Retrospective correction of image nonuniformities

Transcription

1 Retrospective correction of image nonuniformities We will read & discuss some influential papers in the field: Axel et al. AJR Lim et al. JCAT 1

2 Axel et al. AJR Introduction The use of surface coils in MR imaging has made it possible to obtain images of superficial structures with improved signal- to-noise ratio relative to conventional circumferential receiver coils. However, the signal intensity distribution in surface-coil images is inherently nonuniform; the decreased sensitivity to signal from more distant regions implies a corresponding increased sensitivity to noise from those regions. As the total noise detected is uniformly distributed over the image, there is a net increase in the signal-to-noise ratio in the image of regions closer to the coil as compared with the conventional receiver coil. The resulting range of relative intensity over surface-coil images makes it difficult to display them properly or to analyze them quantitatively. In addition, the local image contrast will decrease proportionally to the average local intensity. We report a method for correcting such surface-coil images so as to produce a uniform relative intensity over the region being imaged. 2

3 Axel, Methods 1. Surface-coil images were obtained both of the desired object and of a uniform phantom placed in the same position on the coil. The image of the phantom is normalized (so as to serve as a calibration of the surface-coil response pattern) and divided into the image of the object. Thresholding can be used to mask out the background. Conventional two-dimensional Fourier transform MR images have uniformly distributed noise. Although the relative intensities will be made uniform in the resulting corrected image, the local noise in the corrected image will now be nonuniform, owing to the relative boost in the region of lower-coil sensitivity in proportion to the amount the signal has been boosted. If the calibration phantom image is relatively low in noise, the actual local signal-to-noise ratio will be essentially unchanged from the original image. 3

4 Axel, Methods, contd 2. An alternative method can be used to approximately correct a surface-coil image of an object for the pattern of nonuniform coil response without acquiring an additional image of a uniform phantom. The original image of the object is blurred to suppress the details of the object and thus to serve as an approximation of the image of a uniform phantom. This blurred image of the object can is divided into the original image to produce a corrected image. For the white noise typical of an MR image, the blurring process will produce a low-noise correction image. This technique was implemented on images obtained with a simple 3-cm-wide rectangular surface coil used as a receiver coil, magnetically orthogonal to a circumferential exciting coil in a 1.4 T small-bore MR imaging system.. 4

5 Axel, Results 1. An image of the wrist (Fig.1) was obtained with the surface coil, with TR = 500 msec, TE = 16 msec, four averages, 256 x 256 matrix, 8-cm FOV and 5-mm slice. An image of a bag of saline (Fig.2) containing 2 mm CuSO 4 was obtained with the same coil in the same position. The result of dividing the first image by the second (masking out the background) is shown in Figure 3. 5

6 2. The original image was blurred (Fig. 4) by convolution using a 9 x 9 pixel kernel and divided into the original image (Fig. 5). 6

7 Axel, Discussion 1. This technique of surface-coil image intensity correction can produce good-quality corrected images. Although it flattens out the overall effective sensitivity, it cannot improve the local signal-to-noise ratio (for a low noise calibration, it also will not harm signal-to-noise ratio). Thus, the relative noisiness of the image of regions distant from the coil may be apparent in the corrected image. If this is objectionable, these regions can be easily masked out in the image correction process, just as the background was here. If the background in the calibration image is not masked out of the final image, the noisiness of these regions may be objectionable. Thus, the uniform phantom used to make the calibration image should cover at least the region likely to be of interest in the object imaged with the surface coil. 7

8 Axel, Discussion, contd 2. In making the overall intensity more uniform, this technique also corrects the relative contrast in different regions. In approximately correcting the surface-coil image by dividing the image by a blurred version of itself, the overall intensity variations of the original object will be flattened and the intensity profiles of edges somewhat distorted, although local contrast will be approximately corrected. 3. With reproducible surface-coil positioning, the pattern of response of the coil can be stored and used to correct subsequent surface-coil images without repeated phantom calibration images. Alternatively, a simplified form of the coil response can be stored and used to approximately correct the images in a similar manner with less attention to precise positioning. Because the surface-coil response pattern is usually a relatively slowly changing function of position, small amounts of mis-registration will generally not be a problem. 8

