Shape preserving local histogram modification

220 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 Shape Preserving Local Histogram Modification Vicent Caselles, Associate Member, IEEE, Jose-Luis Lisani, Jean-Michel Morel, and Guillermo Sapiro, Member, IEEE Abstract— A novel approach for shape preserving contrast enhancement is presented in this paper. Contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level-sets of the image. This basic property is violated by common local schemes, thereby introducing spurious objects and modifying the image information. The scheme is based on equalizing the histogram in all the connected components of the image, which are defined based both on the grey-values and spatial relations between pixels in the image, and following mathematical morphology, constitute the basic objects in the scene. We give examples for both grey-value and color images. Index Terms—Connected components, histogram equalization, level-sets, local operations, mathematical morphology. I. INTRODUCTION I MAGES ARE captured at low contrast in a number of different scenarios. The main reason for this problem is poor lighting conditions (e.g., pictures taken at night or against the sun rays). As a result, the image is too dark or too bright, and is inappropriate for visual inspection or simple observation. The most common way to improve the contrast of an image is to modify its pixel value distribution, or histogram. A schematic example of the contrast enhancement problem and its solution via histogram modification is given in Fig. 1. On the left, we see a low contrast image with two different squares, one inside the other, and its corresponding histogram. We can observe that the image has low contrast, and the different objects cannot be identified, since the two regions have almost identical grey values. On the right we see what happens when we modify the histogram in such a way that the grey values corresponding to the two regions are separated. The contrast is improved immediately. Histogram modification, and in particular histogram equalization (uniform distributions), is one of the basic and most useful operations in image processing, and its description can be found in any book on image processing. This operation is a Manuscript received March 27, 1997; revised April 28, 1998. This work was supported in part by DGICYT Project under Reference PB94-1174, by the Math, Computer, and Information Sciences Division, Office of Naval Research under Grant ONR-N00014-97-1-0509, by NSF-LIS, and by the ONR Young Investigator Program. V. Caselles and J.-L. Lisani are with the Department of Mathematics and Informatics, University of Illes Balears, 07071 Palma de Mallorca, Spain (e-mail: dmivca0@ps.uib.es). J.-M. Morel is with the Department of Applied Mathematics, Ecole Normale Superieure Cachan, 94235 Cachan Cedex, France (e-mail: jeanmichel.morel@cmla.ens-cachan.fr). G. Sapiro is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: guille@ece.umn.edu). Publisher Item Identifier S 1057-7149(99)00936-7. Fig. 1. Schematic explanation of the use of histogram modification to improve image contrast. particular case of homomorphic transformations: Let be the image domain and : the given (low be a given function contrast) image. Let : which we assume to be increasing. The image is called a homomorphic transformation of . The particular case of histogram equalization corresponds to selecting to be the distribution function of : Area (1) Area If we assume that of variables is strictly increasing, then the change (2) gives a new image whose distribution function is uniform in , , . This useful and basic the interval operation has an important property which, in spite of being obvious, we would like to acknowledge: it neither creates nor destroys image information. As argued by the mathematical morphology school [1], [6], [7], the basic operations on images should be invariant with respect to contrast changes, i.e., homomorphic transformations. As a consequence, it follows that the basic information of an image is contained in the family of its binary shadows or level-sets, that is, in the family of sets (3) for all values of in the range of . Observe that, under fairly general conditions, an image can be reconstructed from its . If is a level-sets by the formula does strictly increasing function, the transformation not modify the family of level-sets of , it only changes its index in the sense that 1057–7149/99$10.00  1999 IEEE for all (4) CASELLES et al.: LOCAL HISTOGRAM MODIFICATION Although one can argue if all operations in image processing must hold this principle, for the purposes of the present paper we shall stick here to this basic principle. There are a number of reasons for this. First of all, a considerable large amount of the research in image processing is based on assuming that regions with (almost) equal grey-values, which are topologically connected (see below), belong to the same physical object in the three-dimensional (3-D) world. Following this, it is natural to assume then that the “shapes” in an given image are represented by its level-sets (we will later see how we deal with noise that produces deviations from the levelsets). Furthermore, this commonly assumed image processing principle will permit us to develop a theoretical and practical framework for shape preserving contrast enhancement. This can be extended to other definitions of shape, different from the level-sets morphological approach here assumed. We should note that the level-sets theory is also applicable to a