Morphological Algorithms for Color Images Based on a Generic-programming Approach

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1 Morpholoical Alorithms for Color Imaes Based on a Generic-prorammin Approach MARCOS CORDEIRO D ORNELLAS 1, REIN VAN DEN BOOMGAARD 1, JAN-MARK GEUSEBROEK 1 1 ISIS - Intellient Sensory Information Systems University of Amsterdam, Faculty WINS Kruislaan, SJ Amsterdam, The Netherlands fornellas,rein,mark@wins.uva.nl Abstract. The main purpose of imae processin and analysis is, often, to sement the imae into objects in order to analyze the eometrical properties and the structure of the objects and reconize them. The analysis of the eometric objects must be quantitative, because only such an analysis and description of the eometric objects can provide a coherent mathematical framework for describin the spatial oranization. The quantitative description of eometrical structures is the objective of mathematical morpholoy. So far, the use of such framework has allowed the development of a class of alorithms to deal with binary and rayscale imaes. Multi-channel processin has been of rowin interest recently, especially in color imae processin. In this paper, the extension of mathematical morpholoy to color imaes by treatin multi-channel data as vectors are presented. An overview of several approaches to this extension usin different orderin relations is also provided. Basic vector morpholoical alorithms are introduced based on reduced orderin. We also present experimental results and show that these alorithms can be easily mapped into the enericprorammin framework. Keywords: Multivalued morpholoy, Color imae processin, Generic-prorammin, Orderin. 1 Introduction Mathematical morpholoy is a powerful and unified approach for eometrical shape analysis and description based on set theoretic notions such as inclusion, union and intersection. Morpholoical operations require the selection of a probe called a structurin element. An imae is analyzed by inspectin its interaction with the structurin element as this is moved around the imae plane. Not only the choice of the structurin element but also the operation bein performed with this structurin element determines the effect of a morpholoical operator. Imae analysis usin morpholoy usually consists of two steps: a transformation step, in which morpholoical operations are performed on an imae, extractin some features from the imae, and a measurement step, in which certain eometrical characteristics of the transformed imae are measured as presented in Serra [19], and Haralick et all [11]. Initially, mathematical morpholoy was constructed for binary imaes, which can be represented mathematically as sets. When applied to a binary imae, a morpholoical operation (dilation or erosion) produces an increase or decrease in spatial extent, respectively. These operations are based on the eometrical relationships or connectivities from pixels of the same type. Binary mor- Formerly at Universidade Federal de Santa Maria, Departamento de Eletrônica e Computação, Santa Maria-RS, Brazil. Supported by CAPES Foundation under rant BEX 2780/95-0. pholoical operations are naturally expanded to rayscale morpholoy by usin an infimum or supremum operation, as defined in Heijmans [13] [14]. Several techniques developed for use with rayscale imaes can be extended to color imaes by applyin the alorithm to each one of the imae components separately. The most straihtforward way to process multichannel imaes is to treat each channel independently usin sinle-channel techniques. However this approach does not exploit the correlation between channels. Multichannel techniques that take into account this correlation have been reported to be more effective. These alorithms are based on the concept of rankin multivariate data introduced by Barnett [2]. In this paper we describe some approaches for applyin mathematical morpholoy to color imaes. We present the concept of vector morpholoical alorithms, which are based on vector rankin concepts proposed by Goutsias et all. [9], Serra [20], and Comer and Delp [5]. We compare the performance of some vector morpholoical alorithms usin reduced orderin and show that these alorithms can be straihtforwardly mapped into the eneric-prorammin framework suested by Ornellas and Boomaard [7]. The oranization of the rest of this paper is as follows. In section 2, we ive a brief introduction to mathematical morpholoy and complete lattice theory. In sec- Anais do XI SIBGRAPI (1998) 1 8