9 Axel, Discussion, contd 4. In systems that use the surface coil for both excitation and receiving, the variation of flip angle with distance from the coil will result in a corresponding variation of saturation that will depend on the local relaxation time. This cannot be fully corrected for with the image of the uniform phantom, even if the relaxation time of the phantom is adjusted to match the average value in the object. The correction produced by dividing the image of the object by a blurred version of itself will still be useful for reducing the overall range of intensities in the image while approximately correcting local contrast. Lim & Pfefferbaum, JCAT 9

10 Introduction One of the major advantages of magnetic resonance imaging over CT for the study of the brain has been the production of images in which gray and white tissue are visually differentiable. Techniques to delineate cortical and subcortical gray matter areas from CT are limited by the poor soft tissue contrast and artifact affecting cortical gray matter areas. With MR, collection parameters can be manipulated to weigh differentially the contribution of three variables affecting signal intensity: proton density, spin-lattice relaxation time, and spin-spin relaxation time. This provides images that approach neuroanatomy textbook photographs of sectioned brain in visual clarity. The full exploitation of this visual resolution requires quantitative image analysis techniques. Quantification of gray and white matter volumes would be of particular value in clarifying the relative contribution of gray and white matter pathology to tissue loss inferred from enlargement of fluid-filled spaces seen in several neuropsychiatric disorders. Lim, Introduction, contd 10

11 In vivo quantitative analysis of white & gray matter volumes would be helpful in documenting the pathophysiology of conditions believed to involve primarily white matter (e.g., multiple sclerosis) or gray matter (e.g., Alzheimer disease). Delineation of gray and white matter areas would also be valuable for providing structural templates against which functional activity, such as glucose metabolism or specific ligand uptake, measured by positron emission tomography, can be assessed. We present techniques for segmentation of MR images into fluid, white matter, and gray matter compartments. Algorithms for stripping the skull, correcting radiofrequency (RF) inhomogeneity, and resolving partial voluming ambiguities are also described. The technique had been applied to brain scans from five healthy, young, normal men, and resulting gray and white tissue volumes compared with published data from a postmortem study. Lim, Methods 11

12 A standard spin echo sequence was used with a field of view of 24 cm. Echoes were obtained at 20 and 80 ms. Seventeen to twenty 5 mm slices were collected with an interslice skip of 2.5 mm. The slices were oriented in an oblique plane, parallel to a line connecting the anterior and posterior commissures. These landmarks were identified on a mid-sagittal slice, collected with a TR of 600 ms, TE of 20 ms, with one excitation for each of 256 phase encodes. Anchoring slice orientation to standard neuroanatomical landmarks allows reproducibility between images within and across subjects, controls for variations in head positioning in the scanner, and provides images comparable with standard neuroanatomic atlases. 1. The first step is to outline the brain by identifying and stripping away pixels representing skull and scalp. On MRI, bone has a low signal, which can be confused with fluid and tissue in many acquisition sequences. For skull stripping, we use a late echo image in which fluid, with its longer T2, has a higher signal than tissue, bone, or flowing blood. The peripheral cortical gray matter and the CSF around the periphery of the brain provide a sharp transition with the signal void of the skull in axial sections. 12

13 Once the skull margin has been identified, it is applied to the early echo of the same slice. We sampled 256 radii, each consisting of 128 pixels, from the center of the basis image, and laid them in rows to form a new "radial" image. The basis image is shown in Fig.1a and the radial image in Fig.1b. Skull margins were determined for each row by identifying the pixel in that row which was 40% of a running average of the previous 10 pixels in that row. 13

14 We started the search for the brain/bone transition close to its target-at a pixel number representing 90% of the median pixel number identified as skull margins for the last five rows. Once the 256 brain/bone transitions have been determined, (Fig. 1c) their locations are converted back to the rectangular coordinates of the original image. The points are connected to outline the brain area (Fig. 1d). The algorithm fails on lower slices (cerebellum); however, it is effective for the more superior sections. Following automatic stripping, an interactive program allows each stripped image to be reviewed and any erroneous transition lines to be corrected. The automatic algorithm succeeded in all sections including the lateral ventricles and centrum semiovale. Lim, Methods, contd 14