large number of problems beyond image processing [5], [10]. In this paper, we want to design local histogram modification operations that preserve the family of level-sets of the image, that is, following the morphology school, preserve shape. Local contrast enhancement is mainly used to further improve the image contrast and facilitate the visual inspection of the data. As we will see later in this paper, global histogram modification does not always produce good contrast; small regions, especially, are hardly visible after such a global operation. On the other hand, local histogram modification improves the contrast of small regions as well, but since the level-sets are not preserved, artificial objects are created. The theory developed in this paper will enjoy the best of both words: the shape-preservation property of global techniques and the contrast improvement quality of local ones. The recent formalization of multiscale analysis given in [1] leads to a formulation of recursive, causal, local, morphological, and geometric invariant filters in terms of solutions of certain partial differential equations of geometric type, providing a new view on many of the basic mathematical morphology operations. One of their basic assumptions was the locality assumption, which aimed to translate into a mathematical language the fact that we considered basic operations which were a kind of local average around each pixel or, in other words, only a few pixels around a given sample influence the output value of the operations. Obviously, this excluded the case of algorithms as histogram modification. This is why operations like those in [8] and [9] and the one described in this paper are not modeled by these equations, and a novel framework must be developed. It is not the goal of this paper to review the extensive research performed in contrast enhancement. We should only note that basically, contrast enhancement techniques are divided in the two groups mentioned above, local and global, and their most popular representatives can be found in any basic book in image processing and computer vision. An early attempt to introduce shape criteria in contrast enhancement was done in [3]. To the best of our knowledge, none of the variations to histogram modification reported in the literature have formally approached the problem of shape preserving contrast enhancement as done in this paper. 221 II. GLOBAL HISTOGRAM MODIFICATION: A VARIATIONAL FORMULATION We call representatives of all images of the form , where is a strictly increasing function. The question is, which representative of is the best for our purposes? That will depend, of course, in what our purposes are. We have seen above which is the function we have to select if we want to normalize the contrast making the distribution function of uniform. In addition, it was shown in [8] and [9] that when on the range we are equalizing an image : minimizing the functional (5) The second term of the integral can be understood as a measure of the contrast of the whole image. Thus, when we are distributing the values of so that we minimizing maximize the contrast. The first term tries to keep the values of as near as possible to the mean . When minimizing on the class of functions with the same family of binary shadows as , we get the equalization of . We will see below how to modify this energy to obtain shape preserving local contrast enhancement. III. CONNECTED COMPONENTS To be able to extend the global approach to a local setting, we have to insist in our main constraint: we have to keep the same topographic map, that is, we have to keep the same family of level-sets of but we have the freedom to assign them a “convenient” grey level. To make this statement more precise, let us give some definitions (see [11]). be a topological space. We say that Definition 1: Let is connected if it cannot be written as the union of two nonempty closed (open) disjointsets. A subset of is called is a maximal connected subset a connected component if of of , i.e., is connected and for any connected subset such that , then . of which This definition will be applied to subsets , are topological spaces with the topology induced from is the intersection of an open set of i.e., an open set of with . We shall need the following observation which follows from the definition above: Two connected components of a topological space are either disjoint or they coincide; thus, the topological space can be considered as the disjoint union of its connected components. Remark: There are several notions of connectivity for a topological space. One of the most intuitive ones is the notion of arcwise connected (also called connected by arcs). is said to be connected by arcs if any A topological space of can be joined by an arc, i.e., there exists two points such that , a continuous function : . In a similar way as above we define the connected components (with respect to this notion of connectivity) as the maximal connected sets. These notions could be used below instead of the one given in Definition 1. 