A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> Image reconstruction from projections can be viewed as a linear inverse problem with discrete data if the number of projections is small. The input data corresponds to integral equations in the mathematical model for numerical reconstruction. As this problem does not have unique solution, these equations may be viewed as the equality constraints for maximization of a functional such as entropy. Applying the dual formulations in optimization of entropy, one finds that they correspond to the well known algorithms MART and MENT respectively. This equivalence applies only if the natural basis is adopted in MART. For MENT, in this case, the non-discrete formulation is identical to the discrete dual one. <s> BIB001 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> At its simplest, voxel-based morphometry (VBM) involves a voxel-wise comparison of the local concentration of gray matter between two groups of subjects. The procedure is relatively straightforward and involves spatially normalizing high-resolution images from all the subjects in the study into the same stereotactic space. This is followed by segmenting the gray matter from the spatially normalized images and smoothing the gray-matter segments. Voxel-wise parametric statistical tests which compare the smoothed gray-matter images from the two groups are performed. Corrections for multiple comparisons are made using the theory of Gaussian random fields. This paper describes the steps involved in VBM, with particular emphasis on segmenting gray matter from MR images with nonuniformity artifact. We provide evaluations of the assumptions that underpin the method, including the accuracy of the segmentation and the assumptions made about the statistical distribution of the data. <s> BIB002 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> Outlines a fully automatic method for the correction of intensity nonuniformity in MR images. This method does not require any a priori model of the tissue classes. The basic idea is that entropy is a good measure of the image quality which can be minimized in order to overcome the bias problem. Therefore, the optimal correcting field is defined by the minimum of a functional combining the restored image entropy and a measure of the field smoothness. This measure stems from the usual physical analogy with membranes. A third term added to the functional prevents the optimal field from being uniformly null. The functional is minimized using a fast annealing schedule. The performance of the method is evaluated using both real and simulated data. <s> BIB003 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> Shading is a prominent phenomenon in microscopy, manifesting itself via spurious intensity variations not present in the original scene. The elimination of shading effects is frequently necessary for subsequent image processing tasks, especially if quantitative analysis is the final goal. While most of the shading effects may be minimized by setting up the image acquisition conditions carefully and capturing additional calibration images, object-dependent shading calls for retrospective correction. In this paper a novel method for retrospective shading correction is proposed. Firstly, the image formation process and the corresponding shading effects are described by a linear image formation model, which consists of an additive and a multiplicative parametric component. Secondly, shading correction is performed by the inverse of the image formation model, whose shading components are estimated retrospectively by minimizing the entropy of the acquired images. A number of tests, performed on artificial and real microscopical images, show that this approach is efficient for a variety of differently structured images and as such may have applications in and beyond the field of microscopical imaging. <s> BIB004 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> In this paper the problem of retrospective correction of intensity inhomogeneity in 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. <s> BIB005 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> Abstract The desire to correct intensity nonuniformity in magnetic resonance images has led to the proliferation of nonuniformity-correction (NUC) algorithms with different theoretical underpinnings. In order to provide end users with a rational basis for selecting a given algorithm for a specific neuroscientific application, we evaluated the performance of six NUC algorithms. We used simulated and real MRI data volumes, including six repeat scans of the same subject, in order to rank the accuracy, precision, and stability of the nonuniformity corrections. We also compared algorithms using data volumes from different subjects and different (1.5T and 3.0T) MRI scanners in order to relate differences in algorithmic performance to intersubject variability and/or differences in scanner performance. In phantom studies, the correlation of the extracted with the applied nonuniformity was highest in the transaxial (left-to-right) direction and lowest in the axial (top-to-bottom) direction. Two of the six algorithms demonstrated a high degree of stability, as measured by the iterative application of the algorithm to its corrected output. While none of the algorithms performed ideally under all circumstances, locally adaptive methods generally outperformed nonadaptive methods. <s> BIB006 </s> A Review on MR Image Intensity Inhomogeneity Correction <s> Entropy minimization <s> An image restoration approach based on a Bayesian maximum entropy method (MEM) has been applied to a radiological image deconvolution problem, that of reduction of geometric blurring in magnification mammography. The aim of the work is to demonstrate an improvement in image spatial resolution in realistic noisy radiological images with no associated penalty in terms of reduction in the signal-to-noise ratio perceived by the observer. Images of the TORMAM mammographic image quality phantom were recorded using the standard magnification settings of 1.8 magnification/fine focus and also at 1.8 magnification/broad focus and 3.0 magnification/fine focus; the latter two arrangements would normally give rise to unacceptable geometric blurring. Measured point-spread functions were used in conjunction with the MEM image processing to de-blur these images. The results are presented as comparative images of phantom test features and as observer scores for the raw and processed images. Visualization of high resolution features and the total image scores for the test phantom were improved by the application of the MEM processing. It is argued that this successful demonstration of image de-blurring in noisy radiological images offers the possibility of weakening the link between focal spot size and geometric blurring in radiology, thus opening up new approaches to system optimization. <s> BIB007
As a frequently used criterion to characterize the intensity distribution of an image, entropy has been employed to design algorithms for image restoration, thresholding, or classification BIB007 BIB001 . Also, it has been utilized to quantify the image property with IIH present and guide the parameter searching for IIH removal BIB002 BIB003 BIB004 BIB005 . It is assumed that the intensity distribution of the original image is multimodal, and the presence of IIH causes the intensity overlapping between objects. Figure 3 shows the histograms of brain tissue in a BrainWeb-simulated image (Figure 3(a) ) and the head image in Figure 1 (Figure 3(b) ). On Figure 3(a) , the solid line is the histogram without inhomogeneity, where the modes corresponding to different tissues are very distinctive. With the presence of IIH (dashed-dotted line), the valleys between different modes are markedly flattened. For the head image with severe IIH, the histogram (Figure 3(b) ) is so flat that the modes corresponding to the gray matter and the white matter are difficult to distinguish. The flattening of the histogram leads to the increase in the entropy of the image, therefore, the IIH correction can be achieved through searching the parameter space of the IIH model such that the entropy of the image is reduced. It should be pointed out that direct minimization on the entropy would lead to the null field BIB003 BIB006 . To avoid this pitfall, constraints over the solution space are necessary. Mangin BIB003 constrained the solution to minimize the distance between the mean values of the restored and the original image. In BIB005 , the restored image was constrained to have the same mean value as the original one. Evidently, other quantities relevant to image features, variance for example, can also be applicable in a similar fashion. Again, constraints upon the solution space are necessary.