Color Image Enhancement via Chromaticity Diffusion

Transcription

1 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 5, MAY Color Image Enhancement via Chromaticity Diffusion Bei Tang, Guillermo Sapiro, Member, IEEE, and Vicent Caselles Abstract A novel approach for color image denoising is proposed in this paper. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color pixel is considered as an -dimensional vector. The vectors direction, a unit vector, gives the chromaticity, while the magnitude represents the pixel brightness. The chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data such as chromaticity. Both isotropic and anisotropic diffusion flows are presented for this -dimensional chromaticity diffusion flow. The brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement. We present the underlying theory, a number of examples, and briefly compare with the current literature. Index Terms Brightness, chromaticity, color image denoising, directions, harmonic maps, isotropic and anisotropic diffusion, liquid crystals, partial differential equations. I. INTRODUCTION IN a number of disciplines, directions provide a fundamental source of information. Examples in the area of image processing are gradient directions, optical flow directions, surface normals, principal directions, and color. In the color case addressed in this paper, the direction is given by the normalized vector in the color space, and it represents the pixels chromaticity. Formally, an direction defined on an image in is given by a vector such that the Euclidean norm of is equal to one, that is,, where are the components of the vector. The notation can be simplified by considering, where is the unit ball in. When smoothing the data, or computing a multiscale representation of a direction ( stands for the scale), it is crucial to maintain the unit norm constraint. That is, the smoothed direction must also sat- Manuscript received May 13, 1999; revised January 31, This work was supported in part by the TMR European project Viscosity solutions and their applications, reference FMRX-CT , the European Network PAVR FMRXCT960036, the Office of Naval Research under Grant ONR-N , the Office of Naval Research Young Investigator Award, the Presidential Early Career Awards for Scientists and Engineers (PECASE), a National Science Foundation CAREER Award, and the National Science Foundation Learning and Intelligent Systems Program (LIS). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Scott T. Acton. B. Tang and G. Sapiro are with Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN USA ( guille@ece.umn.edu). V. Caselles is with Escola Superior Politecnica, Universitat Pompeu Fabra, Barcelona, Spain. Publisher Item Identifier S (01) isfy.or,. The same constraint holds for a multiscale representation of the original direction. This is what makes the smoothing (denoising) of directions in general, and chromaticity in particular, different from the smoothing of ordinary vectorial data: The smoothing is performed in instead of ( for RGB color spaces for example). In [33], we introduced a new framework for the diffusion of directional data. This extends to images in classical results for images in, e.g., [3], [23], [25] [27], [36], [38], [39]. That is, from the original unit norm vectorial image we showed how to construct a family of unit norm vectorial images that provides a multiscale representation of directions. The method intrinsically takes care of the unit norm constraint. The approach follows results from the literature on harmonic maps in liquid crystals, and is obtained from a system of coupled partial differential equations that reduces a given (harmonic) energy. The framework includes both isotropic and anisotropic (preserving chroma discontinuities) diffusion, works for directions in any dimension, it supports nonsmooth data (edges), applies also to directions defined on general manifolds, and it is based on a substantial amount of existing theoretical results that help to answer a number of relevant image processing questions. In this paper we extend and apply this direction diffusion framework to the problem of color image enhancement (more precisely, denoising). As mentioned above, we consider each pixel of the color image as a vector in. 1 The unit direction, representing chromaticity, is obtained by normalizing this vector. The brightness is given by the vectors magnitude. While the brightness is enhanced using classical scalar anisotropic diffusion flows, Section II, the chromaticity is processed with the novel direction diffusion framework, Section III. A. Related Previous Work The use of partial differential equations (PDEs) has already been established as an important area in the image processing community [8]. A number of authors have addressed color image enhancement using PDEs [6], [27], [29], [37], [38], [40]. All these works deal with (anisotropic) diffusion of vectors, basically extending in different forms the already well established scalar diffusion flows, e.g., [3], [5], [8], [25], [26], [37]. 2 These works do not separate the vector into its direction 1 In this paper we consider n =3 and work on RGB (we found that the results are worse in Lab), although the theory is general and can be used for other dimensions as well. We can for example obtain n>3 from wavelets-type decompositions of the color space. 2 A vectorial approach helps for example in the detection of color edges and in controlling the introduction of spurious colors /01$ IEEE

