Retrospective Correction of MR Intensity Inhomogeneity by Information Minimization

Transcription

1 1398 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 Retrospective Correction of MR Intensity Inhomogeneity by Information Minimization Bo stjan Likar, Member, IEEE, Max A. Viergever, Member, IEEE, and Franjo Pernu s* Abstract In this paper, the problem of retrospective correction of intensity inhomogeneity in magnetic resonance (MR) images is addressed. A novel model-based correction method is proposed, based on the assumption that an image corrupted by intensity inhomogeneity contains more information than the corresponding uncorrupted image. The image degradation process is described by a linear model, consisting of a multiplicative and an additive component which are modeled by a combination of smoothly varying basis functions. The degraded image is corrected by the inverse of the image degradation model. The parameters of this model are optimized such that the information of the corrected image is minimized while the global intensity statistic is preserved. The method was quantitatively evaluated and compared to other methods on a number of simulated and real MR images and proved to be effective, reliable, and computationally attractive. The method can be widely applied to different types of MR images because it solely uses the information that is naturally present in an image, without making assumptions on its spatial and intensity distribution. Besides, the method requires no preprocessing, parameter setting, nor user interaction. Consequently, the proposed method may be a valuable tool in MR image analysis. Index Terms Information minimization, intensity inhomogeneity, magnetic resonance imaging, nonuniformity correction, segmentation. I. INTRODUCTION IN magnetic resonance imaging (MRI), image intensity inhomogeneity is an adverse phenomenon which manifests itself by slow intensity variations of the same tissue over the image domain. Intensity inhomogeneity may be caused by a number of factors including poor radio-frequency (RF) coil uniformity, static field inhomogeneity, RF penetration, gradient-driven eddy currents, and overall patient anatomy and position [1], [2]. Spurious intensity variations, which may reach up to 30% of the image intensity amplitude [3], [4] usually do not affect the visual impression of the image significantly, but may have serious implications for magnetic resonance (MR) image analysis, e.g., in segmentation, registration, or quantification. Consequently, Manuscript received July 10, 2000; revised October 15, This work was supported in part by the Ministry of Science and Technology of the Republic of Slovenia under Grant J and in part by the European Commission under Project IST The Associate Editor responsible for coordinating the review of this paper and recommending its publication was R. Leahy. Asterisk indicates corresponding author. B. Likar is with the Department of Electrical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia. M. A. Viergever is with the Image Sciences Institute, University Hospital Utrecht, 3508 GA Utrecht, The Netherlands. *F. Pernu s is with the Department of Electrical Engineering, University of Ljubljana, Tr za ska 25, 1000 Ljubljana, Slovenia ( franjo.pernus@fe.unilj.si). Publisher Item Identifier S (01) the correction of the adverse spatial intensity variations, which are particularly severe when using surface-coil MR scanners, is a necessary preprocessing step in many image analysis tasks. Methods for correction of intensity inhomogeneities may be prospective or retrospective. The former require an acquisition protocol tuned to inhomogeneity correction, while the latter can be applied to any MR image, since they only use the information naturally occurring in an image. Prospective methods generally require additional in vitro or in vivo acquisitions. In vitro methods cannot correct for patient-induced inhomogeneities. On the other hand, retrospective methods cannot distinguish between variations induced by scanner properties and those resulting from patient-related sources. A number of prospective methods have been introduced in MRI, which either use phantom acquisitions [1], [2], [5] [8] or measure the excitation field [9] or the static uniformity of the reception coil [10] in vivo. However, the patient-independence requirements and/or extended scan time makes them impractical for clinical use. As an alternative, a number of retrospective methods have been proposed. The most intuitive retrospective method for correcting multiplicative and smooth intensity variations is homomorphic filtering, see e.g., [11]. However, high image contrasts that generate filtering artifacts and the overlapping of the patient data and the bias field frequency spectrums limit the efficiency and feasibility of this approach [6], [12]. Image blurring [7], smoothing [13], averaging [14], and Fourier domain filtering [15] were also considered, while the use of homomorphic unsharp masking, matching the local and global median or mean, was reviewed and optimized in [16]. The histogram matching method, based on minimizing the windowed square residual difference, was applied to correct for interexamination variations [17]. A number of methods were proposed that fit polynomials or thin-plate splines to manually or automatically selected points [12], to regions defined by inhomogeneity-tolerant preliminary segmentation [3], or directly to image data by using multiple-valley criterion functions instead of least squares [18]. Sled et al. presented an iterative optimization method for MRI, which seeks the smooth multiplicative field that maximizes the frequency content of the distribution of tissue intensity [19]. A segmentation-based method, using the expectation-minimization (EM) algorithm to iteratively classify and correct the image, was introduced in [20] and improved in [4]. A further extension of the EM algorithm, which interleaves classification, bias field estimation, and estimation of class-conditional intensity distribution parameters, was proposed recently [21]. Segmentation-based methods that use fuzzy -means clustering can be found in [22] [24], while the usage of Markov random fields /01$ IEEE

