to right. Back projection image is shown for reference. performance at low regularization strengths is quantumnoise while at large regularization strengths it is back-ground variability. The nonuniform background variabil-ity limits the extent to which increasing regularizationcan improve task performance, as indicated by the lowerregularization strengths at which the curves saturate forSKE/BKS tasks relative to SKE/BKE tasks.
LSQD - Figure 15 shows the corresponding results forLSQD reconstruction for the two SKE/BKS detectiontasks. The observed behavior is largely similar to theLSQI results, with saturating behavior again occurringat large regularization strengths and a signal dependenttransition region. We do note that the SNR drop fordisk detection is again greater than that for calcificationdetection.
III.C. Real data study: ACR phantom dataIII.C.1. No added background
LSQI - ROIs of reconstructions of the ACR phan-tom with LSQI at three different levels of regularization,alongside a back-projection reconstruction, are shown inFigures 16 and 17. The ROIs are shown on a diagramof the ACR phantom in Figure 4 for reference. Displaywindows for each reconstruction were manually chosenbased on subjectively maximizing visibility of the disksor specks. We note that the values of λ used in the realdata reconstructions may not precisely match those usedin the 2D simulation studies.As a general trend the reconstructed disks and specksappear more conspicuously as the noise-level is reducedby increasing regularization strength for the shown valuesof λ . Furthermore, there is good correspondence betweenthe LSQI image for the largest shown λ -value and theback-projection image.2 Fig.
18 ROIs of LSQD reconstructions containing 0.25, 0.5, and0.75cm disks of ACR phantom. Regularization increases from leftto right.
The background noise texture differs significantly be-tween the four reconstructions, with the magnitude offluctuations decreasing with increasing regularizationstrength. It is evident from the figures how the detec-tion of large low-contrast and small high-contrast objectscould be complicated by this texture in under-regularizedreconstructions.
LSQD - Figures 18 and 19 show the analogous resultsfor LSQD reconstruction except that no back-projectionimage is shown. As with LSQI, higher levels of regulariza-tion reduce the magnitude of background noise variation;however, there is a clear difference in noise texture. In-creasing regularization of course also blurs the disk andspeck signals.
III.C.2. Added nonuniform background
LSQI - Figures 20 and 21 show ROIs of reconstructionsof the ACR phantom in which the projection of a nonuni-form background was added to the data as described insection II.D. Note the the regularization parameter val-ues for the displayed images are not identical to thoseshown in the corresponding results of Figures 16 and 17.A general trend of increasing conspicuity with increas-ing regularization is still apparent in the presence ofnonuniform background structure. Relative to the unal-tered data reconstructions, the background texture, par-ticularly at large regularization strengths, contains morelow-frequency structure, hindering visualization of thedisk and speck signals. These observations are consistentwith trends observed in the ROI-HO SKE/BKS simula-tion studies.
LSQD - Figures 22 and 23 show the corresponding re-sults for LSQD reconstruction. As with LSQI, the trendof increasing conspicuity with regularization strength ismaintained in the presence of a nonuniform background,though there is an apparent reduction in conspicuity of
Fig.
19 ROIs of LSQD reconstructions containing 0 .
54, 0 . , and0 .
32 mm specks of ACR phantom. Regularization increases fromleft to right. the disks relative to the unaltered data case. Conspicuityof the specks appears to be less affected by the nonuni-form background (compare Figures 19 and 23).
IV. DISCUSSION image RMSE - The sensitivity of image RMSE tochanges in mean pixel value appears to make it inap-propriate for meaningful characterization of image recon-struction parameters for DBT. From the results shownthe image RMSE does not follow the subjective visualquality of the reconstructed images from the 2D DBTsimulation. Perhaps the most egregious discrepancy oc-curs in the image series comparing use of various reg-ularization strengths for LSQI, in Figure 6, and LSQD,Figure 9, with FBP. The FBP result has an image RMSEwhich is more than twice that of any of the shown LSQIor LSDQ images, yet it is subjectively competitive withany of these images. gradient image RMSE - We observe that the gradi-ent image RMSE is not as sensitive to changes in meanpixel value and appears to reflect visual changes betweendifferent regularization strengths and aspect ratios. Inparticular, an increase in gradient image RMSE is ob-served at large aspect ratios for λ = 10 − . , reflectingthe appearance of high frequency artifacts in both theLSQI and LSQD images shown in Figures 7 and 10, re-spectively. Moreover, in the image sequences varying reg-ularization strength in Figures 6 and 9, the rank orderingsuggested by the gradient image RMSE seems plausibleeven including the FBP images.We caution, however, that these studies are prelimi-nary and that many more empirical studies are needed tosupport the utility of the gradient image RMSE metricfor image reconstruction parameter characterization inDBT. One peculiar aspect of this metric is that the rela-tive magnitude of the variations of gradient image RMSE3 Fig.