15 Lim, Methods, contd 2. At later echo times, such as TE = 80 ms, the fluid signal is greater than tissue, whereas at an early echo of TE = 20 ms, tissue is higher than fluid. Adding or subtracting images acquired at different TEs can differentially enhance fluid/tissue or white/gray contrast. Subtracting the late echo from the early echo enhances the fluid/tissue contrast by subtracting out the fluid signal. The results of these operations are shown in Fig. 2. Figure 2a and b are the early and late echo images from the same section. Figure 2c is the early-plus-late echo image and Fig. 2d is the early-minus-late echo image. The improvement in contrast is particularly pronounced for the fluid/tissue image, the early-minus-late echo image. 15

16 Lim, Methods, contd 3. Before these combination images can be segmented, it is necessary to deal with the influence of nonuniform RF coil sensitivity on signal intensity. This RF inhomogeneity introduces a gradient or low frequency variation in the signal level across a given image. Although the human visual system can maintain contrast detection in the face of a low frequency or direct current (DC) shift, simple thresholding techniques assume a spatially invariant baseline level. Figure 3a presents an image with a typical RF inhomogeneity intensity gradient. Figure 3b illustrates how a single gray/white threshold for this image, set in the frontal pole, fails to differentiate completely these compartments in the posterior pole. This RF inhomogeneity is present to some degree in almost all scans and must be removed before thresholding can be performed. 16

17 Radiofrequency inhomogeneity is relatively low in spatial frequency compared with the anatomical information of interest. We first created a low pass image using a 33 point averaging filter (Fig. 4a). The original image was then divided by the low frequency version. Unfortunately, when the original image is used to create its low-frequency version, sharp signal transitions, especially those at the brain-skull interface. are enhanced. This resulting brightening of the cortical rim of gray matter (Fig. 4b) poses another challenge to image quantification. 17

18 Correction of this artifact was accomplished by developing a technique for softening brain-skull transitions. This involved extending the intensity of the brain adjacent to the brain-skull margin out to the edge of the image matrix to create a "feathered" image. The feathering operation uses the brain-skull interfaces as reference points, takes the mean intensity value of three pixels inside each margin, and extends it to the edge of the image matrix. This is done radially, using 1,024 spokes to ensure that the edges of the image matrix are completely filled. The resultant feathered image (Fig. 5a) is used to create a low frequency version (Fig. 5b) with which to correct RF inhomogeneity in the original image with the skull removed (Fig. 5c). The final step is the application of a 3x3 gaussian matrix filter to attenuate high frequency noise. The thresholded image of Fig. 5d demonstrates the success of these procedures in eliminating RF inhomogeneity artifact. 18

19 Lim, Results In Fig. 7 the randomized fluid/total area ratios are plotted for each section done by each rater. The Spearman rank correlation coefficient for the two raters was In Fig. 8 the analogous plot for the gray matter/total tissue ratio is shown with a Spearman rank correlation coefficient of Because total tissue equals white matter plus gray matter, the white matter/total tissue comparison between raters has the identical correlation (rho = 0.729). 19

20 In Table I the age and whole brain gray/white ratios of the subject are shown. The average of the two raters' values for each subject are used. The mean gray/white ratio for the five subjects was We compared these results to Miller et al. who used fixed brain sections, treated to enhance gray/white contrast, and a digital image analyzer to perform an analogous quantitative postmortem study. They analyzed 91 brains from subjects dying between the ages of 20 and 98 years. They had four subjects comparable in age with those in our study, whose gray/white ratios varied between 0.9 and 1.3 with a mean of 1.06 (data read from graph). 20

21 Lim, Discussion One of the major obstacles to any thresholding approach to segmentation of MR images is the influence of RF inhomogeneity. Such inhomogeneity is characteristic of most commercial MR scanners. Although it poses minimal problems to clinicians making qualitative assessments of MR images, it is problematic for quantitative analysis. Various approaches to correcting this artifact have been previously described. One approach has been the use of phantoms. Such techniques require that the inhomogeneity be well characterized prior to imaging and that the slice position be well defined. Image-specific homomorphic filtering techniques have been described for use with surface coil images. To deal with the problem of postfiltering edge brightening, Fuderer and van Est took advantage of the expected sensitivity of the surface coil and used an intensity threshold to segment the image into object and background, applying filtering correction to only the object. In our technique the outlined brain was the object and the skull and periphery were the background. 21