2 2 D ORNELLAS, M.C. AND BOOMGAARD, R. V.D. AND GEUSEBROEK, J. tion 3 we describe the morpholoical development framework. We also introduce the eneric-prorammin approach and how to represent color imaes as well as their color models. In section 4 we present the concept of multivalued orderin applied to color imaes. We discuss alternative approaches and discuss their advantaes and drawbacks. Section 5 refers to color morpholoy and their operators. The evaluation performance and alorithm results are shown in section 6. Conclusion and further research are iven in section 7. 2 Mathematical Morpholoy and Complete Lattice Theory One of the basic ideas in mathematical morpholoy is that the set of all possible imaes constitutes a complete lattice. The followin definition will be used in this paper: Definition 2.1 (Complete Lattice) A set L with a partial orderin is called complete lattice if every subset P of L has a least upper bound (supremum) _H and a reatest lower bound ^P. Let L,M be two lattices. The set of all operators mappin L into M forms a complete lattice under the partial orderin iven by:, (X) (X); for all X 2 L (1) The set L is anti-symmetrical, reflexive, and transitive. The concept of an infimum and a supremum stems from the partial orderin relation. If the supremum and infimum exist for any collection of imaes taken from the lattice of all imaes, then that lattice is called a complete lattice as presented in Heijmans and Ronse [15]. An imae filter is considered as a mappin performed within a complete lattice; the filter maps one lattice element (the input imae) to another lattice element (the output imae). The key insiht provided by mathematical morpholoy is to construct these imae filters from what are called erosions and dilations. Erosions and dilations have proved to be very powerful and can be combined to yield effective imae filters with desirable filterin properties. From a theoretical point of view, dilation, erosion, adjunction, openin, and closin are the most important alebraic notions in mathematical morpholoy to characterize operators on lattices as defined as follows: Definition 2.2 (Dilation) An operator : L! M is called a dilation if (_ i2i X i ) = _ i2i (X i ), for every collection fx i ji 2 I. Definition 2.3 (Erosion) An operator : L! M is called an erosion if (^i2i X i ) = ^i2i (X i ), for every collection fx i ji 2 I. Definition 2.4 (Adjunction) Let : L! M and : M! L be two operators. We say that the pair (; ) is an adjunction between L and M if (Y ) X $ Y (X), for all X 2 L; Y 2 M. refers to erosion while refers to dilation. Definition 2.5 (Openin and Closin) The operators and form an openin on L and a closin on M, respectively. Grayscale morpholoical operations are an extension of binary morpholoical operations and serve as a basis to expand mathematical morpholoy to color imaes. For rayscale operations, the imae will be represented by the function f (x; y), where (x; y) 2 R 2 or Z 2, or simply f, and the structurin element will be the function (r; s) or. Grayscale dilation, erosion, openin, and closin are defined as follows: (f )(x; y) = sup (r;s)2gff (x? r; y? s) + (r; s) (2) (f )(x; y) = inf (r;s)2gff (x + r; y + s)? (r; s) (3) f = (f ) (4) f = (f ) (5) where sup and inf denote the supremum and infimum operators, respectively, and G R 2 or Z 2 is the support of (x; y). A special class of rayscale morpholoical filters when (x; y) = 0 for every (x; y) 2 G are referred to flat morpholoical operators as proposed by Crespo and Schafer [6]. 3 Morpholoical Development Framework 3.1 Generic-prorammin Approach The eneric-prorammin paradim within the C++ context was first proposed by Stepanov and Musser [21] and became popular with the inclusion of their Standard Template Library (STL) into the C++ standard. In C++, the template mechanism is used to build eneric alorithms by means of compile-time polymorphism. Generic-prorammin represents efficient alorithms independently of any particular data structure and facilitates the development of more abstract alorithms. Moreover, all eneric data structures brin the most advanced type of interface they can implement efficiently. In this way, an unique interface between morpholoical alorithms and the data structures can be defined so that the same alorithm implementation can be applied to any number of different imae types (e.. binary, rayscale, RGB color, HSI color, etc.). The intention is to assure that one can have alorithms defined as enerically as possible without loosin efficiency as proposed by Ornellas and Boomaard [7]. 