2 LIKAR et al.: RETROSPECTIVE CORRECTION OF MR INTENSITY INHOMOGENEITY BY INFORMATION MINIMIZATION 1399 and a greedy algorithm based on iterative conditional modes was considered in [25]. Filtering out and weighted re-integration of well-characterized local derivatives to estimate the bias field was proposed by Vokurka et al. [26]. In similar vein, in [27] a new variational shape-from-orientation approach to correcting intensity inhomogeneities, in which a bias field surface is reconstructed from sparse orientation constraints in a regularization framework, was suggested. A review and evaluation of MRI nonuniformity corrections for brain tumor response measurements was provided by Velthuizen et al. [28], reporting that different methods give significantly different correction images and thus concluding that nonuniformity correction is not yet well understood. In a recent paper [29], we described a retrospective shading correction method based on entropy minimization, designed for correcting the intensity inhomogeneity in two dimensional (2-D) images. In this paper, we generalize the method to three-dimensional (3-D) MR images. The derivation of our algorithm is based on the assumption that an image corrupted by intensity inhomogeneity contains more information than the corresponding uncorrupted image. The method was quantitatively evaluated and compared with two recently proposed methods, [19] and [26], on a large number of simulated and real MR images. Section II formulates the problem of retrospective correction of intensity variations and outlines the proposed correction strategy. Section III covers the implementation of the correction strategy using a linear model of image degradation. Section IV presents the experiments for testing the performances of the here proposed and the two recently published methods and outlines the corresponding results. The obtained results, method s features, and related methods are discussed in Section V. II. THEORY A. Problem Formulation Let denote the acquired image and let denote the true image of the imaged object. The two images are related by or (1) with denoting the image degradation model (bias field) that introduces a spatially dependent intensity degradation to the true image and representing the inverse of the degradation model. The problem of retrospective correction of intensity inhomogeneity is to find the true image from the acquired image. We will pursue a model-based solution to this problem, which involves two steps. First, a parametric correction model, which is the estimate of the inverse of the degradation model, has to be selected. This can be done either theoretically, by mathematically describing the process of image degradation, or experimentally, by analyzing the acquired images of well-defined calibration objects. Next, the parameters defining the optimal correction model have to be found so that the application of to the acquired image will yield the image that has minimal intensity inhomogeneity. For this purpose, a quantitative intensity uniformity criterion and a have to be spec- strategy for finding the corrected image ified. B. Correction Strategy The proposed correction strategy is based on the assumption that because of the image degradation process the information of the acquired image will be higher that the information of the true image The image degradation process is, thus, treated as an additional source of information that increases the information of the true image. The above assumption should hold when spurious intensity variations induced in the degradation process are spatially different, i.e., smoother, than the natural intensity variations in the true image. This is often true in medical images and enables retrospective correction of intensity inhomogeneity by using the information of the acquired image and an appropriate correction model. The correction model should describe only the intensity variations due to degradation process and not the intensity variations of the true image. Besides, the correction model must be constrained so that it cannot change the global intensity characteristics of the input image, such as brightness or contrast, but can only introduce smooth, spatially dependent, intensity variations that are needed for correction of inhomogeneity. Having such a correction model, an estimate of the true image may be obtained from the acquired image as By changing the parameters of the correction model and transforming the acquired image (3) in such a way that the information of the transformed image is minimized, i.e., that the correction model best reduces additional information induced in the degradation process, leads to the optimal correction model which defines the transformation of the acquired image the corrected image The flowchart shown in Fig. 1 illustrates the image degradation process and the proposed retrospective correction strategy. The image degradation process increases the information of the true image due to the intensity degradation induced by. The information of the acquired image is then minimized by the correction model, which leads to the corrected image. III. CORRECTION BY A LINEAR MODEL In this section, we describe the implementation of the proposed retrospective correction strategy by using a linear image degradation model. First, we present the modeling step in which we define the parametric correction model. Second, (2) (3) (4) to (5)

3 1400 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 same. The domain comprises the relevant part of the image, i.e., the tissue data. See Section III-C for a sharper definition. We further normalize the parameters so that equal change of any parameter or of basis function will produce intensity transformation of the same order and for all (11) we use polyno- As the smoothly varying basis functions mial terms in the following form: Fig. 1. Image degradation and retrospective correction of intensity inhomogeneities based on information minimization. we explain the procedure for finding the optimal correction model and computing the corresponding corrected image. Third, we give implementation details. A. Modeling The linear model of image degradation, which is a relatively general model that can characterize the image degradation processes in MR and in many other modalities, consists of a multiplicative and an additive intensity degradation component By the inverse of the degradation model, the estimation the true image is obtained as where (6) of (7) and (8) are the multiplicative and additive correction components, respectively. These two components are described by a linear combination of smoothly varying basis functions and that are uniquely defined by parameters