Method and system for fuzzy clustering of images

Abstract

An approach to clustering a set of images based on similarity measures employs a fuzzy clustering paradigm in which each image is represented by a node in a graph. The graph is ultimately partitioned into subgraphs, each of which represent true clusters among which the various images are distributed. The partitioning is performed in a series of stages by identifying one true cluster at each stage, and removing the nodes belonging to each identified true cluster from further consideration so that the remaining, unclustered nodes may then be grouped. At the beginning of each such stage, the nodes that remain to be clustered are treated as all belonging to a single candidate cluster. Nodes are removed from this single candidate cluster in accordance with similarity and connectivity criteria, to arrive at a true cluster. The member nodes of this true cluster are then removed from further consideration, prior to the next stage in the process.

Claims

1. An apparatus for clustering a set of N images into P final clusters, where N and P are integers, comprising: means for calculating at least one similarity measure S ij between members of each possible pair of images, wherein S ij represents the similarity measure between the ith and jth images with i and j being image indices and S i,j =S j,i ; means for calculating a total connectivity value for each of the images remaining to be clustered, a total connectivity value for each image being defined as a sum of a function f of the similarity measures associated with that image; means for identifying, from among said images remaining to be clustered, a maximum total connectivity value t max corresponding to an image I max , image I max belonging to a current cluster C which initially includes all images remaining to be clustered; means for removing, from the current cluster C, at least one image based on at least one of its similarity measure with image I max and its total connectivity value within current cluster C; means for adding, to current cluster C, images having a similarity measure that is grater than a threshold T 3 with any image currently in C; means for calculating, for each image within current cluster C, a total connectivity value based on those images within C; means for removing, from current cluster C, those images having a total connectivity value less than some threshold T 4 , wherein said total connectivity value calculating step and said image removing step are repeated until no further images are removed to thereby establish current cluster C as one of the final clusters; and means for removing all images in current cluster C from further consideration, wherein one or more of said calculating steps, said identifying step, said removing steps, and said adding step are repeating until all N images are assigned to a final cluster. 2. The apparatus of claim 1 , further comprising: means for applying a non-linear function to each similarity measure, prior to calculating the total connectivity value. 3. The apparatus of claim 2 , wherein the non-linear function is a transcendental function. 4. The apparatus of claim 3 , wherein the transcendental function is a hyperbolic tangent. 5. The apparatus of claim 4 , wherein each similarity measure S i,j is transformed according to the formula tanh(5×S i,j ) prior to calculating a total connectivity value. 6. The apparatus of claim 1 , wherein the function f is the identify function. 7. The apparatus of claim 1 , wherein the function f is a nonlinear function. 8. The apparatus of claim 1 , wherein said at least one image removing means removes at least one image from the current cluster based on its similarity measure with image I max , and then at least one other image is removed from the current cluster based on its total connectivity value within the current cluster. 9. The apparatus of claim 1 , wherein said at least one image removing means removes at least one image from the current cluster based on its total connectivity value within the current cluster, and then at least one other image is removed from the current cluster based on its similarity measure with image I max . 10. The apparatus of claim 1 , wherein said at least one image removing means removes images having a similarity measure with I max that is less than some threshold T 1 from the current cluster. 11. The apparatus of claim 1 , wherein said at least one image removing means removes a number N 1 images having the N 1 lowest similarity measures with I max from the current cluster. 12. The apparatus of claim 1 , wherein said at least one image removing means removes images having a total connectivity value which differs from t max by more than some threshold T 2 from the current cluster. 13. The apparatus of claim 12 , wherein images have a similarity measure with I max that is less than some threshold T 1 are removed from the current cluster. 14. The apparatus of claim 13 , wherein at least one image is removed from the current cluster based on its similarity measure with image I max and then at least one other image is removed from the current cluster based on its total connectivity value within the current cluster. 15. The apparatus of claim 12 , wherein a number N 1 images having the N 1 lowest similarity measures with I max are removed from the current cluster. 