Abstract:
A fuzzy logic approach to the problem of distinguishing cerebrospinal fluid, gray and white matter pixels in magnetic resonance images of the brain. An unsupervised fuzzy clustering procedure based on a variant of the fuzzy c-means algorithm computes automatically, with virtually no operator intervention, the percentage area of each of these three compartments in each image. Each volume element represented in the image can belong to all compartments in varying degrees. The procedure requires input of the number of different compartments in the image, as well as a parameter which determines the amount of overlap of compartment boundaries. Preliminary data processing involves noise removal from images by highpass filtering. The final solution is substantially independent of required initial estimates of the gray scale values that are most representative of the white, gray and cerebrospinal fluid compartments in the image. The method is useful in the diagnosis of hydrocephalus.

Description:
The government has certain rights in the invention. A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to methods and apparatus for estimating relative volumes of tissue types distinguishable in magnetic resonance (MR) images of the tissues. 
     2. Distinguishing Tissue Types in Brain Images 
     In vivo estimates of white matter, gray matter and cerebrospinal fluid (CSF) volumes are useful in diagnosis and treatment of several conditions affecting the brain, including Alzheimer&#39;s disease, Down&#39;s syndrome, senile dementia, and hydrocephalus. In the latter condition, communicating and non-communicating hydrocephalus can be distinguished by estimating CSF volume in images which include the brain ventricles. In other cases, the need for surgical intervention in hydrocephalic patients to establish or repair a CSF drainage shunt can be assessed with serial estimates of CSF contained within the brain ventricles. Diagnosis may also be aided because it has been observed that the ratio of white matter to gray matter in hydrocephalic children is much lower than in control (normal) children. 
     Computerized tomography (CT) scans and radionuclide ventriculography have been used to estimate ventricular volumes in the brain, but estimate of the error associated with single measurements using either of these techniques are in the range 20-30%. Additionally, reliable in vivo estimates of extraventricular cranial CSF volumes can not be obtained with CT or radionuclide techniques. Reasons for this include the inherently poor resolution of radionuclide techniques and, in the case of CT, the relatively poor contrast of CSF with other brain tissue in CT images. 
     Superior brain images, on the other hand, are obtained with MR because gray and white matter and CSF are usually visually differentiable in MR images. MR imaging is the first technique to allow in vivo investigations of gross brain structural variation with sufficient resolution to be clinically useful. MR images, however, require careful interpretation due to possible ambiguities in interpretation of the gray scale value of a particular image point or element (pixel). 
     Ambiguities may arise because a single gray scale pixel value from an MR image of the brain can represent mixed tissue types (e.g., gray matter and CSF). This occurs because of partial volume averaging, in which a given pixel value represents a volume element (voxel) containing more than one tissue type. For example, a given pixel in the image could be classified as 0.5 (50%) gray and 0.5 (50%) white, or 0.4 fluid and 0.6 gray. 
     The most common techniques to solve this volume averaging problem use some variation of a classification procedure which tries to estimate or find a boundary in the image to allow separation of tissue compartments. Another (rule-based) approach to classifying tissue types requires the choice of two empirical thresholds which may change from subject to subject. Pixel classification in this latter system, therefore, requires a specific model of the data which a particular patient may not fit optimally. Attempts to overcome these problems through use of statistical models and statistical pattern recognition approaches may succeed in avoiding reliance on specific models for the data, but their validity rests on assumptions regarding data distribution which cannot be confirmed in the case of particular patients. Additionally such schemes generally require clinical or operator judgment and this cannot be readily automated. 
     SUMMARY OF THE INVENTION 
     The present invention substantially avoids requirements for patient-specific assumptions and operator judgment in interpreting MR brain images. It allows accurate volumetric estimation of white matter, gray matter and CSF in suitable MR images through use of fuzzy logic techniques to classify image elements (pixels) by their gray scale values. In contrast to prior methods for MR image interpretation, the presently claimed fuzzy classification process for MR image pixels is fully automatic and executes very quickly on a modern personal computer or work station. The classification scheme does not require determination of a boundary in the image to separate tissue compartments, but instead recognizes that individual image pixels may represent two or more distinct tissue types at the same time. 