20 ROIs of LSQI reconstructions of altered ACR phantomdata with added nonuniform background. ROIs contain 0.25, 0.5,and 0.75cm disks of ACR phantom. Regularization increases fromleft to right. Back projection image is shown for reference. shown in Figures 5 and 8 amount to 1% to 2% over theshown parameter ranges. Yet, the preliminary resultsfrom subjective visualization seem to indicate that thesesmall variations are meaningful. That gradient imageRMSE exhibits such behavior for DBT image reconstruc-tion is perhaps not too surprising, because the metricis still essentially comparing reconstructed images fromvery limited angular-range scanning with perfect tomo-graphic image reconstruction. By its design, the gradientimage RMSE is most sensitive to discrepancy at edge dis-continuities at the borders of various tissues within thesubject. It is known that image reconstruction from alimited scanning-angular range such as in DBT can onlyrecover a small subset of these edges . ROI-HO efficiency for signal detection - The resultsof the second simulation study demonstrate that theproposed ROI-HO implementation can be used to accu-rately calculate ROI-HO efficiencies for the SKE/BKEand SKE/BKS detection tasks in linear optimization-based image reconstruction for the 2D-DBT model in thescanning-arc plane. Due to non-locality of image recon-struction in general, it is not obvious that restricting theimage reconstruction operator to an ROI with the imagewill yield accurate image domain HO SNR values. Thatwe observe a saturation of efficiency values (hence im-age domain SNR) with ROI length (cid:96) in Figures 11 and12 indicates that the ROI approximation can yield accu-rate results. We expect that ROI-HO efficiency for the2D DBT model should reflect HO efficiency for 3D DBT,because the 2D model captures the most important scan-ning geometry features of 3D DBT. This correspondence,however, needs to be demonstrated, and we will do so infuture work.Examining the SKE/BKE task efficiency curves inmore detail, both Figures 11 and 12 indicate low effi-
Fig.
21 ROIs of LSQI reconstructions of altered ACR phantomdata with added nonuniform background. ROIs contain 0 .
54, 0 . , and 0 .
32 mm specks of ACR phantom. Regularization increasesfrom left to right. Back projection image is shown for reference. ciency at low regularization strength. This efficiency lossis primarily due to the fact that we are modeling single-slice viewing with the decimation operation that limitsthe reconstruction to the line in the 2D DBT model (cor-responding to a plane in 3D DBT), where the signal fordetection exists. In this line, the signal is reconstructedat an amplitude lower than the true signal due to DBTdepth blur, yet noise from the data model is amplified inthe viewing line due to low regularization. Both LSQIand LSQD saturate at nearly a perfect efficiency of 1.0with increasing λ , where LSQI and LSQD images limitto back-projection and the inverse Laplacian of back-projection images, respectively.In comparing FBP and LSQI/LSQD for the SKE/BKEdetection task, there are substantial differences in howthe respective regularization parameters impact signaldetection efficiency, highlighting the differences betweenthese algorithms.Turning to the SKE/BKS tasks, one might expect thatwith the incorporation of a nonuniform background vari-ability model, the need for the reconstruction algorithmto resolve the signal from the background would cause apenalty to SNR at large regularization strengths for LSQIand LSQD. Contrary to this expectation, we observedmonotonically increasing efficiency with increasing reg-ularization as in the SKE/BKE tasks. The nonuniformbackground variability instead appeared to play the roleof limiting the extent to which increasing regularizationcould improve task performance, while not penalizing theSNR for further increases in regularization beyond thepoint at which efficiency saturates.We also noted that the decrease in SNR relative tothe SKE/BKE tasks was much greater for disk detectionthan calcification detection. This result suggests that thestationary process used to model nonuniform background4 Fig.
22 ROIs of LSQD reconstructions of altered ACR phantomdata with added nonuniform background. ROIs contain 0.25, 0.5,and 0.75cm disks of ACR phantom. Regularization increases fromleft to right. variability interferes more with detection of large, low-contrast objects than small, high-contrast objects. Thisis consistent with the slowly spatially varying appearanceof realizations of this background model (see, e.g., ref.32).In relation to other work on using the HO for im-age quality assessment in DBT, which employ noiserealizations and/or assumptions of stationarynoise in estimating image statistics, the currentapproach yields highly accurate values subject only tominor numerical error due to computer precision. Thedrawback of our ROI-HO implementation is that it isbased on simulation and cannot be applied directly toreconstructed images from actual DBT systems.
3D DBT image reconstruction of the ACR phantom -The ACR phantom study provides a realistic visual as-sessment of images reconstructed in the parameter rangesinvestigated in the second simulation study for the ROI-HO signal detection efficiency. The viewing conditionsfor the unaltered data reconstructions are similar to theSKE/BKE signal detection task design. The propertiesof the signals, disks and speck clusters, are known to theobserver, see Figure 4; also the background is uniformand therefore also known to the observer. The proper-ties of the signals of the 2D simulation study are modeledloosely on the disk and speck objects of the ACR phan-tom.The main trend seen in the unaltered ACR data re-constructions for LSQI and LSQD is that increasing reg-ularization appears to increase the conspicuity of bothdisk and speck cluster signals. This trend appears to co-incide with the SKE/BKE ROI-HO efficiency trends forLSQI and LSQD. For ACR results with LSQI, in partic-ular, the λ = ∞ images, a.k.a. back-projection, appearto show similar conspicuity for both disks and specks asthe results from the largest finite regularization strength Fig.