16. The apparatus of claim 15 , wherein at least one image is removed from the current cluster based on its similarity measure with image I max and then at least one other image is removed from the current cluster based on its total connectivity value within current cluster. 17. An apparatus for clustering a set of N items into P final clusters, where N and P are integers, comprising: means for calculating at least one similarity measure S i,j between members of each possible pair of items, wherein S i,j represents the similarity measure between the ith and jth items with i and j being items indices and S i,j =S j,i ; means for calculating a total connectivity value for each of the items remaining to be clustered, a total connectivity value for each item being defined as a sum of a function f of the similarity measures associated with that item; means for identifying, from among said items remaining to be clustered, a maximum total connectivity value t max corresponding to an item I max , item I max belonging to a current cluster C which initially includes all items remaining to be clustered; means for removing, from the current cluster C, at least one item based on at least one of its similarity measure with item I max and its total connectivity value within current cluster C; means for adding, to current cluster C, items having a similarity measure that is greater than a threshold T 3 with any item currently in C; means for calculating, for each item within current cluster C, a total connectivity value based on those items within C; means for removing, from current cluster C, those items having a total connectivity value less than some threshold T 4 , wherein said total connectivity value calculating step and said item removing step are repeated until no further items are removed to thereby establish current cluster C as one of the final clusters; and means for removing all items in current cluster C from further consideration, wherein one or more of said calculating steps, said identifying step, said removing steps, and said adding step are repeating until all N items are assigned to a final cluster. 18. A computer-readable medium having stored thereon a plurality of instructions, the plurality of instructions including instructions which, when executed by a processor, cause the processor to perform the steps of a method for clustering a set of N images into P final clusters, where N and P are integers comprising of: (a) calculating at least one similarity measure S i,j between members of each possible pair of images, wherein S i,j represents the similarity measure between the ith and jth images with i and j being image indices and S i,j =S j,i ; (b) calculating a total connectivity value for each of the images remaining to be clustered, a total connectivity value for each image being defined as a sum of a function f of the similarity measures associated with that image; (c) identifying, from among said images remaining to be clustered, a maximum total connectivity value t max corresponding to an image I max , image I max belonging to a current cluster C which initially includes all images remaining to be clustered; (d) removing, from the current cluster C, at least one image based on at least one of its similarity measure with image I max and its total connectivity value within current cluster C; (e) adding, to current cluster C, images having a similarity measure that is greater than a threshold T 3 with any image currently in C; (f) calculating, for each image within current cluster C, a total connectivity value based on those images within C; (g) removing from current cluster C, those images having a total connectivity value less than some threshold T 4 ; (h) repeating steps (f) and (g) until no further images are removed to thereby establish current cluster C as one of the final clusters; (i) removing all images in current cluster C from further consideration; and (j) repeating steps (b)-(i) until all N images are assigned to a final cluster. 19. The computer-readable medium of claim 18 , comprising the step of applying a non-linear function to each similarity measure, prior to calculating the total connectivity value. 20. The computer-readable medium of claim 19 , wherein the non-linear function is a transcendental function. 21. The computer-readable medium of claim 20 , wherein the transcendental function is a hyperbolic tangent. 22. The computer-readable medium of claim 21 , wherein each similarity measure S i,j is transformed according to the formula tanh(5×S i,j ) prior to calculating a total connectivity value. 23. The computer-readable medium of claim 18 , wherein the function f is the identify function. 24. The computer-readable medium of claim 18 , wherein the function f is a nonlinear function. 25. The computer-readable medium of claim 18 , wherein in step (d), at least one image is removed from the current cluster based on its similarity measure with image I max , and then at least one other image is removed from the current cluster based on its total connectivity value within the current cluster. 26. The computer-readable medium of claim 18 , wherein in step (d), at least one image is removed from the current cluster based on its total connectivity value within the current cluster, and then at least one other image is removed from the current cluster based on its similarity measure with image I max . 