and, respectively. To neutralize the global transformation effect of the correction components and with respect to the given image (7), we introduce the mean-preserving condition (9) (10) which ensures that the mean intensity values of and, defined over the domain of size, will be the and (12) with neutralization constants, needed to fulfil the mean-preserving condition (10), and normalization constants, needed to normalize the parameters (11). A detailed description of the derivation of the constants and is given in the Appendix. The two correction components and, which define the correction model (7), are modeled by the linear combination of neutralized and normalized polynomials and (13) B. Correction Let us now consider the linear image degradation model (6) from the information perspective. The components and can be viewed as multiplicative and additive intensity degradation images. The true image contains solely the information about the imaged object. Owing to the image degradation process, the true image is first multiplied by the image, and next, the image is added to it. Consequently, the information of the two intensity degradation images, and, is merged with the information of the true image. Because the true and the two degradation images contain distinct spatial variations, the information content of the resulting image will be higher than the information of the true image. In the retrospective correction, we try to minimize the information of by using (7) and changing the parameters and of the correction components defined in (13). The optimal parameters and are found by Powell s multidimensional directional set method and Brent s one-dimensional (1-D) optimization algorithm [30] (14) The optimal parameters and define the optimal components and, respectively, which transform the acquired image into the optimally corrected image (15)

4 LIKAR et al.: RETROSPECTIVE CORRECTION OF MR INTENSITY INHOMOGENEITY BY INFORMATION MINIMIZATION 1401 Fig. 2. (a) Entropy as a function of individual multiplicative and additive bias field parameters. (b) C. Implementation Details Three variations of the proposed information minimization method were implemented. In the first method, named MA2, the multiplicative and additive component are modeled by a second-order polynomial. The total number of parameters for the two components in three dimensions is 18. The other two methods, named M2 and M4, consist solely of a multiplicative component, which is modeled by a second- and fourth-order polynomial, respectively. The number of parameters is nine for the M2 method and 34 for the M4 method. For each image, a correction domain, containing the tissue data and not the unaffected background, has to be defined. The domain is extracted by thresholding the image and subsequently eroding the obtained nonbackground area, using a simple sixvoxel structuring element that represents the neighboring voxels in three dimensions. Prior to the optimization process, the neutralization constants and normalization constants of the correction components and (13) are calculated by the discrete versions of (A8), (A12), (A14), and (A16). Effective calculation of image information content is crucial for the performance of the proposed method. The information of any image can be quantitatively expressed by the Shannon entropy as (16) where is the probability that a point in image has value. The magnitude of the entropy is a measure of the uncertainty described by a set of probabilities. The entropy is nonnegative and takes on its maximum value when all gray levels are equally likely. If a fast local optimization process (14), yielding a global optimum, is to be achieved, the variation of the parameters of the correction components should result in smooth changes of the entropy. The entropy is defined (16) from a set of probabilities that can be obtained from the intensity histogram of the current estimate. Since is obtained by an intensity transformation applied to image, an integer gray value is transformed to a new real value, which in general lies between two integer values, say and. An intensity interpolation is, thus, needed to update the corresponding histogram entries. We use partial intensity interpolation by which the histogram entries and are fractionally updated by and, respectively. Prior to the calculation of the set of probabilities, the histogram is slightly blurred to reduce the effects of imperfect intensity interpolation (17) where the parameter defines the size, of a triangular window. Because the intensity transformation may in general transform the gray values out of the image gray-level range, the histograms are formed in a considerably wider range of gray levels. To provide a high statistical power, which is proportional to the number of samples used to form a 1-D histogram from a 3-D image, we use 8-bit intensity quantization ( 256) for which we set to two ( 5). Due to the high statistical power, histograms may be formed by substantially subsampling the image data, which increases the speed of entropy calculation. Only voxels lying on a 3-D lattice of size are used to construct a histogram. Our experience shows that using approximately 5000 image samples is sufficient to form the 8-bit 1-D histogram that is statistically enough powerful and enables efficient calculation of entropy. IV. EXPERIMENTS AND RESULTS A. Validating the Correction Strategy In this section, we tested the proposed correction strategy using the MA2 method and showed that the proposed method can retrospectively correct the intensity variations and that it is not sensitive to subsampling and image noise. For this purpose, simulated MR images of a normal brain from the BrainWeb MR simulator [31] [33] were used. The images had a resolution of , 8-bit quantization, 1-mm slice thickness. To test the assumption that the linear image degradation process increases the information of the imaged object, the T1 image with 0% intensity nonuniformity and 3% noise was used as the true image (Fig. 3). This image was degraded by multiplicative and additive component. The parameters and, were each changed from 20 to 20, with a step size of one, and the entropy of the transformed image was measured. The obtained entropy functions are illustrated in upper frame of Fig. 2. The magnitudes of the individual entropy functions are very similar due