     Fuzzy Classification 
     The principle behind fuzzy classification is very simple. A good example of the need for such a system would be the concept of &#34;age&#34; as applied to a particular person. When is someone to be considered &#34;young&#34; or &#34;old?&#34; A classification scheme having a cutoff age above which a person would be considered (classified) as &#34;old&#34; would clearly have limited usefulness if applied uniformly to individuals. It is apparent that there is a continuum of ages through which a person passes in changing from &#34;young&#34; to &#34;old.&#34; At age 80, a person is clearly old, and at age 20 clearly young, but at an intermediate age (perhaps 50), a person might be considered as being &#34;partly old&#34; and &#34;partly young.&#34; Fuzzy logic, as applied in the present invention, recognizes the condition wherein a pixel in an MR image may represent elements from more than one group (e.g., white matter and gray matter). By considering the (possibly mixed) classification(s) of each pixel, methods of the present invention can produce accurate estimates of the total amount of tissue of each type comprising the image. In such applications, the present invention has several advantages over prior methods. 
     First, methods of the present invention are relatively fast, requiring about two minutes or less to analyze a single brain MR image to yield a decision on the percent of gray matter, white matter and CSF represented in the image. Second, the methods are substantially automatic, requiring no choice of tissue boundaries in the image and little human operator intervention. Third, the methods solve the volume averaging problem by allowing for mixed tissue representation in a single voxel. Fourth, the methods do not assume any a priori statistical or heuristic model of the data; a model for each image analyzed is estimated from the gray scale values for pixels in that image. Fifth, use of two or more different spectral channels of the same MR image (i.e., T1 and T2 weighted images and proton density images) provides additional information over that available from a single channel image, allowing more accurate discrimination of tissue types. Sixth, spatial-location cues are not required. There is no need, when using methods of the present invention, to know where pixels are located relative to anatomic landmarks; the methods are designed to recognize differences in image intensity alone, without regard to image shape. 
     Preferred Embodiments 
     The present invention includes a method of estimating tissue volumes in a pixelized magnetic resonance image, the method comprising: providing magnetic resonance images including pixel gray scale values; selecting a current centroid of a cluster representing each tissue; computing a Euclidean distance from each pixel vector to each current centroid; computing a membership value for each pixel vector in each cluster; computing a new centroid for each cluster; returning to the step for computing Euclidean distance after substituting the new centroid for the current centroid for each cluster if testing reveals insufficient convergence due to separation of the new and current centroids for each cluster greater than a desired threshold (preferably about 0.1); and computing tissue volumes based on pixel vector membership values. 
     The present invention also comprehends non-iterative techniques. For example, a method of estimating tissue volumes in a magnetic resonance image, the method comprising: providing a magnetic resonance image including pixel gray scale values, the values being obtained from proton density (PD) and T2-weighted magnetic resonance images and, in some embodiments, from a T1-weighted magnetic resonance image; selecting a current centroid of a cluster representing each tissue; computing a Euclidean distance from each pixel vector to each current centroid; computing a membership value for each pixel vector in each cluster; and estimating tissue volumes based on pixel vector membership values by summing up and averaging membership values to compute tissue volumes. 
     The present invention also includes a method of operating a machine to estimate tissue volumes in a magnetic resonance image, the machine comprising a processor operatively connected to a memory, the method comprising: selecting a current centroid of a cluster representing each tissue; storing the current centroid of each cluster in the memory; storing a magnetic resonance image including pixel gray scale values in the memory; computing with the processor a Euclidean distance from each pixel vector to each current centroid; computing with the processor a membership value for each pixel vector in each cluster; computing with the processor a new centroid for each cluster; returning to the step for computing Euclidean distance after substituting in the memory the new centroid for the current centroid for each cluster if testing with the processor reveals insufficient convergence due to separation of the new and current centroids for each cluster greater than a desired threshold (preferably about 0.1); and estimating with the processor tissue volumes based on pixel membership values. 