27. The computer-readable medium of claim 18 , wherein in step (d), images having a similarity measure with I max that is less than some threshold T 1 are removed from the current cluster. 28. The computer-readable medium of claim 27 , wherein in step (d), a number N 1 images having the N 1 lowest similarity measures with I max are removed from the current cluster. 29. The computer-readable medium of claim 18 , wherein in step (d), images having a total connectivity value which differs from t max by more than some threshold T 2 are removed from the current cluster. 30. The computer-readable medium of claim 29 , wherein images having a similarity measure with I max that is less than some threshold T 1 are removed from the current cluster. 31. The computer-readable medium of claim 30 , wherein at least one image is removed from the current cluster based on its similarity measure with image I max and then at least one other image is removed from the current cluster based on its total connectivity value within the current cluster. 32. The computer-readable medium of claim 29 , wherein a number N 1 images having the N 1 lowest similarity measures with I max are removed from the current cluster. 33. The computer-readable medium of claim 32 , wherein at least one image is removed from the current cluster based on its similarity measure with image I max and then at least one other image is removed from the current cluster based on its total connectivity value within current cluster. 34. A computer-readable medium having stored thereon a plurality of instructions, the plurality of instructions including instructions which, when executed by a processor, cause the processor to perform the steps of a method for clustering a set of N items into P final clusters, where N and P are integers comprising of: (a) calculating at least one similarity measure S i,j between members of each possible pair of items, wherein S i,j represents the similarity measure between the ith and jth items with i and j being item indices and S i,j =S j,i ; (b) calculating a total connectivity value for each of the items remaining to be clustered, a total connectivity value for each item being defined as a sum of a function f of the similarity measures associated with that item; (c) identifying, from among said items remaining to be clustered, a maximum total connectivity value t max corresponding to an item I max , item I max belonging to a current cluster C which initially includes all items remaining to be clustered; (d) removing, from the current cluster C, at least one item based on at least one of its similarity measure with item I max and its total connectivity value within current cluster C; (e) adding, to current cluster C, items having a similarity measure that is greater than a threshold T 3 with any item currently in C; (f) calculating, for each item within current cluster C, a total connectivity value based on those items within C; (g) removing, from current cluster C, those items having a total connectivity value less than some threshold T 4 ; (h) repeating steps (f) and (g) until no further items are removed to thereby establish current cluster C as one of the final clusters; (i) removing all items in current cluster C from further consideration; and (j) repeating steps (b)-(i) until all N items are assigned to a final cluster.
CROSS-REFERENCE TO RELATED APPLICATIONS This application is a continuation of U.S. patent application Ser. No. 09/819,557, filed Mar. 28, 2001 now U.S. Pat. No. 6,798,911. The aforementioned related patent application is herein incorporated by reference. BACKGROUND OF THE INVENTION Field of the Invention This application is directed to a system and method for automatically clustering images into similar groups using fuzzy theory. Clustering is a well known technique for partitioning a set of N objects into P groups, N>P, based on some set of metrics. Typically, the set of metrics includes one or more similarity measures in the form of either quantitative or qualitative factors which pertain to the similarity between these objects. Examples of known clustering methods are disclosed in B. Merkin, Mathematical Classification and Clustering, Kluwer Academic Publishers, 1996, and references cited therein. The automatic clustering of images based on the content within the images has applications in indexing and retrieval of visual information. However, visual similarity is not very well defined since it is a subjective phenomenon. Distinguishing the similarity of two images using computers is difficult. Indeed, even humans may not agree on the similarity of two images. Furthermore, the definition of similarity depends on the goal of the clustering process. For example, two portraits of different people may be considered as similar if the goal of the clustering algorithm is to separate human faces from other images. On the other hand, the same two images are not similar if the goal is to find all pictures of one particular person from among a collection of portraits of different people. And while any similarity measure applicable to two images may be used, the particular similarity measure selected can affect the outcome of the clustering process. Consequently, similarity measures are selected based on their ability to provide effective discriminatory metrics for the variety of images to be clustered. Prior art methods for clustering images are based on defining a similarity measure between two images. If X i