2 702 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 5, MAY 2001 (chromaticity) and magnitude (brightness), and although good results have been reported, the chromaticity is not always well preserved and color artifacts are frequently observed. A number of authors have proposed to separate the chromaticity and the brightness, and to somehow process them separately. The most recent results, and some of the most interesting ones, are related to the extensions of scalar medians to vectorial data. One possibility to obtain this extension is to consider the ( ) distance in vector space between the image vectors, obtaining a vector median filter [4]. Another possible option is to consider an order in the vector space [9], [42]. A third class of filters considers the vector directions, that is, the chromaticity, obtaining vector directional filters [22], [34], [35]. In this case, the median is obtained as the vector that minimizes the sum of the angles with all the other vectors in the set. This filter is in general combined or cascaded with a scalar filter that enhances the vector magnitude (brightness). These filters are therefore smoothing the chromaticity and the brightness, in a discrete fashion and inspired by scalar median filters, and were shown to be very effective in preserving the image color (chromaticity). Moreover, they have been shown to outperform, in general, the independent scalar filtering of the channels. We can consider the approach here presented as an extension of this discrete technique to the PDEs (diffusion) framework. Note that for the scalar case, there is a close relationship between median filtering and anisotropic diffusion obtained from PDEs [20] (basically, scalar median filtering is a nonconsistent implementation of curvature-based anisotropic diffusion, thereby producing worse results). Therefore, it is natural to study the extension of discrete vectorial algorithms, like the vector directional filter, to this PDEs arena as well. In the examples section we will present explicit comparisons between our algorithm and some of these discrete approaches. The general framework of partial differential equations for denoising of general data on general manifolds is also studied in [10], [29]. In [29], the authors present the general framework and then enhance color images by treating them as surfaces in 5-D (three dimensions for the color and two for the space). Color artifacts are introduced in this formulation. In [10], the authors deal mainly with a discrete formulation of the problem, presenting a number of important contributions and relations with our work [33]. The authors of [10], [29] have also recently showed, inspired by [33] and this work, examples on the denoising of chroma plus brightness using their framework. See also [30] for an example of introducing perceptual color metrics in the framework introduced in [29]. Finally, we should note that Perona, in his pioneering work [24], also proposed an algorithm for diffusing directions, and applied it to the regularization of two-dimensional (2-D) image gradients (see also [19] and [36]). His algorithm is developed only in 2-D, and processes the direction by diffusing the angle the vector makes with a fixed coordinate system. Of course, in the case of color images, the directional vector is at least in 3-D, and this angle diffusion framework needs to be extended. See [33] for a detailed discussion of the advantages of the directional diffusion scheme here proposed over the angle diffusion one introduced by Perona. II. BRIGHTNESS DIFFUSION Let represent the color data (normally ), and and its magnitude (brightness) and unit direction (chromaticity) respectively. In other word and The basic idea in image enhancement via partial differential equations is to use as the initial datum of a given PDE. This PDE will produce, the enhanced image. As pointed out in the introduction, a number of a PDEs have been proposed in the literature for scalar image denoising. All of them attempt, in a different form, to remove the noise while preserving the image edges. This is in contrast with Gaussian filtering, which is equivalent to the Laplacian flow or heat equation given by which destroys the edges end thereby the image content. This is an isotropic flow, and then those preserving edges are commonly denoted as anisotropic flows. It is outside the scope of this paper to review the literature related to anisotropic image diffusion. We just proceed to present the PDE corresponding to the flow that will be used in this paper together with the novel directional diffusion framework introduced in the following section. In particular, in this paper we denoise the brightness via (subscripts denote derivatives) which corresponds to the affine invariant anisotropic flow introduced in [2], [28]. The basic idea here is that each level set of the brightness is deforming according to a curvature based affine invariant flow, while the whole diffusion is being stopped by the edge dependent term. It can be shown that basically this flow is smoothing the image in the direction parallel to the edges. See the mentioned references for more details on this classical filter or on any of the other proposed scalar anisotropic diffusion flows. 3 III. CHROMATICITY DIFFUSION Following the previous section, the (noisy) color vector has been separated in its brightness and its chromaticity. Let stand for each one of the components of. We search for a family of images, a multiscale representation, of the form 3 In addition to this flow, in Section IV we will also use a discrete 323 median filter to denoise the scalar brightness. (1) (2)