     The present invention also includes a program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to perform the method steps of the invention. 
     A further embodiment of the invention is a method of estimating a volume of brain tissues in a patient, comprising: providing a magnetic resonance image of a patient&#39;s brain tissues; estimating a volume of gray matter in the image using a fuzzy clustering process; estimating a volume of white matter in the image using a fuzzy clustering process; and estimating a volume of cerebrospinal fluid in the image using a fuzzy clustering process. 
     Clinical applications of the present invention include a method of diagnosing hydrocephalus in a patient, comprising: providing a magnetic resonance image of a patient&#39;s brain tissues; estimating a volume of cerebrospinal fluid in the image using a fuzzy clustering process; and diagnosing hydrocephalus if estimated cerebrospinal fluid volume is greater than a predetermined percentage of about ten percent, the actual value depending on the location of the image plane in the brain. 
     Another clinical application of the present invention is a method of diagnosing hydrocephalus in a patient, the method comprising: providing magnetic resonance images of a patient&#39;s brain tissues including pixel gray scale values; selecting a current centroid of a cluster representing gray matter; selecting a current centroid of a cluster representing white matter; selecting a current centroid of a cluster representing cerebrospinal fluid; computing a Euclidean distance from each pixel vector to each current centroid; computing a membership value for each pixel vector in each cluster; computing a new centroid for each cluster; testing for separation of new and current centroids for each cluster; substituting the new centroid for the current centroid for each cluster; returning to the step for computing Euclidean distance if testing reveals separation greater than a desired error; estimating tissue volumes based on pixel vector membership values; and diagnosing hydrocephalus if cerebrospinal fluid volume is greater than a predetermined percentage of about ten percent, the actual value depending on the location of the image plane in the brain. 
     Another embodiment of the present invention is apparatus for estimating tissue volumes in magnetic resonance images, the apparatus comprising: a magnetic resonance unit for producing pixelized magnetic resonance images; a memory for storing the pixelized images; program storage for central processing unit programs; a central processor unit to perform the steps of: selecting a current centroid of a cluster representing each tissue; storing the current centroid of each cluster in the memory; storing a magnetic resonance image including pixel gray scale values in the memory; computing with the processor a Euclidean distance from each pixel vector to each current centroid; computing with the processor a membership value for each pixel vector in each cluster; computing with the processor a new centroid for each cluster; testing with the processor for separation of new and current centroids for each cluster; substituting in the memory the new centroid for the current centroid for each cluster; returning to the step for computing Euclidean distance if testing reveals separation greater than a desired error; and estimating with the processor tissue volumes based on pixel vector membership values; a display to present gray scale data representing pixelized magnetic resonance images to an operator; and an interface to transfer pixelized magnetic resonance image data from a computation bus to a display and to transfer operator instructions to a central processor unit via a computation bus. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of hardware components comprising an apparatus for estimating tissue volumes in MR images according to the present invention. 
     FIG. 2 is a flow chart of the operation of the apparatus of FIG. 1 showing data preprocessing followed by a fuzzy clustering of tissue types in MR images in accordance with the present invention. 
     FIG. 3 is a flow chart showing detailed steps in the fuzzy clustering process realized when the apparatus of FIG. 1, operated according to the steps of FIG. 2, is applied to preprocessed MR image data. 
     FIG. 4 illustrates the membership percentage (vertical axis) for two hypothetical gray scale pixel clusters. 
     FIG. 5A displays an example of the threshold that would be selected by the operator for a typical PD-weighted axial MR image. 
     FIG. 5B displays an example of the threshold that would be selected by the operator for a typical T2-weighted axial MR image. 
     FIG. 6A shows a photograph of an example of a filtered T2-weighted image. 