and X j are two images in an image database containing a total of N images, a similarity measure S i,j between each pair of images (X i , X j ) can be established such that: S i,j =1 if X i and X j are similar, and   Eq. 1 S i,j =0 if X i and X j are dissimilar. Then, one can create a graph of N nodes in which each node corresponds to one of the N images and nodes i and j are connected if and only if S i,j =1. Such a graph can be topologically complex and may have many dimensions. Accordingly, one can define such a graph as a binary symmetric N×N graph matrix A in which an element a i,j =1 if nodes i and j are connected, element a i,j =0 if nodes i and j are not connected, and element a i,j =1. Equivalently, the graph can be defined by a list of the image pairs which are connected. Given such a graph, one can then find the connected subgraphs using known algorithms in graph theory. Such connected subgraphs represent clusters of images within the database. However, the validity of the resulting clusters using the above paradigm depends heavily on the precision and correctness of the similarity measure S i,j . Typically, the first step toward such a similarity measure is the calculation of a distortion measure D i,j between each of the (N)(N−1)/2 pairs of images. The distortion measure may be made using one or more features extracted from each of the original images. Then, using some threshold T, which perhaps is adaptively determined, one may decide if two images are similar, and assign an S i,j value of 1 to those deemed to be similar. Such a thresholding process results in an N×N binary matrix B created as follows: b i,j =1 if a i,j >T, and  Eq. 2 b i,j =0 otherwise. However, one disadvantage of this process is that such thresholding results in the loss of information which may otherwise be used in some manner during the clustering process. SUMMARY OF THE INVENTION The present invention is directed to an approach for automatically clustering images in which the similarity measure is a fuzzy measure, i.e., 0≦S i,j ≦1 based on the original distortion measure D i,j . A fuzzy graph is effectively established, and a fuzzy clustering algorithm is then applied to the fuzzy graphs to find the corresponding connected subgraphs which, in turn, represent the clusters. The method of the present invention begins with the calculation of a fuzzy similarity measure between an initial set of images to be clustered. Total connectivity values are then calculated for each of these images, the total connectivity for an image being the sum of a function of the various similarity measures associated with than image. The image having the highest connectivity, I max , is determined to definitely belong to a current cluster that is being established, the current cluster initially including all the images which remain to be assigned to a true, final cluster. Images are removed from the current cluster based on either their similarity measures with respect to I max , or their connectivity values, or both. Next, images which have just been removed from the current cluster, but have high similarity measures with respect to any of the images remaining in the current cluster are added back into the current cluster. Total connectivity values are then calculated for each of the images remaining in the current cluster, and those with low total connectivity values are removed. This last step is repeated until there is no change in the membership of the cluster. At this point, the current cluster is fixed and thus determined to be a “true” cluster, the images within the current cluster are removed from the initial set, and the remaining images subjected once again to the above process until all images have been assigned to truce clusters. In one aspect of the invention, the similarity measures are first transformed using a non-linear function prior to calculation of the total connectivity measures. The non-linear function may be a transcendental function such as a hyperbolic tangent. In another aspect of the invention, the function f is the identity function so that the similarity measures are simply summed. However, the function f may instead be a non-linear function of the similarity measures. BRIEF DESCRIPTION OF THE DRAWINGS The present invention can better be understood through the attached figures in which: FIG. 1 shows an example of a fuzzy graph created in accordance with the image clustering method of the present invention; FIG. 2 shows the corresponding fuzzy matrix for the fuzzy graph of FIG. 1 ; FIGS. 3 a , 3 b and 3 c show candidate fuzzy subgraphs for the graph of FIG. 1 , based on different threshold values; and FIG. 4 presents a flowchart of the steps entailed in a preferred embodiment of the present invention. DETAILED DESCRIPTION In the context of the present invention, a fuzzy graph is defined much in the same way as a conventional graph, except that the elements in the graph matrix A are real numbers, preferably scaled to between zero and one (0≦a i,j ≦1) to facilitate further processing and comparisons. In addition, the similarity measure is commutative in that a i,j =a ji and the diagonal values a j,i =1, as before. This implies that the “connectedness” between any two nodes i, j, i ≠j, in the graph is a fuzzy relation. FIG. 1 shows an example of a fuzzy graph 100 comprising N=6 six nodes, which have been labeled 110 , 112 , 114 , 116 , 118 and 120 . Non-zero similarity values a 12 , a 13 , a 23 , a 24 , a 34 , a 45 and a 56 are established between certain pairs of these nodes. It is understood, however, that, each node could theoretically be connected to every other node, making for a much more topologically complex graph. FIG. 2 shows the corresponding fuzzy graph matrix 200 for the fuzzy graph 100 of FIG. 1 . Entries in the matrix 200 represent the links a i,j in the fuzzy graph. In addition, as seen in FIG. 2 , the matrix shows bilateral symmetry with a i,j =a j,i . It is understood, however, that the information in fuzzy graph matrix 200 could just as effectively be represented as a list of the node pairs along with their corresponding non-zero similarity values, since the pair-wise similarity values are the basis for the clustering process. FIGS. 3 a , 3 b and 3 c show subgraphs created by applying similarity thresholds of R=0.25, R=0.45 and R=0.55, respectively, to the fuzzy graph 100 of FIG. 1 . As seen in FIG. 3 a , a low similarity threshold of R=0.25 results in no clustering and leaves the original graph intact, without severing any of the links among the nodes. In effect, then, such a low threshold value performs no clustering and so does nothing to help differentiate among the images to which the nodes correspond. As seen in FIG. 3 b , a moderate similarity threshold of R=0.45 partitions the original graph into two subgraphs by severing the link between nodes 116 and 118 , effectively setting link a 45 (and thus also a 54 ) to zero. Thus, an R-value of 0.45 results in the original set of nodes being grouped into two distinct clusters. Finally, as seen in FIG. 3 c , the higher similarity threshold of R=0.55 results in two subgraphs/clusters having the same node membership as in the case of R=0.45, but with link a 34 severed. Thus, setting R=0.55 leads to the same two clusters of images that resulted when R=0.45, with the caveat that one pair of images (those represented by nodes 114 and 116 in FIG. 1 ) is considered to be “dissimilar”, even through each member of this pair is considered “similar” to some other image within the cluster to which the corresponding nodes have been assigned. Whether the two resulting clusters from either R=0.45 or R 0.55 should be the final partition of nodes is determined through further processing. FIG. 4 presents a flowchart 400 for a process to cluster a set of N images, {X 1 , X 2 , . . . , X N } in accordance with the present invention. In step 402 , (N)(N−1)/2 pair-wise initial similarity measures (SN i,j ) are calculated for the N images in a database. The similarity measures may be considered to belong to an N×N similarity matrix. Preferably, the resulting initial similarity measures are real-valued and range from zero to one. As discussed above, the particular similarity measure selected can affect the outcome of the clustering process. In a preferred embodiment of the present invention in which images from a video stream are clustered, the similarity measure that is used is the one disclosed in U.S. Pat. No. 6,055,025, entitled “Method and Apparatus for Detecting Abrupt and Gradual Scene Changes in Image Sequences”, whose contents are incorporated by reference to the extent necessary to understand the present invention. It should be kept in mind, however, that the present invention contemplates that a variety of similarity measures may be employed, depending on the nature of the objects to be clustered. In step 404 , the individual initial similarity measures for the N nodes preferably are subjected to a nonlinear function. The purpose of the nonlinear function is to improve distance separation between the initial similarity measures {SN i,j } and arrive at similarity measures {S i,j } to be used in the remainder of the algorithm. More preferably, the nonlinear function is a transcendental function, and most preferably is a hyperbolic tangent of five times an initial similarity measure: S i,j =tanh(5×SN i,j ). The transformed similarity measures are used in the remainder of the preferred embodiment discussed hereafter. The present invention employs the concept of a “T-connectivity” t i,c of a node i belonging to a subgraph C. For the purposes of the preferred embodiment, t i,c is defined as the sum of a function f( ) of the similarity measures associated with that node. Mathematically, this can be represented as: T i , C = ∑ j = 1 , j ≠ i N ⁢ ⁢ f ⁡ ( S i , j ) Eq . ⁢ 3 Function f( ) can simply be the identity function so that t i,c is simply the sum of the S i,j . Alternatively, function f( ) can be a non-linear function which, for example, takes the square of the individual S i,j before they are summed. Other functions are also possible. Regardless of which such function is used, for the present purposes, we refer to a subgraph C as being “T-connected”, if t i,c >T, T being some connectivity threshold value, for all nodes i in subgraph C. Thus, a T-connected subgraph is one whose nodes all have a connectivity greater than T. Thus, the “T-connectedness” of a subgraph serves as a rough measure of the aggregate similarity of the nodes within that subgraph. Initially, all N nodes are to be clustered and belong to a single graph and thus a single candidate cluster C. In a trivial sense, this single graph is a subgraph unto itself and has a T-connectedness corresponding to the lowest connectivity among all N nodes. And though all N nodes are