3.2 Imae Representation and Color Models Color is a visual object attribute that results from the combined output of three sets of retinal cones each sensitive to

3 MORPHOLOGICAL ALGORITHMS FOR COLOR IMAGES BASED ON A GENERIC-PROGRAMMING APPROACH 3 different portions of the visible part of the electromanetic spectrum. A lot of psychophysioloy studies have shown that color information is enerally tri-variant. The cones have peak sensitivities in the red, reen and blue portions of the spectrum respectively. Any perceived color may usually be created by a variety of sets of colors when combined in the correct proportions as reported by Travis [22]. So, three variables are necessary to describe the color sensation. Serra [20] and Goutsias et all introduced the definition for a multivalued imae in the scope of mathematical morpholoy. [9] as follows: Definition 3.1 (Multivalued Imae) Given n totally ordered complete lattices T 1 ; : : : ; T n of product T; T = T 1 T 2 : : : T n, a multivalued imae is a mappin f : E! T, where E is a multidimensional diital space Z m. A color model is a means of representin the three components of a color in terms of a position in a three dimensional space. RGB, HSI, and Lab are some of the color models which miht be used, dependin on the particular purpose of the analysis The RGB Color Model Most video monitors use red, reen, and blue color sources as the primary colors for color imae eneration. RGB color model uses a rectanular coordinate system with one coordinate axis assined to each of three color components, red, reen, and blue. The RGB system is the classical physical color model used to diitize color imaes and many alorithms have been developed in the RGB space such as filterin, ede detection or reion sementation. Usin the RGB color model insures that no distortion of the initial information is introduced as suested by Golland and Bruckstein [8]. However, RGB representation is far from the human concept of color. In the RGB space, color features are hihly correlated, and it is impossible to evaluate the similarity of two colors from their distance in this space The HSI Color Model The HSI color model is cylindrical with the intensity axis coincidin with the achromatic diaonal of the RGB system. Saturation is the radius from the intensity axis and hue is the anle with respect to the red direction. From a perceptual point of view, color can be described in HSI color model by three attributes. The Hue H is a value which represents the main color of the pixel in the RGB triplet; the saturation S describes the pureness of the color, and I represents the amount of liht received by the sensor. It depends on the lihtin conditions and on the liht source emissivity. Accordin to Carron and Lambert [3], HSI is appealin color system for sementation because it separates the intensity information from the chromatic information RGB versus HSI Color Model If we consider HSI color model, as lon as the colors in the imae are reasonably well saturated hue will tend to remain relatively constant in the presence of shadows and other lihtin variations. In such cases an imae based on hue alone may work better than traditional ray scale analysis. However, low saturation hue may be difficult to determine accurately and hence irrelevant. If saturation is hih, Hue is then very relevant. When saturation is zero hue is undefined. Its sensitivity to the imae noise is even lower than that of intensity. Serra [20] also commented that by chanin the system of coordinates from a vector space (e.. RGB) into a sort of a spherical one, we find that the HSI representation, which is also a complete lattice. Nevertheless, it is partly discrete and partially continuous. For applications that must be able to differentiate all colors, saturated and unsaturated, HSI representation can introduce sinificant problems. Also, McConnell [17] suested that for sementation and classification of objects, which may be multicolored, the disadvantaes of HSI space will almost always outweih any possible advantaes. In this paper, without loss of enerality, we will restrict ourselves to the RGB color model. 