22 Lim, Discussion, contd We chose to use an image-specific correction approach because it avoids the need for characterizing the RF inhomogeneity and accurately determining section position. This approach gives greater flexibility in slice selection and is applicable to sections obtained in coronal or sagittal planes. 22

23 Pham et al. Pattern Recognition Letters Introduction Image segmentation plays an important role in a variety of applications such as robot vision, object recognition, and medical imaging. There has been considerable interest recently in the use of fuzzy segmentation methods, which retain more information from the original image than hard segmentation methods. The fuzzy C means algorithm (FCM), in particular, can be used to obtain a segmentation via fuzzy pixel classification. Unlike hard classification methods which force pixels to belong exclusively to one class, FCM allows pixels to belong to multiple classes with varying degrees of membership. This approach allows additional flexibility in many applications and has recently been used in the processing of magnetic resonance (MR) images. The FCM algorithm, however, does not address the intensity inhomogeneity artifact that occurs in nearly all MR images. In MR imaging, intensity inhomogeneities may be caused by nonuniformities in the RF field during acquisition as well as other factors. 23

24 Pham, Introduction, contd The result is a shading effect where the voxel intensities of the same tissue class vary slowly over the image domain. This shading can cause severe errors when attempting to segment corrupted images using intensity-based pixel classi cation methods. It has been shown that intensity inhomogeneities are well modeled by the product of the original image and a smooth, slowly varying multiplier field. There are two general approaches to segmenting images with intensity inhomogeneities. The first approach is to separately apply a correction algorithm, followed by a segmentation algorithm. This approach allows flexibility, in that once the image has been corrected, the intensity inhomogeneities can essentially be ignored in any subsequent processing. In their work, Dawant et al. used manually selected reference points in the image to guide the construction of a spline correction surface. Johnson et al applied a homomorphic filter in an attempt to remove the multiplicative effect of the inhomogeneity. Their method, however, is effective only on images with relatively low contrast. Meyer et alused an edge-based segmentation scheme to find uniform regions in the image followed by a polynomial surface fit to those regions. 24

25 Pham, Introduction, contd The second approach used to segment images with intensity inhomogeneities is to simultaneously compensate for the shading effect while segmenting the image. This approach has the advantage of being able to use intermediate information from the segmentation while performing the correction. A number of methods have used segmentation algorithms based on Markov random fields that account for inhomogeneities by allowing the centroids of each class to vary independently. Unser et al. proposed an adaptive K-means algorithm that also allowed the centroids to vary independently according to a first order regularization term. These methods only yielded hard segmentations. However, Wells et al. used an expectationmaximization algorithm that modeled the inhomogeneities as a bias field of the image logarithm. Although their results are impressive, their proposed method is supervised, requiring manual interaction to provide training data. 25

26 Pham, Introduction, contd In this paper, we adopt the second approach of segmenting images while simultaneously compensating for inhomogeneities and propose a new algorithm, called the adaptive fuzzy C-means algorithm (AFCM), which produces a fuzzy segmentation while compensating for intensity inhomogeneities. AFCM incorporates a multiplier field term into the standard FCM objective function and, except for the initial specification of two parameters, is completely automated. We propose our new objective function, describe the steps of the AFCM algorithm, apply it to several test images and draw comparisons between the standard FCM algorithm and AFCM. The objective function contains a multiplier field term that models the brightness variation caused by the inhomogeneities. We describe a method based on nonparametric density estimation for automatically obtaining initial values for the centroids needed in the algorithm. 26

27 Pham, Methods The standard FCM algorithm seeks the membership functions u k and the centroids v k, such that the following objective function is minimized: k --- tissue class, ex k=1 for brain white matter, k=2 for gray matter The total number of classes C is assumed to be known. i, j --- voxel image coordinates u k (i, j) --- the membership value (class probability) at pixel location i, j for class k y(i,j) --- the observed image intensity at location i, j v k --- the average (centroid) intensity of class k. 27