23 ROIs of LSQD reconstructions of altered ACR phantomdata with added nonuniform background. ROIs contain 0 .
54, 0 . , and 0 .
32 mm specks of ACR phantom. Regularization increasesfrom left to right. of λ = 10 . in Figures 16 and 17. This result is some-what indicative of the detection efficiency plateau seenat large λ in Fig. 11. We do point out, however, thatthe growing magnitude of background variations relativeto the blurred signal complicates visualization at highregularization strengths for LSQD.The apparent trend of increasing conspicuity with reg-ularization for both disk and speck cluster signals is alsoobserved in the ACR reconstructions in which a nonuni-form background was incorporated into the data. Thisobservation is consistent with the SKE/BKS ROI-HO ef-ficiency trends. We note in particular that the conspicu-ity of the signals does not appear to be reduced at largeregularization strengths, contrary to what one may ex-pect with a nonuniform background.We note that the correspondence between the 2D DBTROI-HO results and 3D DBT ACR reconstructed im-ages represent only preliminary indications as many moreempirical results are needed. Again, the connection be-tween the 2D DBT ROI-HO with a 3D DBT ROI-HOneeds to be established. We also point out the SKE/BKSresults represent a single parameter setting of a singlebackground variability model. Other background vari-ability models may yield different detection efficiencycurves. Another important point is that the ROI-HOyields image domain signal detection SNR, while for theresults with the ACR phantom we only discussed signalconspicuity. Signal conspicuity and detection, althoughrelated, are not the same thing. Closer visual corre-spondence with ROI-HO SNR can be investigated bydesigning and performing two-alternative forced choiceexperiments (ref. 35, p. 819). We also note that LSQIand LSQD are not commonly used in the clinic. Application of image quality metrics - The metricsadapted and investigated here yield useful informationfor setting parameters in linear iterative image recon-5struction, but they alone do not provide a complete pic-ture and they are meant to compliment other image qual-ity metrics developed for DBT systems parameter char-acterization/optimization. In particular, consider the re-sults of the ROI-HO SKE/BKE signal detection task ef-ficiency for LSQD and LSQI; following the curves shownin Figures 11 and 12 alone suggests use of LSQI andLSQD with extremely large regularization parameter λ .We point out, however, that the ROI-HO signal detec-tion efficiency exhibits a plateau at large λ , and consid-ering these curves together with other metrics such asthe gradient image RMSE suggests instead to employ aregularization strength at the low λ end of the plateau. V. CONCLUSIONS AND FUTURE WORK
We have investigated three simulation-based imagequality metrics for use in the selection of parameters forlinear iterative image reconstruction in DBT. In the pro-cess, we have adapted an RMSE-based metric to DBT,and developed a task-based metric for linear iterative im-age reconstruction in DBT. The imaging properties ofDBT led us to modify image RMSE to a gradient imageRMSE. We have also extended our work on ROI-HO sig-nal detection efficiency for filtered back-projection tolinear iterative image reconstruction. Finally, we showedimages reconstructed from simulated 2D DBT data andACR phantom 3D DBT data to provide a subjective vi-sual assessment of image reconstruction algorithm pa-rameter trends. The results demonstrated that sensi-tivity to mean pixel value is a weaknesses of the imageRMSE in the context of DBT. Eliminating dependence onmean pixel value by the proposed gradient image RMSEappears to rectify the short-comings of image RMSE. Theresults also demonstrated a signal-dependent saturatingbehavior of the ROI-HO efficiency for two SKE/BKE andSKE/BKS signal detection tasks with increasing regular-ization strength.Future work will focus on extending the simulationstudies from 2D to 3D in order to quantify the correspon-dence of the 2D DBT model in the scanning arc planewith full 3D DBT. While the RMSE-based metrics canbe applied to any image reconstruction algorithm includ-ing non-linear iterative image reconstruction, the currentversion of the ROI-HO applies only to linear image re-construction. Methodology on developing noise proper-ties for non-linear image reconstruction, may allowextension of the ROI-HO to non-linear image reconstruc-tion algorithms. Finally, we intend to extend the ROI-HO to more realistic signal detection models where onlystatistical knowledge is available for the signal is avail-able. Such a model is expected to be more sensitive toDBT depth resolution than the ROI-HO for a SKE/BKEor SKE/BKS detection task. VI. ACKNOWLEDGEMENTS
The authors would like to thank Hologic for providingthe phantom data. This work was supported in part byNIH R01 Grants Nos. CA158446, CA182264, EB018102,and NIH F31 Grant No. EB023076. The contents of thisarticle are solely the responsibility of the authors and donot necessarily represent the official views of the NationalInstitutes of Health.
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