4 Multivalued Orderin Applied to Color Imaes The problem of orderin multivariate data is not straihtforward to mathematical morpholoy. Althouh that there is no universally accepted multivariate orderin scheme for total orderin of multivariate samples, much work has been done to define concepts such as median, rane, and extremes in multivariate analysis. Barnett [2] proposed the classification of the orderin principles in four roups: marinal orderin, reduced orderin, partial orderin, and conditional orderin. The key point to extend mathematical morpholoy to color imaes is to define a well-suited orderin relation. In mathematical morpholoy, we consider an imae filter as a mappin performed within a complete lattice framework as proposed by Ronse [18]. Any morpholoical operators we apply to the color imaes can be applied to the components separately, due to the fact that these filters commute with infimum and supremum respectively. This kind of marinal processin is equivalent to the vectorial approach defined by the canonic lattice structure when only supremum and infimum operators and their compositions are involved and induces a totally ordered lattice as presented by Gu [10] and Goutsias et all. [9]:

4 4 D ORNELLAS, M.C. AND BOOMGAARD, R. V.D. AND GEUSEBROEK, J. X Y, X(i) Y (i); 8i 2 1; : : : ; M (6) With these relations, the supremum of a family fx j is the vector _X where each component _X(i) is the supremum of the fx j (i). Respectively, the infimum of a family fx j is the vector ^X where each component ^X(i) is the infimum of the fx j (i). However, this morpholoical procedure fails because every color can be seen as a vector in a spatial domain, which can not be ordered and so the supremum or infimum of the two is a mixture of both the colors. Besides, imae components are hihly correlated. Usin this procedure, the lattice structure ives the same results as the simple marinal processin of the data and new colors not contained in the input imae will appear even for flat structurin elements. It is indeed possible to process the imae by usin only the most sinificative component but this method ives too much emphasis to the selected component (e.. intensity in the HSI model). Moreover, the minimum morpholoical theoretical backround is not respected anymore since a fundamental constraint has been reduced. Other techniques based on reduced orderin consist in considerin any distance metric between two vectors X and Y as explained by Comer and Delp [5]. The output of the vector filter will depend not only on the input imae and the structurin element, but also on the scalar-valued function used to perform the reduced orderin. The selection of specific distance metric is very important when local statistics are taken into account as is the Mahalanobis distance [(x i? a) T??1 (x i? a)] 1=2 (? bein the covariance matrix of the input data) suested by Hardie and Arce [12]. The arithmetic mean, the marinal or vector median can be used as a reference vector a as proposed by Astola et all. [1]. Goutsias et all. [9] introduced another approach rounded on the use of a vector transformation from R M into R Q followed by a marinal orderin on R Q. If Q > 1, the marinal orderin induces a partial orderin on the vectors. Q = 1 is required to obtain a total orderin with an h-adjunction defined as follows: Definition 4.1 (h-adjunctions) Assume that R is a complete lattice and that T is a nonempty set. Furthermore, let h : T! R be a surjective mappin. Define an equivalence relation =h and h on T as follows: t =h t 0, h(t) = h(t 0 ); t; t 0 2 T (7) t h t 0, h(t) h(t 0 ); t; t 0 2 T (8) Let ; : T! T be two mappins with the property that for St 2 T : The key point is to transform the imae data by means of a surjective transformation h. The underlyin assumption is that the transformed imae data is better suited for the morpholoical approach but typically, h is neither bijective nor injective: two vectors can