222 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 be a given image and , Definition 2: Let : , . A section of the topographic map of is a set of the form (6) where each , is a connected component of , , the set such that for (7) is also connected. be a given image and let Definition 3: Let : : be the family of its level-sets. We shall say is a local contrast change that the mapping : if the following properties hold. is continuous in the following sense: P1: when being a connected component of . is an increasing function of for all . for all , are in the same , . connected component of not reduced to P4: Let be a connected set with . Then is not a point. Let reduced to a point. be a section of the P5: Let , and let , topographic map of , . Then . be a given image. We shall Definition 4: Let : say that is a local representative of if there exists some , . local contrast change such that We collect in the next proposition some properties which follow immediately from the definitions above. and let Proposition 1: Let : , , be a local representative of . Then, we have the following. : . We have that 1) if and only if , , . 2) is a continuous function. (resp. 3) Let ( ) be a connected component of ) containing , . Then . be a section of the topographic map of . 4) Let Then is also a section of the topographic map of . Proof: 1) Is a simple consequence of P2 in Definition 3. 2) Is a consequence of P1 in Definition 3. 3) By P3 of Definition 3, we have . Since and is connected, then . Now, suppose that P2: P3: Thus, 3, is not reduced to a point. By P4 of Definition is not reduced to a point, a contradiction since on . It follows that . Since is , then . connected and 4) Let be a section of the topoand . By graphic map of . Let coincides with the connected component of Part 3, containing which we denote by . Let , . Since, using P5, : , then we may write . Now it is easy to see that is a section of the topographic map of . Remarks: 1) The previous proposition can be phrased as saying that the set of “objects” contained in is the same as the set of “objects” contained in , if we understand the “objects” of as the connected components of the level, , and, respectively, for . sets 2) Our definition of local representative is contained in the notion of dilation as given in [6] and [7], Th. 9.3. Let be a lattice of functions : . A mapping : is called a dilation of if and only if it can be written as where is a function assigned to each point and is possibly different from point to point. Thus, let be a local contrast change and let . Let us denote by the conwhich contains if , nected component of . Let otherwise, let if ; and if . Then . 3) Extending the definition of local contrast change to include more general functions than continuous ones, i.e., to include measurable functions, we can state and prove a converse of Proposition 1, saying that the topographic map contains all the information of the image which is invariant by local contrast changes [2]. IV. SHAPE PRESERVING CONTRAST ENHANCEMENT We can now state precisely the main question we want to address: what is the best local representative of , when the goal is to perform local contrast enhancement while preserving the connected components (and level-sets). For that we shall be a use the energy formulation given in Section II. Let connected component of the set , , , . Write (8) We then look for a local representative of that minimizes for all connected components of all sets of the , , , , or, in other words, form CASELLES et al.: LOCAL HISTOGRAM MODIFICATION 223 the distribution function of in all connected components of is uniform in the range , for all , , . We now show how to solve this problem. Let us introduce some notation that will make our discussion easier. Without loss of generality we assume that : . Let , , . We need to assume that , the distribution function of , is continuous and strictly increasing. For that we assume that is continuous and1 for all Area (9) We shall construct a sequence of functions converging be the to the solution of the problem. Let histogram equalization of . Suppose that we already con. Let us construct . For each structed , let (10) and let ( be the connected components of can be eventually ). Define Lemma 2: Let , , : , histogram in such that . Let : be two functions with uniform , respectively. Assume that (15) Let : be given by if if (16) . Then has a uniform histogram in Proof of Theorem 1: The first part of the statement follows immediately from the two lemmas above. Now, consider . Observe that the sequence for all , Indeed, if (17) , then , , then . The estimate (17) follows. , while, if , Now, since , , . (18) (11) is a continuous strictly By our assumption (9), increasing function in , and we can equalize the in . Thus, we define histogram of and the series on the right-hand side is absolutely convergent, converges absolutely and uniformly to some continthen . satisfies the statement above. uous function : Indeed, since for all (12) and (13) (19) and is the uniform limit of some such that , then for all there is (20) We will then prove the following. Theorem 1: Under the assumption (9), the functions have a uniform histogram for all connected components of all where , : “dyadic” sets of the form , . Moreover, as , converges to a function that has a uniform histogram for all connected , for all , , components of all sets . Theorem 2: Let be the function constructed in Theorem 1. Then is a local representative of . The proof of Theorem 1 is based in the next two simple lemmas. , such that . Let Lemma 1: Let : , , be two functions with uniform . Let : be given by histogram in if if Then, has a uniform histogram in , . for all , , it follows that for all Thus, by approaching . 