28 Pham, Methods, contd The FCM objective function (eq.1) is minimized when high membership values are assigned to pixels whose intensities are close to the centroid for its particular class, and low membership values are assigned where the pixel data is far from the centroid. An advantage of FCM is that if a pixel is corrupted by noise, then the segmentation will be changed by some fractional amount, while in hard (non-fuzzy) segmentations, the entire classification may change. Furthermore, in the segmentation of medical images, fuzzy membership functions can be used as an indicator of partial volume averaging, which occurs where multiple classes are present in a single pixel. Taking the first derivatives of Eq. (1) with respect to u k (i, j) and v k and setting those equations to zero yield necessary conditions for (eq.1) to be minimized. Performing an iteration through these two necessary conditions leads to an iterative scheme for minimizing the objective function. 28

29 Pham, Methods, contd This is the standard FCM algorithm. The resulting fuzzy segmentation can be converted to a hard or crisp segmentation by assigning each pixel solely to the class that has the highest membership value for that pixel. This is known as a maximum membership segmentation. We propose a new objective function which preserves the advantages of FCM while being applicable to images with intensity inhomogeneities. In our approach, we model the brightness variation by multiplying the centroids by some unknown multiplier field m(i,j), which we assume is smooth and slowly varying with respect to i and j. 29

30 We define the two-dimensional AFCM algorithm to be the algorithm that seeks to minimize the following objective function with respect to u(), v and m(): D i, D j --- forward finite difference operators (like derivatives) along rows/columns, D ii --- ˆ D i * D i, * is a one-dimensional discrete convolution D ij ˆ --- D i ** D j ** two-dimensional discrete convolution D jj ˆ --- D j * D j are second-order finite difference. The last two terms, controlled by the parameters λ 1 and λ 2, are first and second order regularization terms operating on the multiplier field (not the membership functions). 30

31 Pham, Methods, contd The first order regularization term penalizes multiplier fields that have a large amount of variation. The second order term also penalizes, to a certain degree, the amount of variation but especially penalizes multiplier fields that possess discontinuities. If we assume that the membership functions u k (i, j) and the centroids v k are known, then the multiplier field that minimizes J AFCM is the field that makes the centroids close to the data, but is also slowly varying (as governed by the 1st and 2nd regularization terms) and smooth (as governed by the 2nd order term). Without the regularization terms, a multiplier field could always be found that would set the objective function to zero. The parameters λ 1 and λ 2 should be set according to the magnitude and the smoothness of the intensity inhomogeneity in the image. For an image with little or no inhomogeneities, larger values for parameters λ 1 and λ 2 should be used, thereby reducing AFCM to standard FCM. 31

32 The objective function J AFCM is minimized by taking the first derivatives of J AFCM with respect to u k (i, j), v k, m (i, j) and setting them equal to zero, resulting in three necessary conditions for J AFCM to be at a minimum. The AFCM algorithm iterates steps 1,2,3 until convergence, when the maximum change in u k (i, j) over all classes & pixels < Step 0. Provide initial values for centroids v k and set the multiplier m(i, j) = 1 Step 1. Compute memberships: Step 2. Compute new centroids: Step 3. Compute new m(i,j) by solving: here H 1 (i,j) = D i *Ď i + D j *Ď j ; H 2 (i,j) = D ii *Ď ii + 2D ij **D ij + D jj *Ď jj and Ď(i) = D(-i). Remember: D i D j are forward finite difference operators (like derivatives) along rows/columns, D ii = D i *D i. 32

33 Pham, Results We implemented AFCM on a Silicon Graphics Indigo2 with 150 MHz R4400 processor using MATLAB (Mathworks, Natick MA). Execution time for AFCM ranged 5-15 minutes for a 256x256 image, and was typically proportional to the amount of inhomogeneity present in the image. For comparison, execution time for FCM was approximately ten seconds and was independent of image size since it can be performed purely on the image histogram. In all examples shown, the regularization parameters λ 1 and λ 2 were set empirically and convergence was achieved in < 50 iterations. Our first experiment applies the FCM and AFCM algorithms to a checkerboard test image shown in Fig. 2(a). This is a two class image that has been corrupted by a sinusoidal intensity inhomogeneity. 33

34 Fig. 2(b) shows that standard FCM yields an incorrect maximum membership segmentation. The inhomogeneity causes the intensity of the darker class to vary towards the brighter class in such a way that they are indistinguishable in terms of pure pixel intensity. Fig. 2(c) shows the AFCM result.t he AFCM algorithm converged in 4 iterations with λ 1 = and λ 2 =