have the same imae h. This stands in need of a new correspondin equivalence relation =h. A major drawback in practice is that the extrema of a family fx j are not necessarily unique. As a consequence, many different vectors can lead to the same result h(x) = sup i fx i. The concept of anamorphoses and conditional lattices were introduced by Serra [20]. In this way, a color imae should be morpholoically processed, by introducin priorities in processin its individual components. This idea involves functional relationships whether we keep or not the symmetry amon the components. Recently, Chanussot and Lambert [4] introduced the bit-mixin paradim, followin the idea of h-adjunctions. A mappin h : R M! R is used to rank the vectors that means that each vector pixel is represented by a sinle scalar value. The main idea is that a bijective mappin is used, inducin a total orderin and determines clearly the infimum and supremum of each set of vectors. With the bit-mixin representation, it is possible to perform any classical morpholoical filter on the coded imae, and to decode the result afterwards to et the color filtered imae. The proposed order can be considered as an extension to a total order. Two vectors are always comparable. Moreover, the output vector of any morpholoical filter will necessarily be one of the input vectors. 5 Color Morpholoy 5.1 Orderin Color as Vectors To extend the vector approach to color imaes, it is necessary to define an order relation which orders the colors as vectors, rather than orderin the individual components as suested by Jones and Talbot [16]. This can be done usin reduced orderin. This kind of orderin imposes a total orderin relationship that can be accomplished by the lexicoraphical orderin 1 as reported by Woods [23]. The structurin element for the vector morpholoical operations defined here is the set G, and the scalar-valued function used for the reduced orderin is h : R 3! R. The operation of vector dilation is represented by the symbol v. The value of the vector dilation of f by G at the point (x; y) is defined as: and (f v G)(x; y) 2 ff (r; s) : (r; s) 2 G(x;y) (10) h((f v G)(x; y)) h(f (r; s))8(r; s) 2 G(x;y) (11) (s) h t, s h (t) (9) then the pair (; ) is called an h-adjunction. 1 An ordered pair (i; j) is lexicoraphically earlier than (i 0 ; j 0 ) if either i i 0 or i = i 0 and j j 0. It is lexicoraphic because it corresponds to the dictionary orderin of two-letter words.

5 MORPHOLOGICAL ALGORITHMS FOR COLOR IMAGES BASED ON A GENERIC-PROGRAMMING APPROACH 5 Similarly, vector erosion is represented by the symbol v, and the value of the vector erosion of f by G at the point (x; y) is defined as: and (f v G)(x; y) 2 ff (r; s) : (r; s) 2 G(x;y) (12) h((f v G)(x; y)) h(f (r; s))8(r; s) 2 G(x;y) (13) Vector openin is defined as the sequence of vector erosion and vector dilation, and vector closin is defined as the sequence of vector dilation and vector erosion. With the above definitions for vector morpholoical operations we must impose the restriction that the set G is a finite set, because if G is not finite then it is possible that no value of satisfies the above equations. It is not difficult to see that usin erosion and dilation in conjunction with a total orderin induced by a reduced orderin will enerate no new colors. Under the total orderin relation, the infimum or supremum will be one of the actual colors (and not a mix of all the colors). Therefore, the only colors in the output imae will be those obtained from translations of the input colors. A similar result holds for combinations of erosions and dilations such as openins and closins. 5.2 Scalar-valued Functions and Alorithm Representation Since the output of the vector filter depends on the scalarvalued function used for reduced orderin, the selection of this function provides flexibility in incorporatin spectral information into the multivalued imae representation. For example, certain linear combinations of the tristimulus values can be used. This can be written as: ( h(t) = a 1t1 + a2t2 + a3t3 t t 0 $ h(t 0 ) = a1t a 2t a 3t 0 3 (14) h(t) h(t 0 ) if the imae is filtered in the RGB color model. For the case where a 1 = 0:299; a 2 = 0:587, and a 3 = 0:114, h(t) and h(t 0 ) become the luminance component. Multiscale openin in