5 1402 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 Fig. 3. Simulated uniform and nonuniform (40% intensity inhomogeneity) images, and the corresponding retrospective correction of the nonuniform image illustrated by the corrected image, correction field, and intensity histograms. to the parameter normalization (11). The functions are smooth, have wide capturing ranges, and minima close to the neutral parameters ( 0) at which the original image remains unchanged. It is obvious that the simulated image degradation process increases the entropy and, thus, the information of the uniform test image. This example supports the fundamental assumption of information increase, which is the basis of our correction strategy. Next, we test if the proposed correction method can reduce the information of the nonuniform image. For this purpose, we used the simulated T1 image with 40% intensity nonuniformity and 3% noise as the acquired image (Fig. 3). The correction process was simulated by individually varying each of the parameters and, from 20 to 20, with a step size of one, transforming the nonuniform image by the correction model (7), and calculating the entropy of the transformed image. The corresponding entropies [Fig. 2(b)] show that the entropy of the original nonuniform image, given at neutral parameters ( 0), is reduced by changing the values of some parameters and transforming the image. This example, illustrating that the correction model is capable of reducing the information of a nonuniform image, again supports the correction strategy. Further, we illustrate in Fig. 3 that the proposed method can retrospectively correct the simulated image with 40% nonuniformity. The recovery of the intensity distribution can be observed by comparing the histograms of the uniform, nonuniform, and corrected image. The means of the nonuniform and corrected image are equal due to the mean preserving condition (10). The obtained optimal correction components and are visualized by applying them (15) to the uniform field. The obtained correction field also shows the corresponding image domain, which was used for the above experiments. In Fig. 4, we illustrate the effect of subsampling on the performance of the MA2 correction method applied to the simulated T1, T2, and PD images with 40% nonuniformity and 3% noise. The effect of subsampling was expressed by the change of the coefficient of joint variations cjv(gm, WM) (described in more Fig. 4. Sensitivity to subsampling. detail in the following section) of the gray and white matter. The change of cjv(gm, WM) at each integer subsampling parameter is given relative to the corresponding cjv(gm, WM) obtained without subsampling ( 1). It can be seen that the images can be substantially subsampled when computing the entropy without noticeably affecting the method s performance. Finally, we have tested the performance of the method at various levels of noise on T1 images from the BrainWeb MR simulator [31] [33]. For this purpose, the coefficients of joint variations cjv(gm, WM) of the corrected images with 40% nonuniformity were compared to the cjv(gm, WM) of the corresponding images with 0% nonuniformity. Regardless to the level of noise, being 0%, 1%, 3%, 5%, 7%, or 9%, the cjv-s of corrected nonuniform images were almost identical to the cjv-s of corresponding uniform images, indicating that the method performs well also in the presence of noise. B. Quantitative Evaluation 1) Methods: Five methods were included in the quantitative evaluation, i.e., the so-called MA2, M2, and M4 variations of the information minimization method described in this paper, the nonparametric nonuniform intensity normalization

6 LIKAR et al.: RETROSPECTIVE CORRECTION OF MR INTENSITY INHOMOGENEITY BY INFORMATION MINIMIZATION 1403 (N3) method proposed by Sled et al. [19], and the fast model independent (FMI) method introduced by Vokurka et al. [26]. The N3 method seeks a smooth multiplicative field that maximizes the frequency content of the intensity distributions. The problem is solved in logarithmic space where the multiplicative bias field becomes additive and, consequently, the probability distribution of the acquired image is given by the convolution of the probability distributions of the bias field and the true uniform image, which are supposed to be independent or uncorrelated random variables. The correction strategy is based on deconvolving the bias field distribution, which is supposed to be Gaussian, from the distribution of the acquired image. The correction process is iterative, deconvolving narrow Gaussian distributions from the subsequent estimates of the distribution of the corrected image. A corresponding smooth bias field is estimated by the B-spline approximation. The FMI method is a nonparametric method for correcting intensity nonuniformity, which is based on the assumption that various sources of nonuniformity in MR imaging give rise to smooth variations in image intensity, and that these variations can be extracted and corrected for. First, intraslice variations are corrected by re-integration of well-characterized local derivatives that are obtained by estimating the additive noise, local intensity gradients, and smooth local derivatives using statistical averaging. Second, interslice intensity variations are corrected by robust estimation of consistent slice-to-slice relative intensity shifts, which is carried out by comparing the adjacent slices and using the central slice as the reference to which every other slice is corrected. The N3 and FMI methods were implemented at our site by using the code available via internet. 1 The parameters of the N3 and FMI methods were set to their default values. The implementation of the N3 method was validated by comparing the results of the nonuniformity correction on the same images as originally published by the authors [19]. The implementation of the FMI method was validated by comparing the correction of a MR volume performed at our site to the correction of the same volume conducted by the first author [26]. 