     FIG. 6B shows a photograph of an example of the mask produced from the filtered T2-weighted image of FIG. 6A. 
     FIG. 6C shows a photograph of an example of a solution image showing the final classification of each pixel in the T2-weighted image of FIG. 6A and its corresponding PD-weighted image (not shown), calculated using the mask of FIG. 6B. 
    
    
     DETAILED DESCRIPTION 
     Preprocessing of Pixelized MR Image Data 
     Preferred embodiments of the present invention comprise applications of data preprocessing steps 17 and 18 (see FIG. 2 and computer code printout below), followed by fuzzy clustering 19 of the remaining data. Referring to the apparatus illustrated in FIG. 1, preprocessing is applied to data from pixelized MR images which are produced in MR unit 74 (e.g., General Electric Signa Magnetic Resonance Imaging System) and stored in the memory 70 (which may comprise, e.g., magnetic memory media and volatile or non-volatile semiconductor memory). MR unit 74 comprises MR image section 78 which produces MR images, and MR interface section 76 which receives MR image data over MR internal bus 75 from MR image section 78. MR interface section 76 produces and archives to memory (with a file directory) pixelized MR images in a format usable in methods of the present invention. 
     Working through the interface 64 (e.g., a keyboard), a human operator can observe data transmitted to the display 60 (e.g., an SVGA monitor) over display path 62 and visually determine a threshold, which effectively determines the characteristic of a highpass filter through which pixelized MR image data are passed. Application of the filter will eliminate substantially all of the noise in the MR image data. 
     Alternatively, this process can be performed by the central processing unit (CPU) 68 (e.g., an 80486, 33 MHz computer) using a thresholding program drawn from memory 70 over computation bus 66, the program having been previously loaded into memory 70 from program storage media 72. 
     Program storage media 72 may be, for example, magnetic media including tape, disc or drum, semiconductor media such as semiconductor read only memory, optoelectronic media such as a CD ROM, mechanical storage media such as punched cards, or any other information storage media which is capable of storing a program of instructions that may be executed by the apparatus of FIG. 1 to perform the methods of the present invention. 
     As illustrated in FIG. 2, preprocessing also comprises thresholding 17 (see computer code subroutine Hi --  Threshold --  Image() ) and treating a brain mask 18 (see computer code subroutine Fill --  Brain() ) from a T2 weighted image to simplify application of the fuzzy clustering process 19 (see computer code subroutine Cluster() ). Methods of the present invention include the above preprocessing steps because of the requirement for a high signal-to-noise (S/N) ratio in digital MR image data applied to the fuzzy clustering process. 
     Thresholding is performed interactively by the operator as the gray scale histogram of each slice image is viewed individually. The operator selects a threshold at a point in the histogram just before the beginning of the cerebral density distribution of the data. FIGS. 5A and 5B display examples of the thresholds that would be selected by the operator for a typical PD-weighted (5A) and T2-weighted (5B) axial MR image. Preprocessing is applied individually to the PD and T2-weighted MR images and to any other image (spectral channel) available for each tissue slice to be considered (e.g., a T1-weighted image in certain embodiments). 
     Creating a Brain Mask 
     Following highpass thresholding of the T2-weighted axial image, the cerebrum will have been isolated from skull, meninges, and scalp. A brain mask is then created by using a polygon filling subroutine (subroutine Fill --  Brain() ) to isolate all nonzero pixels within the cerebrum itself (pixels of zero gray scale are not filled). The filling algorithm is a region growing procedure that starts from the middle of the image and scans outward in all directions to find all nonzero pixels. FIGS. 6A, 6B and 6C show photographs of examples of a filtered T2-weighted image (6A), the mask produced from it (6B) and the solution image (6C) showing the final classification of each pixel. The brain mask is used to identify all pixel locations to be included in the actual fuzzy clustering procedure. The maximum number of pixels included in the procedure is generally about one-third of the total number present (i.e., about 20,000). 