initially considered to belong to a single candidate cluster C, the N nodes will ultimately be partitioned into some number P of smaller clusters, P<N. Thus, in step 406 , all nodes which remain to be clustered are considered to belong to a candidate cluster C, the connectivities of all of these nodes are calculated, and the node I max having the maximum connectivity t max,c is identified. Node I max is assumed to be a critical member of candidate cluster C, meaning that I max will always be a member of candidate cluster C and also the true cluster which is formed as a result of the process described below. In step 408 , nodes which are dissimilar to node I max , the dissimilarity being based on their similarity measures with node I max , are removed from candidate cluster C. One approach to this is to remove nodes j which have similarity measures with node I max lower than some threshold T 1 . Thus, nodes j for which S imax,j <T 1 applies, are removed from candidate cluster C. Threshold T 1 can be adaptively set, such as by taking a predetermined proportion p 1 , such as p 1 =75%, of the maximum similarity measure S max =Max{S i,j }. Another approach may be to remove a total of N 1 nodes having the N 1 lowest similarity measures with node I max . In such case, N 1 can be a predetermined proportion p 2 , such as p 2 =50%, of the total number of nodes remaining to be clustered. Other approaches to eliminating dissimilar nodes based on the similarity measures are also possible. In step 410 , nodes j which have a total connectivity t j,c which differs from t max,C by more than some threshold value T 2 , are also removed from candidate cluster C. Thus, node j is removed if t max,C −t j,c >T 2 . The threshold T 2 can be adaptively determined, such as by taking a predetermined proportion p 3 , such as p 3 =25%, of t max,C . It should be noted here that the order in which steps 408 and 410 are carried out can make a difference in the set of nodes that remain in candidate cluster C. The present invention also encompasses the situation is which these two steps are reversed. In step 412 , nodes j which have been removed from candidate cluster C due to the actions takes in steps 408 and 410 may be added back to candidate cluster C. The criterion for adding these nodes back to candidate cluster C is that their similarity measures with at least one node i which remains in candidate cluster C after steps 408 and 410 be greater than some threshold T 3 . Thus, a node j which had been removed may be added back to candidate cluster C if S j,i >T 3 , for some i still belonging to C (i 0 C). Threshold T 3 can be adaptively set, such as by taking a predetermined proportion p 4 , such as p 4 =75%, of the maximum similarity measure S max =Max {S i,j }. In step 414 , the connectivities t i,c for each of the nodes remaining in candidate cluster C, as it then appears, are calculated. Next, in step 416 , those nodes which have connectivities lower than some threshold T 4 are removed from candidate cluster C to arrive at modified candidate cluster CN. Threshold T 4 preferably is formed by taking a predetermined proportion p 5 , such as p 5 =50%, of the maximum connectivity among the nodes in candidate cluster C. Steps 414 and 416 are repeated with CNN replacing C, as seen in step 418 , until there are no more changes and C=CN, as determined in comparison step 420 . If, however, the membership in C and CN oscillates, then threshold T 4 may be adaptively adjusted, preferably by gradually lowering it with each pass until convergence is achieved. If it is determined in step 420 that the node membership in candidate cluster C has been finalized, control flows to step 422 in which candidate cluster C is established as a true cluster formed by a subset of the N original nodes. Next, in step 424 , a determination is made as to whether the entire set of N nodes has been clustered. If so, the process is finished. If not, however, then at step 426 , the nodes of the most recently formed true cluster are deleted from the set of nodes remaining to be clustered, and the process continues with step 406 using the nodes which remain to be clustered. In this manner, a single true cluster is established during each stage of the overall process, a stage comprising a single pass of the algorithm represented by steps 406 through 424 . In the foregoing analysis, steps 404 through 426 were discussed without specific reference to the fact that, in the preferred embodiment, the initial similarity measures were for a set of images. Accordingly, it should be recognized that the foregoing algorithm of the preferred embodiment may have applicability in clustering similarity measures, regardless of the underlying items to be clustered. Furthermore, it should be evident to one skilled in the art that the above-described embodiment is readily implemented in computer software, using some high-level programming language to run on a general purpose computer. And finally, while the above invention has been described with reference to certain preferred embodiments, it should be kept in mind that the scope of the present invention is not limited to these. One skilled in the art may find variations of these preferred embodiments which, nevertheless, fall within the spirit of the present invention, whose scope is defined by the claims set for the below.

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