2) Criteria: The performance of an intensity nonuniformity correction method is commonly evaluated by comparing the coefficients of intensity variations within the individual tissue classes in the original and corrected images, see, e.g., [12], [19], and [21]. Such an evaluation requires that the individual tissues are segmented either manually or by some inhomogeneity-tolerant segmentation method. Because a complete manual segmentation of MRI stacks is a time-consuming and tedious task, a simplified partial manual segmentation is often used, see, e.g., [12], [19], and [21]. After the image is segmented, the coefficient of variations cv of the tissue class is computed as cv (18) where and are the standard deviation and mean intensity of class, respectively. The quantity cv, which expresses the normalized standard deviation in a given class, is invariant to the uniform multiplicative intensity transformation. If there is a single tissue class present in the image, the quality of the 1 ftp://ftp.bic.mni.mcgill.ca/pub/mni_n3/ and respectively Fig. 5. Original images of (top left) a normal volunteer and of (bottom left) a brain tumor patient. (right column) Corresponding manually segmented images. nonuniformity correction method, using solely a multiplicative component, can be quantitatively expressed by the change of the quantity cv. A drawback is that cv does not provide any information on the overlap between the intensity distributions of distinct tissue classes. For example, a correction method may transform a given image in such a way that the coefficients of variations of both gray and white matter are reduced, but the overlap between the intensity distributions of the two tissues is increased. In this case, the correction method did not, in the classification sense, improve the given image. To estimate the overlap between the two classes, we introduce the so-called coefficient of joint variations cjv cjv (19) which is the sum of the standard deviations of two distinct classes; and, normalized by the difference of means. It is easy to show that cjv is invariant to the uniform linear, i.e., multiplicative and additive, intensity transformation. As such, cjv is an appropriate quantitative measure of the quality of nonuniformity correction, using either the multiplicative or additive component, or both components. 3) Experimental Data: One set of simulated and three sets of real MRI brain data were used. Set 1: Twelve (four T1, four T2, and four PD-weighted) MRI volumes from the BrainWeb MR simulator [31] [33]. The volumes had the resolution of , 8-bit quantization, 1-mm slice thickness, and 3% noise. Six volumes with 40% and six volumes with 0% nonuniformity were used, three of each were normal and three with MS lesions. Set 2: Six (two T1, two T2, and two PD-weighted) MRI volumes (Fig. 5), three of a normal volunteer ( voxels, 8-bit) and three of a brain

7 1404 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 TABLE I CORRECTION OF SIMULATED AND REAL DATA OF IMAGE SETS 1 AND 2 tumor patient with multiple malignant meningioma lesions ( voxels, 8-bit), were used. A complete gray and white matter manual segmentation was available for each volume [34]. Set 3: The set contained 33 (11 T1, 11 T2, and 11 PDweighted) real MR scans acquired at 11 different sites from 11 multiple sclerosis patients having a moderate number of white matter lesions [19]. The volumes ( voxels, 8-bit) were transformed into a stereotaxic space, so that the same manual labeling of the regions of pure gray and white matter could be applied to each of the three modalities. Set 4: The set contained 18 normal coronal 3-D T1-weighted spoiled gradient echo MRI scans acquired on two different imaging systems. Segmentation was performed on the positionally normalized (3.0-mm coronal, 1.0-mm axial, and 1.0-mm sagittal) [35], [36] scans by trained investigators using a semi-automated intensity contour mapping algorithm [37] and also using signal intensity histograms. The images were all 8-bit, they were cropped from coronal resolution to either or to exclude the neck data, and they had coronal slices. 4) Results: We performed the evaluation of the MA2, M2, M4, N3, and FMI methods by computing the coefficients of variations cv(gm) and cv(wm) of the gray and white matter, respectively, and the coefficient of joint variations cjv(gm, WM) of the two tissues for all images from the four experimental sets. Table I contains the results of nonuniformity correction of the images from the first two sets. Ideally, the cv(gm), cv(wm), and cjv(gm, WM) of the simulated uniform and corrected uniform images of Set 1 should be identical. This was approximately true for all methods except the FMI method. The latter method even corrupted the images with 0% nonuniformity. The corrected images with simulated 40% intensity nonuniformity indicate that all methods, except the FMI, can efficiently minimize the intensity variations of the gray and white matter and reduce the overlap of their intensity distributions. The cv-s and cjv-s of all corrected images with 40% nonuniformity are similar to the cv-s and cjv-s of the images with 0% nonuniformity for all methods, except for the FMI method and the N3 method when applied to a normal T2 image. The FMI method yielded the worst corrections. In two out of six cases, cjv of the corrected nonuniform images was even higher than the cjv of the original image. The methods applied to the real normal and tumor volumes (Set 2) yielded similar reductions of the nonuniformity criteria, with the exception of M4 method, which slightly increased the cjv(gm, WM) of T2 tumor volume, and the FMI method, which increased the cjv(gm, WM) of the three normal volumes. The performance of the correction methods on the image sets 3 and 4 is illustrated in Fig. 6 by the box-whiskers diagrams. The diagrams show the distributions of changes as their minima, 1st quartiles, medians, 3rd quartiles, and maxima. The FMI method performed poorly on nearly all images from the two sets. On T1 images from Set 3, which likely had small nonuniformities, the cjv(gm, WM) were best reduced by the M4 method, slightly less by the MA2 method, while the M2 and N3 method yielded similar, slightly decreased, variations. On T2 images from Set 3, the M4 method performed the best, following by the gradually slightly less effective N3, M2, and MA2 methods. All methods, except FMI, corrected the PD images from Set 3 well. The MA2 method was the most effective in reducing the cjv(gm, WM) although not that effective in reducing the cv(wm). The T1 images from Set 4 were best corrected by the M4 method, while the MA2, M2, and N3 method reduced the variations of gray and white matter but did not change or they even slightly increased the coefficient of joint variations. Table II summarizes the mean changes of the cjv(gm, WM) for the image sets 3 and 4 and suggests similar conclusions on the performances of the methods as the box-whiskers diagrams but in a more transparent and compact form.

