FUZZY WATERSHED FOR IMAGE SEGMENTATION Ramón Moreno, Manuel Graña Computational Intelligene Group, Universidad del País Vaso, Spain http://www.ehu.es/winto; {ramon.moreno,manuel.grana}@ehu.es Abstrat The representation of the RGB olor spae points in spherial oordinates allows to retain the hromati omponents of image pixel olors, pulling apart easily the intensity omponent. This representation allows the definition of a hromati distane. Using this distane we define a fuzzy gradient with good properties of pereptual olor onstany. In this paper we present a watershed based image segmentation method using using this fuzzy gradient. Oversegmentation is orreted applying a region merging strategy based on the hromati distane defined on the spherial oordinate representation. Keywords Refletion model, image segmentation, spherial oordinates, hromatiity 1 Introdution Color images have additional information over graysale images that may allow the development of robust segmentation proesses. There have been works using alternative olor spaes with better separation of the hromati omponents like HSI, HSL, HSV, Lab [1, 2] to obtain pereptually orret image segmentation. However, hromatiity s illumination an blur and distort olor patterns. The segmentation method proposed in this paper has inherent olor onstany due to the olor representation hosen and the definition of the hromati distane. In this paper we use the RGB spherial oordinates representation to ahieve image segmentation showing olor onstany properties [3 5]. We define a hromati distane on this representation. The robustness and olor onstany of the approah is grounded in the dihromati refletion model (DRM) [6]. We propose a hromati gradient operator suitable for the definition of a watershed transformation on olor images and a robust region merging for meaningful olor image segmentation. The baseline hromati gradient operator introdued in [3, 7] is very sensitive to noise in the dark areas of the image. We propose in this paper a fuzzy gradient operator overoming this problem whih is useful to build a watershed transformation on olor images. To ahieve a natural segmentation, we perform region merging on the basis of our proposed hromati distane over the hromati haraterization of the watershed regions. We give a general shema that ombines watershed flooding with region merging in a single proess. Finally, we speify our proposal as an instane of the aforementioned general shema. The paper outline is as follows. Setion 2 presents the RGB spherial interpretation, inluding the definition of the hromati distane. Setion 3 presents the fuzzy watershed image segmentation method. Setion 4 shows experimental results omparing our method to other approahes. Finally, Setion 5 gives the onlusions of this work. 2 Spherial oordinates in the RGB Color Spae There are several works that use a spherial/ylindrial representation of RGB olor spae points looking for photometri invariants [8, 9]. We are interested in the orrespondene between the angular parameters (θ, φ) of the spherial representation of a olor point in the RGB spae and its hromatiity Ψ. 2.1 Chromatiity interpretation of RGB Spherial Coordinates An image pixel s olor orresponds to a point in the RGB olor spae = {R, G, B b }. Chromatiity Ψ of a RGB olor R point is given by two of its normalized oordinates r = G R +G +B, g = B R +G +B, b = R +G +B, that fulfill the ondition r + g + b = 1. That is, hromatiity oordinates Ψ = {r, g, b} orrespond to the projetion of on the hromati plane Π Ψ, whih is defined by the olletion of verties {(1, 0, 0), (0, 1, 0), (0, 0, 1)} of the RGB ube whih define the Maxwell s hromati triangle [10], along the hromati line defined as L = {y = k Ψ ; k R}. In other words, all the points in line L have the same hromatiity Ψ. Chromatiity is a robust olor haraterization, whih is is independent of illumination intensity and preserves the geometry of the objets in the sene. Denoting a olor image in RGB spae as I = { I (x) ; x N 2} = { (R, G, B) x ; x N 2}, where x refers to the pixel oordinates in the image grid domain, we denote the orresponding spherial representation as P = { { P (x) ; x N 2} = (φ, θ, l)x ; x N 2}, so we will use (φ, θ) x as the hromatiity representation of the pixel s olor. Sometimes we use the notation x = (i, j). 1