3 TANG et al.: COLOR IMAGE ENHANCEMENT VIA CHROMATICITY DIFFUSION 703, and once again we use to represent each one of the components of this family. Let us define the component gradient as, where and are the unit vectors in the and directions respectively. From this,, gives the absolute value of the component gradient. The component Laplacian is given by. We are also interested in the absolute value of the image gradient, given by The problem of harmonic maps in liquid crystals is formulated as the search for the solution to where stands for the image domain and. This variational formulation can be re-written as such that This is a particular case of the search for maps between Riemannian manifolds and which are critical points of the harmonic energy (3) (4) (5) (6) dvol (7) where is the length of the differential in. In our particular case, is a domain in and, and reduces to (3). The critical points of (7) are called p-harmonic maps (or simply harmonic maps for ). This is in analogy to the critical points of the Dirichlet energy for real valued functions, which are called harmonic functions. We are then looking, in analogy to the scalar case given by the harmonic functions, a smooth as possible map between the manifolds. (The resulting minimizing map is also a general geodesic.) In this paper we will concentrate on the particular form (4) of the harmonic energy. See [33] for image processing applications of the general form (7). In addition, in our application we are not (just) interested in the harmonic map between the domain in and (the critical point of the energy), which might even be a constant map, but are interested in the process of computing this map via partial differential equations. More specifically, we are interested in the gradient-descent type flow of the harmonic energy (4). This is partially motivated by the fact that diffusion equations for gray-valued images can be obtained as gradient descent flows acting on real-valued data; see, for example [5], [25], [26], and [41]. Isotropic diffusion, (1), is just the gradient descent of the norm of the image gradient, while anisotropic diffusion of the class of (2) can be interpreted as the gradient descent flow of more robust norms acting on the image gradient. In the following sections, we will present the gradient descent flows for our particular energy (4). We concentrate on the cases of, isotropic, and, anisotropic (or in general ). The use of corresponds to the classical heat flow from the linear scale-space theory [23], [39], while the case corresponds to the total variation flow studied in [26]. Most of the literature on harmonic maps deals with in (7) or (4), the linear case. Some more recent results are available for,, [11], [13], and very few results deal with the case [18]. A review of the theoretical results, both for the variational formulation and its corresponding gradient descent flow, which are relevant to the multiscale representation of directions in general and chromaticity in particular, is given in [33]. The papers [14] and [15] are an excellent source of information for regular harmonic maps, while [21] contains a comprehensive review of singularities of harmonic maps (check also [31], a classic on harmonic maps). A classical paper for harmonic maps in liquid crystals, that is, the particular case of (4) (or in general, being a domain in and ), is [7]. A. Isotropic Diffusion It is easy to show that for, the gradient descent flow corresponding to (5) with the constraint (6) is given by the set of coupled PDEs This system of coupled PDEs defines the isotropic multiscale representation of, which is used as initial data to solve (8). (Boundary conditions are also added in the case of finite domains.) The first part of (8) comes from the variational form, while the second one comes from the constraint (see for example [32]). As expected, the first part is decoupled between components, and linear, while the coupling and nonlinearity come from the constraint. Existence results for harmonic mappings were already reported in [16] for a particular selection of the target manifold. Struwe [31] showed, in one of the classical papers in the area, that for initial data with finite energy [as measured by (7)], a 2-D manifold with (manifold without boundary), and, there is a unique solution to the general gradient-descent flow. Moreover, this solution is regular with the exception of a finite number of isolated points and the harmonic energy is decreasing in time. If the initial energy is small, the solution is completely regular and converges to a constant value. (The results actually holds for any compact.) This uniqueness results was later extended to manifolds with smooth and for weak solutions [17]. Recapping, there is a unique weak solution to (8) [weak solutions defined in natural spaces, ], and the set of possible singularities is finite. These solutions decrease the harmonic energy. B. Anisotropic Diffusion The gradient descent flow corresponding to (5), in the range (and formally for ), with the constraint (6) is given by the set of coupled PDEs div This system of coupled PDEs defines the anisotropic multiscale representation of, which is used as initial datum to (8) (9)

4 704 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 5, MAY 2001 Fig. 1. Effects of chroma diffusion. See text for details. solve (8). In contrast with the isotropic case, now both terms in (9) are nonlinear and include coupled components. Formally, we can also explicitly write the case, being the analysis an interpretation of this case much more delicate. The case of in (7) has been less studied in the literature. When is a domain in, and, the function,, is a critical point of the energy for, for [this interval includes the energy case that leads to (9)], and for [21]. Of course, we are more interested in the results for the flow (9), and not just in its corresponding energy. Some results exist for,, showing in a number of cases the existence of local solutions which are not smooth. To the best of our knowledge, the case of, and in particular, has not been fully studied for the evolution equation, and this is part of our future plans. Following the framework for robust anisotropic diffusion introduced in [5], we can also generalize (4) and study problems of the form where is now a robust function like the Tukey biweight. (10) IV. EXAMPLES In this section we present a number of examples for the chromaticity flows for (isotropic) and (anisotropic). We also combine this with median filtering and anisotropic diffusion for the brightness. Although specialized numerical techniques to solve (4) and its corresponding gradient-descent flow, have been developed, e.g., [1], we can basically use the algorithms developed for isotropic and anisotropic diffusion without the unit norm constraint to implement (8) and (9) [12]. Although these equations preserve the unit norm, numerical errors might violate the constraint. Therefore, between every two steps of the numerical implementation of this equations we add a renormalization step [12]. Basically, a simple time-step iteration is used for (8), while for (9) we incorporate the edge capturing technique developed in [26] (we always use the maximal time step that ensures stability). All the diffusions are artificially stopped when the noise has been significantly removed. Adding constraints as those in [26] it is straightforward, and they will permit to run the flows until steady-state is achieved. Fig. 1 shows the effects of chromaticity diffusion. The first and second columns show the original image, noisy one with salt-and-pepper noise added to the chroma only, and the result of isotropic chroma diffusion after 21 iterations respectively