8 LIKAR et al.: RETROSPECTIVE CORRECTION OF MR INTENSITY INHOMOGENEITY BY INFORMATION MINIMIZATION 1405 Fig. 6. Distributions of changes of coefficient of variations (cv) and coefficient of joint variations (cjv) for the five methods and two sets of experimental images. TABLE II MEAN CHANGES OF cjv(gm, WM) IN [%] FOR THE IMAGES FROM SETS 3 AND 4 C. Qualitative Evaluation 1) Experimental Data: Three different MR images were used: Image 1: A coronal three-dimensional T1-weighted spoiled gradient echo MRI scan from Set 4 used for quantitative evaluation, having voxels and 8-bit quantization. Image 2: A breast T1-sagittal image (1T Siemens Magnetom Harmony, body coil, 4-mm slice thickness), having voxels and 8-bit quantization. Image 3: A prostate T1-axial image (1T Siemens Magnetom Harmony, phased array, 3-mm slice thickness), having voxels and 8-bit quantization. 2) Results: Fig. 7 illustrates the successful corrections of intensity inhomogeneities in Image 1 by the M4, N3, and FMI methods. This is the only image from Set 4 on which the FMI method performed better than the other two methods, as shown by original images, corrected images, and corresponding intensity distributions of the gray and white matter. The corresponding coefficients of joint variations, which well estimate the actual overlaps, are also given. The correction fields of the M4 and FMI method are similar, while the correction field produced by the N3 method, which in this case performed less well, is smaller in magnitude. In Fig. 8, we qualitatively compare the performance of the M4, N3, and FMI methods on images 2 and 3. The corrections by the N3 and FMI methods were conducted by the designers, i.e., by J. G. Sled and E. A. Vokurka, respectively, who were asked to tune the parameters to achieve the best performances, while the correction by the M4 method was performed as described in this paper with no image-specific modifications. The histograms of the breast image [Fig. 8(top row)] show only the intensity distributions of the tissues and not of the background intensities. The first mode in the histograms represents lung and breast tissues, the second mode muscle tissue, and the third mode fat tissue. The three modes overlapped significantly in the original images due to the intensity inhomogeneities and are best separated after the correction by the M4 method. The N3 method also reduced the overlap, while the FMI method reduced the contrast and likely increased the overlap between the distinct tissues. The original and corrected images of the prostate and corresponding histograms [Fig. 8(bottom row)] illustrate the distributions of the fat and muscle tissues before and after corrections of inhomogeneities. The M4 and N3 methods reduced the overlap between the distributions of the two distinct tissues, while the FMI method reduced the contrast significantly but only slightly reduced the overlap.