this case would suppress briht objects at each scale. The values of a 1 ; a 2, and a 3 can also be selected to enhance or suppress specific colors. For example, if a 1 = 1; a 2 = 0, and a 3 = 0, then the effect of a multiscale openin would be to suppress objects with hih red content. The same holds to the reen and blue color channels. If we choose a runnin maximum approach, we start with an arbitrary pixel inside the mask and call that one the maximum. Then we compare the next pixels with this maximum. As soon as we found a pixel, which has a bier rank, we assin that pixel as maximum and continue until all pixels are examined within the filter mask. Note that the difference between the red component and the maximum of the other two channels is taken for rankin. This means that the purity of red determines the rankin order. For instance, if we have a yellow pixel [255; 255; 0], the rankin order for it is 0. This can be represented as: ( h(t) = a 1t1? MAX(a2t2; a3t3) t t 0 $ h(t 0 ) = a1t 0 1? MAX(a 2t 0 2 ; a 3t 0 3 ) h(t) h(t 0 ) (15) Serra [20] suested that when linear relationships are employed the partial order relationship associated with a conditional lattice should be iven for the case of three components by (symmetrical and not symmetrical respectively): t t 0 $ ( t 1 t 0 1 t2 t 0 2 t3 t k 1(t 0 1? t 1) + k2(t 0 2? t 2) t t 0 $ ( t 1 t 0 1 t2 t k 0(t 0 1? t 1 t3 t k 1(t 0 1? t 1) + k2(t 0 2? t 2) (16) (17) where k 0 ; k 1, and k 2 are constants. When the bit-mix paradim proposed by Chanussot and Lambert [4] is used, the transform h is based on the representation of each component of T in the binary mode. Let T 2 R M with M components t(i), each one represented on p bits t(i)j 2 f0; 1 with j 2 f0; : : : ; p. The considered mappin h can then be written as follows: t t 0 $ 8 < : p h(t) = Pj=1 h(t 0 p ) = j=1 h(t) h(t 0 ) 2 M:(p?j) P M i=1 2M?i t(i)j 2 M:(p?j) P M i=1 2M?i t 0 (i)j (18) All these scalar-valued functions lead to a family of imaes, parameterized by shape, size, and color, which could be useful for an application such as object reconition. Nevertheless the differences in the values of a 1 ; a 2, and a 3 in some scalar-valued functions stronly influences the resultant representation. Generic vector morpholoical alorithms for dilation and erosion, usin the write formalism are presented in fiure 1 and fiure 2. In our implementations, every object has its pixel s address and a set of keys (t 1 ; t 2 ; t 3 ) that are sorted accordin to one of the scalar valued functions proposed. 6 Experimental Results The result of the dilation alorithm applied to RGB color model usin a flat 5 5 mask can be seen in fiure 3. Fiure 3(a) represents a test imae from the tulip fields in the Keukenhof Gardens. Let us say, for instance, that our main objective is to enlare the red features in the imae. Fiure 3(b) is the result obtained by applyin the bitmix approach to the oriinal imae. We can see from this picture that red, pink, and yellow fields

6 6 D ORNELLAS, M.C. AND BOOMGAARD, R. V.D. AND GEUSEBROEK, J. Input Imae: Color Imae f Input Imae: Structurin Element Output Imae: Color Imae f int write dilation(f, f, ) f for (every object p 2 Df ) f f (p) =?1; for (every object q 2 ) f f (p - q) = max(h(f (p - q)), h(f(p) + (q))); Fiure 1: Dilation Alorithm - Grayscale Imaes (a) (b) Input Imae: Color Imae f Input Imae: Structurin Element Output Imae: Color Imae f int write erosion(f, f, ) f for (every object p 2 Df ) f f (p) = +1; for (every object q 2 ) f f (p + q) = min(h(f (p + q)), h(f(p) - (q))); Fiure 2: Erosion Alorithm - Grayscale Imaes were enlared. In fiure 3(c), the result obtained by the maximum approach is shown. The red and pink fields are larer since the red channel was chosen as the maximum value to the runnin maximum approach. Fiure 3(d) shows the result when the symmetric conditional lattice approach was used. Note that the bitmix approach is stroner than the conditional lattice. Also, all the output imaes became brihter than the oriinal one. Fiure 4 shows the result of the erosion alorithm applied to RGB color model usin a flat 5 5 mask. Fiure 4(a) represents the previous test imae. Conversely, let us say now that our oal is to reduce the red features inside the imae. Fiure 4(b) is the result obtained by applyin the