     The Fuzzy Clustering Process--Choosing Initial Current Centroids 
     The fuzzy clustering process for the present invention (FIG. 3 and step 19 in FIG. 2) requires an initial guess as to the gray scale values that characterize the white, gray and CSF compartments in the image. These guesses represent the choice of current (initial) centroid vector 30 or the most representative gray scale Value for the pixels of each cluster, each cluster representing one of the compartments of the image (e.g., a tissue type). The final solution has been observed in repeated tests to be independent of the choice of initial current centroid values, but initial values substantially different from final centroid values tend to increase the computation time required. Thus, the following strategy is used in selecting initial centroid vectors automatically. 
     For the PD-weighted image, the histogram is computed and its range is quantified (high end minus low end); the range is divided into three substantially equal parts. The approximate center of the lowest third portion of the histogram is designated as the CSF centroid value. The approximate center of the middle third portion is designated as the white matter centroid value, and the approximate center of the highest third portion is deemed the gray matter centroid value. A similar procedure is carried out for the histogram of the corresponding T2-weighted image (the range being divided into three substantially equal parts). The approximate center of the lowest third of the histogram is assigned to white matter, the approximate center of the middle third is designated as gray matter and the approximate center of the highest third is designated as CSF. Centroid vectors are them constructed as the PD and T2 paired values of CSF, white and gray matter as chosen. The portions of each histogram chosen to represent the initial guesses for CSF, white and gray matter are based on empirical observations of many PD and T2-weighted brain images by a trained neurologist and neuropsychologist. 
     The cluster centroids are thus 2-dimensional vectors where the dimensions are the PD and T2-weighted pixel values selected as above. The term &#34;pixel vector&#34; is used to describe a PD and T2 pixel pair at a given spatial location in both PD and T2-weighted images of the same brain slice. Distances between pixel vectors and cluster centroid vectors are thus computed in a 2-dimensional Euclidean space (or in a 3-dimensional space if T1-weighted images are also used). 
     Having initially chosen current centroids for the pixel clusters, one must store 32 in some (preferably digital) memory all pixel values to be used in the clustering process (see FIG. 3). Preferred memory includes magnetic discs or tape, optoelectronic or electronic memory. Pixel values needed temporarily for computations may also be stored in random access electronic or optoelectronic storage means linked to a processor. 
     Using initially chosen current centroid vectors for clusters representing tissue types under consideration, the Euclidean distance of each image pixel vector to each centroid vector is computed 34, and a value representing the membership of each pixel vector in each cluster is also computed 36. The fuzzy clustering procedure thus implemented in the present invention (see FIGS. 2 and 3) is a modified version of an infinite variety of such processes with the general name &#34;fuzzy c-means algorithm&#34; (Bezdek, James, Pattern Recognition with Fuzzy Objective Function Algorithms, New York, Plenum Press (1981); and Pao, Yoh-Han, Adaptive Pattern Recognition and Neural Networks, Addison-Wesley (1989) ). 
     More specifically, the fuzzy clustering procedure of the present invention (FIG. 3) comprises the following steps. After choice of initial centroids 30 (see computer code subroutine Cluster() ) and storage of pixel values 32 (see computer code subroutine get --  hist() ), the procedure iterates: 
     1. For each image pixel vector (consisting of PD and T2-weighted gray scale pairs), the squared Euclidean distance 34 to each Cluster centroid vector is computed as 
     
         μx.sub.k -ν.sub.i μ.sup.2 
    
     where x k  refers to each of the n pixel vectors and ν i  refers to each of the N c  cluster centroid vectors. 
     2. Using the previous calculation for distance, the fractional membership 36 of pixel vector k in cluster i is computed as: ##EQU1## where m is referred to as the &#34;fuzzification parameter&#34; (1&lt;m&lt;∞). This parameter essentially controls the amount of cluster overlap. The value of m for the present invention has been determined to be about 4/3. 