9 1406 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 Fig. 7. Correction of an MR brain image from Set 4 by three different methods. (top row) Original and corrected images. (middle row) Corresponding histograms showing the distributions of gray and white matter. (bottom row) Segmented image and bias fields. V. DISCUSSION All MR images are inherently intensity-inhomogeneous, due to imperfections of the acquisition systems and patient-specific interactions, and often need to be retrospectively corrected to improve the growing number of postacquisition tasks, such as segmentation, registration, and various quantitative analyzes. In this study, a general information-theoretic strategy for correcting intensity inhomogeneity was proposed and the implementation of a linear correction model with polynomial multiplicative and additive correction components was described. The fundamental correction strategy was confirmed by showing that the method does not corrupt the simulated uniform images and could retrospectively correct the simulated nonuniform images. The method, which proved to be insensitive to substantial subsampling of image data and image noise, can perform the correction in less than a minute on the current generation of personal computers. The method is based solely on the assumption that, because spurious intensity variations induced in the degradation process are spatially different than the natural intensity variations in the true image, the information content of the degraded image is higher than the information of the true image. As a consequence, the method should work with a great variety of images that fulfil the above assumption. This was confirmed in our recent paper [29], where we showed that the 2-D method performs well on a great variety of differently structured simulated and real microscopic images. Our experience in various MR images also indicates a high generality of the method. Moreover, pathological variations, such as tumors or other lesions, do not affect the basic assumption, as they are naturally dissimilar to the polynomial correction model. To further quantitatively support the basic assumption, extensive quantitative evaluation on MR brain images, for which gold standard manual segmentations are available, were conducted in this study. Three variations of the information-theoretic correction methods were implemented, i.e., the MA2 method, using the second order multiplicative and additive components, and the M2 and M4 methods, using solely the multiplicative components of the second- and fourth-order, respectively. The three methods were quantitatively evaluated on 69 experimental MRI brain images from four different sets and compared to the N3 method [19] and the recently published model independent method [26], which we abbreviated to FMI. The performances of all methods were expressed by the changes of the coefficient of variations of the gray and white matters and by the changes of the so-called coefficient of joint variations (cjv). The cjv, which estimates the overlap between the two matters, is invariant to linear uniform intensity transformations and does not require a complete segmentation. An alternative to cjv would be to measure the actual overlap between the distributions of the two tissues. However, this would require a complete segmentation and would fail to measure the correction quality if there was no overlap between the two distributions.

10 LIKAR et al.: RETROSPECTIVE CORRECTION OF MR INTENSITY INHOMOGENEITY BY INFORMATION MINIMIZATION 1407 Fig. 8. Correction of an MR image of (top) breast and (bottom) prostate by three different methods and the corresponding intensity histograms. The corrections of the 12 simulated (six uniform and six nonuniform, three of each normal and three with MS lesions) and six real (three normal and three tumor) images, for which 100% segmentations were available, indicated comparable and promising performances of the three information-theoretic methods and the N3 method. The FMI method performed less well as it frequently induced additional intensity inhomogeneities. On 33 real scans (11 T1, T2, and PD-weighted), which were partially manually segmented, and on 18 real T1-weighted scans, 100% semi-automatically segmented, the information theoretic M4 method outperformed the other methods, except when applied to the 11 PD-weighted scans. In this case, the MA2 method was superior. The other methods also performed well and comparable, except the FMI method, which corrupted the majority of the images. The methods for MRI nonuniformity correction commonly use only the multiplicative correction component because this simplifies the task and because the bias field is known to be mostly multiplicative. The proposed correction method is the first that enables correction by a true linear correction model, i.e., by multiplicative and additive component. Such a correction was implemented in the MA2 method. To find out if the retrospective correction may benefit from the additional additive component, the MA2 method was compared to the M2 method that uses solely the multiplicative component of the same order. The MA2 method performed better on PD images, worse on T2, and slightly better on T1 images. Therefore, on the experimental images from this study, no significant improvement was found by adding the additive component but rather by using a more dynamic component as indicated by the better performance of the M4 method. Yet, the possibility to correct the images by both components might prove useful for other types of coils or with other modalities. The M4, N3, and FMI methods were qualitatively evaluated and compared on breast and prostate MR images. The obtained results show that the M4 method works well also on nonbrain images. The proposed method can, thus, be applied to MR images of different organs without task-specific modifications. Concerning the correction times, the proposed information minimization method was found to be faster than the N3 and FMI methods. A typical correction time for an image with a resolution of on a current generation of personal computers was 30 s for the M4 method, 5 min for the N3 method, and 30 min for the FMI method. In general, the M4 variation of the proposed method performed the best from the methods tested. The N3 method also yielded a number of successful corrections, while the FMI method often failed in correcting the images. The main reasons for such an outcome emerge from the differences in the correction strategies. The here proposed methods and the N3 method deal with slow and real 3-D intensity variations in the entire image domain, while the FMI method deals separately with intraslice and interslice variations. Consequently, the proposed and the N3 methods cannot remove rapid interslice variations, sometimes observed with 2-D multislice sequences, but more efficiently deal with smooth 3-D variations. On the other hand, the FMI method, which was designed also with the purpose to reduce rapid interslice variations, proved to be less efficient in removing the smooth 3-D variations. It was noticed that the FMI method often produced rapid spurious interslice offsets when correcting inhomogeneity-free images, suggesting that intraslice corrections are not consistent between adjacent slices. The use of the FMI method should, therefore, as stressed by the method s authors [26], be treated with caution since various regional combinations of anatomy and coil position may generate spurious corrections. The N3 method performed