1 This assumption is mainly theoretical and does not necessarily need to hold for basic practical purposes. . Letting (21) is not dyadic, let . Then If , be such that (22) with dyadic numbers, we prove that for all (23) Let us mention in passing that the above proof also shows that Area (14) and all for all (24) has a uniform histogram in all Similarly, one proves that for connected components of all sets of the form all dyadic numbers , , . Now let , and let be a connected component of . Let , be such that . Let be 224 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 containing a connected component of Then . Let . Observe that (25) . By property (24), we may approach for all by dyadic numbers while . It follows that and , , . Since (27) and (26) The other inequality is proved in a similar way. It follows that has a uniform histogram for all connected components of for all numbers , , all sets of the form . Proof of Theorem 2: We shall use the notation intro( being duced previously. First we define the global histogram of ). Let . Let , . be such that . Then we define Let if if if Let , Suppose that Then either . If and , , (28) then , , . (29) is a local contrast change of . Let us check It is clear that that is a local contrast change of , , i.e., it satisfies P1–P5, for all . To simplify our notation, let us write instead of . P1 : the series in (27) is absolutely and uniformly convergent. for some function . Hence, . Let us now prove that It follows that is a local contrast change for . , , , . Since P1: Let , , and or , then . Hence, , . Then P2: P3: . . for some , and P4: . If P2 : P3 : P4 : P5 : , then one easily checks . that Follows from the definition of . Let , be in the same connected component of , . Let be such that , . Then, for some . Then . Let be a connected set with not reduced to a point. Let . Since , for some , is not reduced to a point. Thus, there exist with . Then, . The : is not reduced to a point. set Let be a section of the and let , . topographic map of Let be such that , , . , , then If since is connected and contains . Thus, . Hence, . Let , be such that , , . Then . P5: Using the corresponding property H1 we see that as , . Follows from P2 and the definition of . Let , be in the same connected component of , . Then and , are in the same connected component of . Then . Proceeding iteratively and using P3 we get that for all . Letting we get that . be a connected set with not reduced Let to a point. For any , let be the containing . connected component of . Since is not reduced Let to a point, it contains an interval. This implies that . Now we observe that . Area . Now, let for Obviously, . Since, by P3, , we have some . It follows that , that hence the equality. If was reduced to a point , then . Hence, Area Area , contradicting (24). Therefore cannot be reduced to a point. Let be a section of the , . First, topographic map of and let using P5 and the fact that each transforms into a section of the topographic map of (Proposition 1), it follows that for , we get that all . Letting . Now, let . Since , are also sections of the topographic map of , then by the previous observation we have . If , then for all CASELLES et al.: LOCAL HISTOGRAM MODIFICATION 225 Fig. 2. Example of the level-sets preservation. The top row shows the original image and its level-sets. The second row shows the result of global histogram modification and the corresponding level-sets. Results of classical local contrast enhancement and its corresponding level-sets are shown in the third row. The last row shows the result of our algorithm. Note how the level-sets are preserved, in contrast with the result on the third row, while the contrast is much better than the global modification. and some constant . Hence Area Area , again a contradiction with (24). Thus, . Proof of Lemma 1: Let . Since Proof of Lemma 2: Let . Since it follows that Now, let Hence, has a uniform histogram. We conclude that . Since has a uniform histogram. 226 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 Fig. 3. Additional example of the level-sets preservation. The first row shows the original image, global histogram modification, classical local modification, and the proposed shape preserving local histogram modification. The second row shows the corresponding level-sets. (a) (b) (c) (d) Fig. 4. Example of shape preserving local histogram modification for real data. (a) Original image. (b) Result of global histogram modification. (c) Intermediate state (d) Steady-state of the proposed algorithm. CASELLES et al.: LOCAL HISTOGRAM MODIFICATION 227 (a) (b) (c) (d) Fig. 5. Additional example of shape preserving local histogram modification for real data. (a) Original image. (b)–(d) Results of global histogram equalization, classical local scheme (61 61 neighborhood), and our algorithm, respectively. 2 V. THE ALOGRITHM AND NUMERICAL EXPERIMENTS The algorithm has been described in the previous section. be an image Let us summarize it here. Let : whose values have been normalized in . Let , , . be the histogram equalizaStep 1) Construct tion of . . Step 2) Construction of , . Let us Suppose that we already constructed construct . For each , let (30) be the connected components of , . Let be the distribution function of with values in the range , . Then we define and let (31) Remark: An interesting variant in practice consists in using the mean of , denoted by , as the value to subdivide : the range of (32) in all connected components of Then we equalize in the range , respectively, in all connected in the range . In this way, components of . Then we compute the mean values of we construct in , . Denote them by , ( ). into four Now we use these values to subdivide again in all pieces and proceed to equalize the histogram of connected components of all these pieces. We may continue iteratively in this way until desired. In Fig. 2, we