     3. The cluster centroid vectors are recalculated 38 using the membership values computed in step 2. The centroid calculation is: ##EQU2## 
     4. Compute the absolute difference 40 (Euclidean distance) between cluster controids calculated in the present iteration and those calculated in the previous iteration. If this distance exceeds a value 0&lt;δ&lt;0.5, preferably δ=0.1, for any cluster, replace the current centroid with the centroid recalculated above 42, then return to step 1 and repeat steps 1-4. Otherwise, convergence in the least mean squares sense has been achieved and the procedure is therefore complete. Finally, the percent volume of the tissue type represented by each cluster is computed 44 as: ##EQU3## 
     Note that this procedure utilizes a different convergence criterion than the standard fuzzy c-means algorithm. In the latter, the convergence rule is: 
     
         if max(|μ.sub.ik,α -μ.sub.ik,α-1 |)&lt;δ then stop. 
    
     where μ ik ,α is the membership of pixel vector k in cluster i for current iteration α. This means that the previous iteration&#39;s membership matrix (which has dimensionality N c  by n) must be retained (stored in memory) during the current iteration, and the N c  by n absolute differences between the current iteration&#39;s membership matrix and the previous one must be computed and the maximum difference determined. Note also that steps 34, 36, 38, 40, 42 and 44 above are represented in the computer code subroutines mem --  sqr() and fuzzcmns() which are listed below. 
     In the present invention, the N c  absolute differences are only calculated between the current iteration&#39;s cluster centroid vectors and the previous iteration&#39;s centroid vectors; the maximum difference is then found. This represents a large saving in memory space and computation time over the usual fuzzy c-means algorithm. 
     Use of the centroid difference approach over the membership difference approach is justified by noting that they are procedurally interchangeable. That is, stability of membership values is reflected in stable positioning of the cluster centroids. 
     For the clinical tests described herein, the value of δ=0.1 for the distance threshold was chosen after many empirical tests of the procedure on normal and hydrocephalic children. This value represents 1/10 of one gray scale unit. If it is assumed that there are at most about 10,000 to 12,000 pixel vectors in a given cluster, then the average error per pixel vector would be on the order of 0.1/10,000 (0.00001) to 0.1/12,000 (0.0000083). This allows a highly accurate determination of cluster centroid locations and membership values. Experimental results have shown that a threshold of about δ=0.1 is an acceptably small error, allowing detection of clinically meaningful differences between normals and hydrocephalics. 
     Following the percentage computation of CSF, white and gray matter, a solution image is constructed in order to provide some verification of the accuracy of the procedure. The final centroid vectors are deemed the most representative prototypical values for each tissue type. The T2 values of these vectors are used to construct the solution image by multiplying the membership values of each tissue type by the T2 gray scale values for each centroid and then summing: 
     
         new gray scale value=A+B+C 
    
     where 
     A=mem{csf} * centroid{csf} 
     B=mem{white} * centroid{white} 
     C=mem{gray} * centroid{gray} 
     and 
     the mem{ } values are the membership percentages computed for each T2-weighted gray scale value in the original image, and the centroid { } values are the final cluster centroid positions in the. T2 space. Finally, this &#34;defuzzified&#34; solution image is displayed simultaneously with the PD, T2 and mask images for visual comparison by a trained neurologist on neuroradiologist. 
     Determining the Fuzzification Parameter 
     Use of the process requires a choice for a parameter &#34;m,&#34; determining the amount of fuzziness of the classification of each pixel. Theoretically, one may choose 1&lt;m&lt;∞, but in analyzing MR images, 1&lt;m≦2 has been found preferable, with a value of m about 4/3 being most preferred. 
     The estimated tissue volumes finally computed are dependent on the chosen value of m. There are no guidelines in the literature for selecting the proper value in the present application, although a value for m of 1.3 has been used in other applications. The optimal value appears to depend to a certain extent on the problem domain (e.g., MR image processing) and the data. There are, however, some important guidelines for determining the optimal m for any given application. The idea is represented by the diagram of FIG. 4, illustrating the membership percentage (vertical axis) for two hypothetical gray scale pixel clusters, c1 and c2. 