11 1408 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 12, DECEMBER 2001 well on brain data and slightly less well on breast and prostate data, indicating that the N3 method is well suited to images with small scale rather than large scale structures as already reported in [19]. Several segmentation-based methods have been proposed that simultaneously compensate for inhomogeneity and segment the image [4], [20], [21], [23] [25]. They have the advantage of being able to use intermediate information from segmentation when performing the correction. For example, in [25] the piecewise contiguous nature of segmentation is modeled by a Markov random field and a greedy algorithm based on iterative conditional modes is used to find an optimal segmentation while estimating model parameters. The method based on fuzzy -means [24], minimizing the sum of class membership function and the first- and second-order regularization terms that ensure the smooth bias field, requires rough initialization of the class centroids but provides unsupervised soft segmentations robust to inhomogeneities. The authors of [24] believe that further research on incorporating spatial information through Markov random fields or atlas information may enhance the robustness. Other segmentation-based methods use the EM algorithm and compute the bias field from the residue image by either spatial filtering [20] or by weighted least-squares fit of the polynomial bias model [21]. These methods interleave classification and bias field estimation [20] and also the estimation of class-conditional intensity distribution [21]. However, the methods require initialization, i.e., the determination of specific class-conditional intensity models. Such models are typically constructed by manually selecting the representative points of each tissue class considered [20]. An alternative approach, using a digital brain atlas with a priori probability maps for each tissue, was recently proposed to construct the intensity modes automatically, but requires rigid preregistration of the digital atlas to each individual scan being processed [21]. Although the segmentation-based methods seem very promising in interleaving the segmentation and nonuniformity correction and also in including a priori knowledge on, for example, the class-conditional intensity models, there are some drawbacks to be heeded. The methods require initialization or inclusion of a priori knowledge, which may be problematic. For example, the determination of specific class-conditional intensity models conducted by manual point selection may not provide objective and reproducible results. The same task performed by automatic rigid or nonrigid preregistration of a digital atlas is also questionable. First, because such a registration is far from trivial, especially in the nonrigid case and in the presence of intensity inhomogeneities, and second because the practical usage is limited by the availability of the appropriate digital atlases. Another drawback of segmentation-based methods lies in the fact that these methods usually assume that the intensity distribution of an image is a mixture of normal distributions of individual tissues, which may often not be valid. The above drawbacks are even more serious when correcting pathological data. Surface-fitting methods, another class of retrospective correction methods, make no specific assumption on the intensity distribution of the individual tissues but rely solely on the spatial variations of the bias field, which are assumed to be smooth. For example, in [3] polynomials were fitted to regions defined by inhomogeneity-tolerant preliminary segmentation, while in [12], thin-plate splines were fitted to manually or automatically selected points. The problem with the first approach lies in the fact that inhomogeneity-tolerant segmentation is hard to achieve prior to nonuniformity correction and can only make use of contiguous regions, which are not always present. The second approach, on the other hand, requires user interaction, classifier training, and uses only a small number of white matter points to fit the field to. The approach described in [18], which fits polynomials directly to image data via a heuristic multiple-valley criterion function, overcomes the problems mentioned above, but requires the specification of a set of expected class means and standard deviations, which is far from trivial and directly influences the correction result. The method proposed in this paper may also be classified as a surface-fitting method. The method requires no presegmentation, selection of control points, or user interaction or initialization. It fits a parametric polynomial model directly to the whole image data by using the entropy as a criterion function. Although the method is optimization based and, thus, cannot generally guarantee the global solution, it was illustrated and confirmed by a number of successful local optimizations that the optimization criterion, i.e., the entropy, is not prone to local optima and has a wide capturing range. On the other hand, the selection of a higher order correction model may result in a difficult optimization. The problem may be solved by a multigrid sampling strategy and/or via multilevel optimization. In the first case, the parameters can be initially optimized with sparse sampling and the optimization could then be continued by gradually increasing the sampling density. In the latter case, the optimization can, for example, start with the first order polynomial whereupon the number of parameters can be sequentially increased, initializing each new order with the results of the previous order. The highest order of the correction model should, however, be chosen carefully owing to the possible interaction of the model with the true image data, which is a limitation of all retrospective methods that use solely the information naturally occurring in the images. Finally, the multigrid sampling and multilevel optimization strategies might be combined to enhance the overall performance. Nevertheless, the simple and transparent implementation, using equidistant subsampling and a fourth-order polynomial correction model, proved to be efficient, robust, and fast in execution. VI. CONCLUSION A novel information-theoretic approach to retrospective correction of intensity inhomogeneities in MRI, based on modeling the bias field and minimizing the information of the acquired images by the parametric polynomial model, was proposed. The modeling approach offers the possibility of incorporating knowledge about the image formation process in the correction algorithm, while the global intensity uniformity criterion, i.e., the entropy, enables the optimization of arbitrarily complex image formation models. The method uses solely the information naturally present in the images and makes no assumption on the shape of either the bias field distribution or the