5 TANG et al.: COLOR IMAGE ENHANCEMENT VIA CHROMATICITY DIFFUSION 705 Fig. 2. Examples of our algorithm and comparison with discrete approaches. See text for details. (the brightness, vector magnitude, was not processed at all). Note how the proposed chroma diffusion removes the color noise while preserving the details in the image. The third column repeats this, but now the noise has been added to the full color image (original on the top). That is, both the chroma and brightness are noisy (middle row). To illustrate the effects of chroma diffusion alone, the bottom figure shows the results of the isotropic direction (chroma) flow, while the noisy magnitude (brightness) was kept without processing. In Fig. 2 we show examples of our algorithm and compare with the approach and [22], where the discrete vector directional and magnitude median filters are combined. In all the columns, originalisshownfirst,followedbythenoisyone,theresultof[22],and the result of our algorithm. In the first column we use Gaussian noise (second row), and the anisotropic chroma diffusion is combined with median filtering for the magnitude (last row). This is repeated in the second column. This time, scalar anisotropic diffusion for the brightness is combined with isotropic chroma dif-

6 706 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 5, MAY 2001 Fig. 3. Additional examples of our algorithm. See text for details. fusion, last row. The last column shows the results for noise obtained from JPEG compression. Once again, the chroma diffusion is combined with scalar anisotropic diffusion for the brightness in our scheme (last row). Note how the proposed algorithm outperforms [22], which by itself was shown in the literature to produce better results than other vector median filters. Additional examples of our algorithm are given in Fig. 3. In all the rows we show, from top to bottom, the original image, noisy one, and enhanced by the proposed algorithm. In the first and second columns, salt-and-pepper noise is used (second row), and the isotropic chroma diffusion flow is combined with median filtering for the brightness denoising. Gaussian noise is used in the third column. This time, scalar anisotropic diffusion is used for the brightness. V. CONCLUDING REMARKS In this paper we have introduced a new algorithm for denoising color images. The framework is based on using PDEs from the area of harmonic maps to smooth the chromaticity data, and then combine this with scalar anisotropic diffusion or median filtering for the brightness. Although all the examples in this paper were presented for 3-D color vectors, the theory applies to any dimension and can be used in any vectorial color representation. A number of questions remain open. First of all, we need to perform a complete analysis of the harmonic energy and gradient descent flow for, the anisotropic case. We need at least results concerning with the existence and uniqueness of (weak) solutions. In addition, it is interesting to study the optimal coupling between the chromaticity and brightness diffusion flows. The coupling of chromaticity and brightness was found helpful for other color image processing tasks like segmentation. We are interested in investigated the diffusion techniques here proposed for this as well. These questions are part of our current research. ACKNOWLEDGMENT G. Sapiro would like to thank Prof. R. Kohn from the Courant Institute, New York University, for encouraging him to think again about filtering vectorial images, Prof. K. Rubinstein from the Technion Israel Institute of Technology, for pointing out

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Sapiro, and D. H. Chung, Vector median filters, inf-sup operations, and coupled PDEs: Theoretical connections, J. Math. Imag. Vis., vol. 12, pp , Apr Bei Tang received the B.S. degree in computer science and engineering from Northwestern Polytechnical University, Xian, China, in 1994, and the M.Sc. degree in computer science from the University of Minnesota, Duluth, in 1997, and the M.Sc. degree in electrical engineering from University of Minnesota, Minneapolis, in Since August 1999, she has been a Research Engineer with the Multimedia Architecture Laboratory, Motorola, Inc. Her research interests include image and video processing, color science, computer vision and CMOS imaging. Guillermo Sapiro (M 95) was born in Montevideo, Uruguay, on April 3, He received the B.Sc. (summa cum laude), M.Sc., and Ph.D. from the Department of Electrical Engineering at the Technion Israel Institute of Technology, Haifa, in 1989, 1991, and 1993, respectively. After postdoctoral research at the Massachusetts Institute of Technology, Cambridge, he became 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, computer graphics, medical imaging, and image analysis. He recently co-edited a special issue of the Journal of Visual Communication and Image Representation. Dr. Sapiro recently co-edited a special issue of IEEE TRANSACTIONS ON IMAGE PROCESSING. 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, the Office of Naval Research Young Investigator Award in 1998, the Presidential Early Career Awards for Scientist and Engineers (PECASE) in 1988, and the National Science Foundation Career Award in Vicent Caselles, photograph and biography not available at time of publication.