     In FIG. 4, the horizontal axis represents increasing gray scale values with the centroid value for c1=105 and for c2=210. The parameter m should be selected in such a way that gray scale values to the left of c1 should have 100% membership with the c1 cluster, and 100% membership to the right of c1 up to a point where the membership values begin to decrease. Eventually the membership values for c1 will decrease more or less smoothly to 0%. For c2, gray scale values to the right of 210 should belong 100% to the c2 cluster, with a similar falloff in values to the left of c2. At a point approximately halfway between the c1 and c2 values (about 157), the membership for a given gray scale value should be approximately 50--50. 
     With this objective in mind, numerous tests over many images were performed by trying out different values for m and examining the membership values for each gray scale value in each image. The value 4/3 for m yielded the desired result in every image tested. Slight variations in this value (about 1.2 to 1.4) were observed to make minimal difference in the final estimates of tissue volumes. Further, once chosen, this parameter does not have to be selected or changed by the operator. We finally set the value of m to 4/3, which then yields considerable savings in computation time (see equations in steps 2 and 3 above). 
     Convergence of the Fuzzy Clustering Process 
     In tests on actual MR images, the fuzzy clustering process (FIG. 3) always converged to a solution, but the solution was sometimes incorrect if the centroid values for two different tissue types were too close in value. This occurred when one of the three tissue types considered (the fluid compartment) was present as a very small percentage of the total tissue. The process was then unable to distinguish the fluid compartment and split the gray matter cluster into two separate clusters. This error is eliminated by modifying the thresholding step to eliminate any one of the tissue types, followed by subtracting the thresholded image from the original. The remaining image, which contains only the remaining two tissue types, may then be subjected to the procedure and the two-cluster solution found. 
     Clinical Tests of the Process 
     An MR image analysis comparing three clinically normal control children with three age matched hydrocephalic children reveals the clinical utility of the present invention. The results are summarized in Table I (all listed results use m=1.3, but a value of m=4/3 is preferred in embodiments requiring substantial calculational efficiency). In the hydrocephalic group, one subject was born prematurely and was diagnosed as hydrocephalic, and the remaining two were diagnosed as having spina bifida with hydrocephalus. For each subject, three contiguous axial image slices were obtained, proceeding superiorly from a slice through both the genu of the corpus callosum and the lateral ventricles, and including the internal capsule, caudate nucleus and putamen comparison of a total of 36 images for the two groups (6 subjects×3 image slices per subject×2 channels [PD and T2]), with volumetric results averaged across the three slices for each subject, is summarized in Table I. 
     
                       TABLE I______________________________________PRELIMINARY RESULTS SYNOPSIS(All subjects 7 years of age)Subject  % CSF    % White    % Gray White/Gray______________________________________Controls1      5.3      45.0       49.7   0.912      2.4      47.4       50.2   0.943      4.5      49.4       46.1   1.07Mean   4.1      47.3       48.7   0.97Hydrocephalus1 (PH) 14.5     29.0       56.5   0.512 (SH) 15.4     35.3       49.3   0.723 (SH) 17.6     35.5       46.6   0.81Mean   15.8     33.3       50.8   0.68______________________________________ 
    
     Overall, the results indicate the expected finding that the ratio of white matter to gray matter in hydrocephalic children is much lower than that of controls. The results also show the increased amount of brain fluid characteristic of hydrocephalus. But more important, the procedure is able to demonstrate that hydrocephalus severely affects the amount of white matter while sparing the gray matter in general. Specifically, hydrocephalus subject number one had asymmetrical and dilated ventricles and, as expected, a more severe white matter decrease as compared to the other two subjects. Hydrocephalus subject number two had symmetrical, non-dilated ventricles, and thus, less of a decrease in white matter. In summary, the procedure is able to clearly identify gross changes in white, gray and fluid compartments in hydrocephalus. ##SPC1##