compare the classical local histogram algorithm described in [4] with our algorithm. In the classical neighborhood algorithm the procedure is to define an and move the center of this area from pixel to pixel. At points each location we compute the histogram of the in the neighborhood and obtain a histogram equalization (or 228 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 Fig. 6. Example of local histogram modification of a color image. The original image is shown on the top. The bottom left is the result of applying our algorithm to the Y channel in the YIQ color space. On the right, the algorithm is applied again only to the Y channel, but rescaling the chrominance vector to maintain the same color point on the Maxwell triangle. (a) (b) (c) Fig. 7. Comparison between the classical local histogram modification scheme with the new one proposed in this paper for a color image. (a) Original image. (b) Image obtained with the classical technique. (c) Result of applying our scheme. Note the spurious objects introduced by the classical local scheme. histogram specification) transformation function. This function is used to map the level of the pixel centered in the neighregion is then moved to borhood. The center of the an adjacent pixel location and the procedure is repeated. In practice, one updates the histogram obtained in the previous location with the new data introduced at each motion step. Fig. 2(a) shows the original image whose level-lines are displayed in Fig. 2(b). In Fig. 2(c) we show the result of the global histogram equalization of Fig. 2(a). Its level-lines are displayed in Fig. 2(d). Note how the level-sets lines are preserved, while the contrast of small objects is reduced. Fig. 2(e) shows the result of the classical local histogram 31 neighborhood), with equalization described above (31 level-lines displayed in Fig. 2(f).2 We see that new levellines appear thus modifying the topographic map (the set of level-lines) of the original image, introducing new objects. Fig. 2(g) shows the result of our algorithm for local histogram 2 All the level sets for grey-level images are displayed at intervals of 20 grey-values. equalization. Its corresponding level-lines are displayed in Fig. 2(h). We see that they coincide with the level-lines of the original image, Fig. 2(b). Fig. 3 repeats the experiments in Fig. 2 for another synthetic image. Fig. 3(a) has been constructed by cutting half of the right side of Fig. 2(a) and putting it at the left side of it. Fig. 3(b) shows the global histogram equalization of Fig. 3(a). Fig. 3(c) shows the result of the classical local histogram equalization described above. Fig. 3(d) presents the result of our algorithm applied to Fig. 3(a). The level-lines off all the figures are given in Fig. 3(e)–(h), respectively. We see how different connected components do not interact in the proposed scheme, and the contrast is improved while preserving the objects in the scene. Results for a real image are presented in Fig. 4. Fig. 4(a) is the typical “Bureau de l’INRIA” image. Fig. 4(b) is the global histogram equalization of Fig. 4(a). Fig. 4(c) shows an intermediate step of the proposed algorithm, while Fig. 4(d) is the steady-state solution. Note how objects that are not visible CASELLES et al.: LOCAL HISTOGRAM MODIFICATION in the global modification, like those through the window, are now visible with the new local scheme. An additional example is given in Fig. 5. Fig. 5(a) is the original image. Fig. 5(b)–(d) are the results of global histogram equalization, classical local scheme (61 61 neighborhood), and our algorithm, respectively. Experiments with a color image are given in Fig. 6, working on the YIQ (luminance and chrominance) color space. In Fig. 6(a) we present the original image. In Fig. 6(b), our algorithm has been applied to the luminance image Y (maintaining IQ) and then we recomposed the RGB color system. In Fig. 6(c), again, we apply the proposed local histogram modification to the color Y channel only, but rescaling the chrominance vector to maintain the same color point on the Maxwell triangle. In the last example, Fig. 7, we compare the classical local histogram modification scheme with the new one proposed in this paper for a color image, following the same procedure as in Fig. 6. Fig. 7(a) shows the original image, Fig. 7(b) the one obtained with the classical technique, and Fig. 7(c) the result of applying our scheme. Note the spurious objects introduced by the classical local scheme. (This figure is reproduced in black and white here.) 229 ACKNOWLEDGMENT The authors wish to thank INRIA for the use of their database. REFERENCES [1] L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, “Axioms and fundamental equations of image processing,” Arch. Rational Mechan. Anal., vol. 16, pp. 200–257, 1993. [2] V. Caselles, J. L. Lisani, J. M. Morel, and G. Sapiro, “The information of an image invariant by local contrast changes,” 1998, preprint. [3] R. Cromartie and S. M. Pizer, “Edge-affected context for adaptive contrast enhancement,” in Proc. Information Processing in Medical Imaging, Lecture Notes in Comp. Science, Wye, U.K., July 1991, vol. 511, pp. 474–485. [4] R. C. Gonzalez and P. Wintz, Digital Image Processing. Reading, MA: Addison-Wesley, 1987. [5] S. J. Osher and J. A. Sethian, “Fronts propagation with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys., vol. 79, pp. 12–49, 1988. [6] J. Serra, Image Analysis and Mathematical Morphology. New York: Academic, 1982. [7] , Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances. New