bitmix approach. It is easy to see that the red, pink, and yellow fields were reduced. In fiure 4(c), the result obtained by the runnin maximum approach is shown. The red pink and yellow fields were reduced and the blue fields were enlared thanks to the influence from the red channel chosen for rankin. Fiure 4(d) shows the result when the symmetric conditional lattice approach is used. Aain, It should be noted that the bitmix approach is more robust than the conditional lattice, producin more reduction in the red, pink, and yellow fields. In addition those output imaes became darker than the test imae. The result of an openin and closin alorithm applied to RGB color model usin a flat 5 5 can be seen (c) (d) Fiure 3: RGB dilation: input imae(a), bitmix(b), maximum(c), and symmetry(d). in fiures 5 and 6. Fiures 5(a) and 6(a) represent a face imae and fiures 5(b) and 6(b) represent the same imae, corrupted by an uniform noise. Fiure 5(c) 6(c) are the results obtained by applyin the bitmix approach. In fiure 5(d) and 6(d), the result obtained by the maximum approach is shown. Fiures 5(e) and 6(e) show the results when the symmetric conditional lattice approach is used and fiures 5(f) and 6(f) represent the result obtained by the luminance orderin 2. 7 Conclusion In this paper, after a brief introduction of mathematical morpholoy and complete lattice theory, a enericprorammin approach where morpholoical alorithms can be easily extended to color imaes has been described. An overview of various multivalued orderin schemes in the literature has been iven. The advantaes as well as drawbacks of usin these techniques have been hihlihted. Based on such procedures, the concept of color morpholoy has been introduced and vector morpholoical alorithms have been implemented. By usin vector erosions and dilations in conjunction with a total orderin induced by a reduced orderin we 2 Due to the prohibitive cost of color printin and the inherent distortions associated with the size reduction and the printin process, the correspondin color imaes will be made available throuh

7 M ORPHOLOGICAL A LGORITHMS FOR C OLOR I MAGES B ASED ON A G ENERIC - PROGRAMMING A PPROACH (a) (b) (a) (b) (c) (d) (c) (d) (e) (f) 7 Fiure 4: RGB Erosion: input imae(a), bitmix(b), maximum(c), and symmetry(d). will enerate no new colors. For example, an infimum or supremum of various translated colors ives the result of an erosion or dilation at any iven point in the output imae. Under the orderin relation, these colors are totally ordered so that infimum or supremum will be one of the actual colors. A similar result holds for combinations of erosions and dilations such as openins and closins. From the experimental results we found that the runnin maximum approach performs better than the other scalar-valued functions when used in conjunction with dilations and erosions. However, the morpholoical openin an closin based on scalar-valued functions like the bitmix paradim and the symmetrical conditional lattice ive better results than the other proposed functions. References [1] J. Astola, P. Haavisto, and Y. Neuvo. Vector median filters. Proceedins IEEE, 78: , [2] V. Barnett. The orderin of multivariate data. Journal of Royal Statistical Society A, 3: , Fiure 5: RGB Openin: input imae(a), noisy imae(b), bitmix(c), maximum(d), symmetry(e), and luminance(f). oy. Proceedins of the International Symposium on Mathematical Morpholoy (ISMM 98), paes 51 58, [5] M. Comer and E. Delp. An empirical study of morpholoical operators in color imae enhancement. Proceedins of the SPIE Conference on Imae Processin Alorithms and Techniques III, February 10-13, 1992, San Jose, California, 1657: , [3] T. Carron and P. Lambert. Color ede detection usin jointly hue, saturation, and intensity. IEEE 1st International Conference on Imae Processin (ICIP 94), paes , [6] J. Crespo and R. W. Schafer. The flat zone approach and color imaes. In Mathematical Morpholoy and Its Applications to Imae and Sinal Processin, Dordrecht, Kluwer Academic Publishers. [4] J. Chanussot and P. Lambert. Total orderin based on space fillin curves for multivalued morphol- [7] M. C. d Ornellas and R. v.d. Boomaard. Generic alorithms for morpholoical imae operators - a case

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