35 Fig. 3 shows the results of applying the FCMand AFCM algorithms to segment an image of a cryosectioned brain. Fig. 3(a) is the original image. Strong intensity inhomogeneities due to nonuniform illumination during the photography are apparent in the image. Both the FCM and AFCM algorithms were used to segment the image into three classes corresponding to background, gray matter (GM) and white matter (WM). Fig. 3(b) and 3(c) show the contours of where the GM membership function is equal to the WM membership function in the FCM and AFCM results, respectively. The FCM contour provides an inaccurate representation of the boundary. The AFCM algorithm, on the other hand, yields a much improved result. The value of λ 1 was set to and λ 2 to in this example. 35

36 Fig. 4(a) is an axial slice taken from a T1-weighted MR brain image. The image was preprocessed to remove extracranial tissue, and the remaining tissue was segmented into three classes corresponding to cerebrospinal fluid (CSF), GM and WM. Figs. 4(b) and 4(c) show the GM±WM boundary as predicted by the FCM and AFCM algorithms. The FCM shows a considerable amount of noise on the right side, which is darker than the left side of the image. The AFCM contour, on the other hand, is significantly cleaner and preserves the complex structure along the cortex more accurately. In this example, the value of λ 1 was set to and λ 2 to

37 In Fig. 5, we show the results of applying AFCM to an MR image that has had an artificial multiplier field applied, governed by the equation: m t (x,y) = (a x+1) 2. where x and y represent the row and column, the origin is at the center of the image, and a is an inhomogeneity parameter. Thus, m t is greater than one near the bottom of the image, is equal to one along the row through the center, and less than one at the top. As a increases, the brightness variation also increases. The image was again segmented into three classes corresponding to CSF, GM and WM using both FCM and AFCM. Fig. 5(a) shows an MR brain image after applying a multiplier field with a=1/512. Figs. 5(b)- (d) shows the GM, WM and CSF fuzzy membership functions computed using the standard FCM algorithm. Once again, poor results are obtained because the FCM algorithm assumes that the centroids are constant throughout the image. In Fig. 5(f)-(h) we show the GM, WM and CSF membership functions computed from the same test image using the AFCM algorithm. A significant improvement over the standard FCM algorithm is achieved. Fig. 5(e) shows the estimated multiplier field from the AFCM algorithm. As expected, the multiplier field takes on the same characteristic shading as the corrupted image. 37

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39 Fig. 6 shows a plot of the performance of the FCM and AFCM algorithms when the inhomogeneity parameter a is varied. The mean squared error (MSE) was computed on the GM membership functions. The FCM computed GM membership function at a= 0 was considered to be the ground truth. As the inhomogeneity increases, the FCM result quickly degrades, while the AFCM result maintains a consistent level of performance. AFCM, when applied to images with intensity inhomogeneities, essentially performs as well as FCM on images with no inhomogeneities. 39

40 Pham, discussion Results based on the AFCM algorithm show great promise. Further validation studies on ground truth data are necessary, however, to better evaluate its performance. A weakness of AFCM (and FCM) is that in the presence of extreme noise, it may perform poorly compared to methods which directly enforce spatial smoothness on the segmentation. In such cases, a prefiltering step may be necessary to smooth the original image. Alternatively, placing additional smoothness constraints on the membership functions may also improve robustness to noise, although at the expense of greater computational complexity. MR images typically have high signal-to-noise ratio, however, and smoothing is often unnecessary to obtain good results. The possibility exists for using AFCM to study the nature of the inhomogeneities in MR imaging. A version for three-dimensional image volumes is currently being implemented in C. This is a straightforward but important extension especially for MR images, where the intensity variations are threedimensional in nature. 40

41 Several further extensions are also possible. AFCM can be generalized to multispectral data, which is common in MR imaging. In this case, the pixel intensities and centroids are assumed to be vectors instead of scalars. Methods for automatically selecting the regularization parameters also require further investigation. Current evidence suggests, however, that the parameters are fairly robust to different images of similar size and inhomogeneity effects. A theoretical analysis of the convergence of the algorithm should also be undertaken. Finally, a number of methods exist for estimating the number of classes based on the data. Additional research is necessary to investigate how these methods might be used in conjunction with AFCM. For MR imaging, this ability would be useful in the automatic detection of pathology or abnormal anatomy. 41