York: Academic, 1988. [8] G. Sapiro and V. Caselles, “Histogram modification via differential equations,” J. Differential Equat., vol. 135, pp. 238–268, 1997. , “Contrast enhancement via image evolution flows,” Graph. [9] Models Image Process., vol. 59, pp. 407–416, 1997. [10] J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge, U.K.: Cambridge Univ. Press, 1996. [11] L. Schwartz, Analyze I: Theorie des Ensembles et Topologie. Paris, France: Hermann, 1991. VI. CONCLUDING REMARKS This paper presented a novel algorithm for the most basic and (probably) most important operation in image processing: contrast enhancement. The algorithm is motivated by ideas from the mathematical morphology school, and it holds the main properties of both global and local schemes: It preserves the level-sets of the image, that is, its basic morphological structure, as global histogram modification does, while achieving high contrast results as in local histogram modifications. A number of problems remain open in this area, and we believe they can be approached with the framework presented in this paper, which complements the results in [8] and [9]. One of the open problems is to extend the algorithm to other definitions of connected components, that is, other definitions of objects. In this paper, we define objects as done by the mathematical morphology school, via level-sets, and since this is not the only possible definition, it remains to be shown that a similar approach can be used for other relevant object descriptions. Note that objects can be defined also via optical flow components in video data, or with a concept of connected components in multivalued images. A general framework for shape preserving contrast enhancement should include these possible definitions as well. From the energy formulation shown in this paper, (5), it is clear that histogram modification is using a measurement of contrast that it is not appropriate at least for human vision. This is because absolute value is not a good model for how humans measure contrast (this value should be at least normalized by the average brightness of the pixel region). The extension of the approach presented in this paper to other models of image contrast is an interesting open area as well. We expect to address these issues elsewhere. Vicent Caselles (M’95–A’96) received the Licenciatura and Ph.D. degrees in mathematics from Valencia University, Spain, in 1982 and 1985, respectively. Currently, he is an Associate Professor at the University of Illes Balears, Palma de Mallorca, Spain. His research interests include image processing, computer vision, and the applications of geometry and partial differential equations to both previous fields. Dr. Caselles was guest co-editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING special issue on partial differential equations and geometry-driven diffusion in image processing and analysis (March 1998). Jose-Luis Lisani was born in 1970. He received the diploma from the Escuela Tecnica Superior de Ingenieros de Telecomunicacion de Barcelona (ETSETB-Spain), Palma de Mallorca, Spain, in telecommunication engineering in 1995. He is currently a Ph.D. student at Ceremade, University, Paris Dauphine, France. His previous research was at the Universitat de les Illes Balears (UIB-Spain), advised by V. Caselles (UIB), concerning the segmentation of 3-D images by minimizing the Mumford–Shah functional. From 1995 to 1997, he worked at the UIB in several European Union projects, including NEMESIS and CHARM. His research fields are mathematical morphology applied to comparison of images, and optical flow estimation. 230 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 2, FEBRUARY 1999 Jean-Michel Morel received the Ph.D. degree in applied mathematics and the Doctorat d’Etat, both from the University Pierre et Marie Curie, France, in 1980 and 1985, respectively. He is currently a Professor of applied mathematics at the Ecole Normale Superieure de Cachan, Cachan, France. He has been working on the mathematical formalization of image analysis problems since 1988, and is co-author (with S. Solimini) of a book on variational methods in image segmentation. Dr. Morel was guest co-editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING special issue on partial differential equations and geometry-driven diffusion in image processing and analysis (March 1998). Guillermo Sapiro (M’95) was born in Montevideo, Uruguay, on April 3, 1966. He received the B.Sc. (summa cum laude), M.Sc., and Ph.D. degrees from the Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, in 1989, 1991, and 1993, respectively. After conducting post-doctoral research at the Massachusetts Institute of Technology, Cambridge, he became a Member of Technical Staff at the research facilities of Hewlett-Packard Laboratories, Palo Alto, CA. He is currently with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis. He works on differential geometry and geometric partial differential equations, both in theory and applications in computer vision and image analysis. Dr. Sapiro was guest co-editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING special issue on partial differential equations and geometry-driven diffusion in image processing and analysis (March 1998). He was awarded the Gutwirth Scholarship for Special Excellence in Graduate Studies in 1991, the Ollendorff Fellowship for Excellence in Vision and Image Understanding Work in 1992, the Rothschild Fellowship for Post-Doctoral Studies in 1993, and the ONR Young Investigator Award in 1998.