Figure 1: A vetor in the RGB spae 2.2 The hromati distane in the RGB spae First, we onvert the RGB artesian oordinates of eah pixel to polar oordinates, with the the blak olor as the RGB spae origin. Let us denote the artesian oordinate image as I (x) = I x = (r, g, b) x ; x N 2 and the spherial oordinate as P (x) = P x = (φ, θ, l) x ; x N 2, where p denotes the pixel position. For the remaining of the paper we disard the magnitude l x beause it does not ontain hromati information, therefore P (x) = P x = (φ, θ) x ; x N 2. For a pair of image pixels x and y, the olor distane between them is defined as: (P x, P y ) = (θ y θ x ) 2 + (φ y φ x ) 2, (1) that is, the olor distane orresponds to the eulidean distane of the Azimuth and Zenith angles of the pixel s RGB olor polar representation. This distane does not take into aount the intensity omponent and, thus, will be robust against speular surfae refletions. It posses olor onstany, beause pairs of pixels in the same refletane surfae will have the same distane regardless of their intensity. 2.3 Chromati gradient operator We formulate a pair of Prewitt-like gradient pseudo-onvolution operations on the basis of the above distane. Note that the (P p, P q ) distane is always positive. Note also that the proess is non linear, so we an not express it by onvolution kernels. The row pseudo-onvolution is defined as CG R (P (i, j)) = and the olumn pseudo-onvolution is defined as CG C (P (i, j)) = 1 r= 1 (P (i r, j + 1), P (i r, j 1)), 1 (P (i + 1, j ), P (i 1, j )), = 1 so that the olor distane between pixels substitutes the intensity subtration of the Prewitt linear operator. The olor gradient image is omputed as: CG(x) = CG R (x) + CG C (x) (2) 2.4 A Fuzzy Color Gradient Inspired in the human vision, diferent retinal ells need diferent energy for its ativation. Cones are vey sensitive to intensity wheras rods need more energy for its ativation, whose an detet the romatiity. Depending of the sourfae radiane, these retinal ells an be ativated. We follow similar approah in order to define our fuzzy hromati distane. For pixels with high intensity we are going to use the hromati gradient defined in equation 2, whereas for pixels with poor illumination are going to use the onventional linear intensity gradient. We use a fuzzy membership funtion α (x) who gives the membership degree to the well illuminated pixels lass, on the other hand, its standard omplement funtion α(x) = 1 α(x) gives the membership degree to the poor illuminated pixels lass. It is defined as follows α(x) = ( exp 0 ) ( I(x) a b a )2 2σ 2 1 2 I(x) < a a I(x) < b b I(x), (3)
where I(x) is the pixel intensity. For intensity values below a threshold a it is an intensity gradient, for values above another threshold b it is a hromati gradient, and for values between both it is a mixture of the two kinds of gradients whose mixing oeffiient is gaussian funtion of the image intensity. This idea is expressed mathematially as a onvex ombination of the two gradient operators: F G(x) = α (x) CG (x) + α (x) G (x) (4) where x is the pixel loation, G(x) is the intensity gradient, alulated using the onvolution mask of Eq.2 but using only the intesity parameter l. This fuzzy gradient does not suffer from noise sensitivity in dark regions of the image, the effet of bright spots is redued beause it is hromatially onsistent in bright image regions, and it detets hromati edges. 3 Fuzzy Watershed Image Segmentation Watershed transformation is a powerful mathematial morphology tehnique for image segmentation. It was introdued in image analysis by Beuher and Lantuejoul [11]. The watershed transform onsiders a bi-dimensional image as a topographi relief map. The value of a pixel is interpreted as its elevation. The watershed lines divide the image into athment basins, so that eah basin is assoiated with one loal minimum in the topographi relief map. The watershed transformation works on the spatial gradient magnitude funtion of the image. The rest lines in the gradient magnitude image orrespond to the edges of image objets. Therefore, the watershed transformation partitions the image into meaningful regions aording to the gradient rest lines. We will use the fuzzy gradient as the topographi relief map. 3.1 General Shema The general shema of the watershed method onsists of a flooding proess whih performs a region growing based on the ordered examination of the level sets of the gradient image. In fat, an ordered suession of thresholds are applied to produe the progression of the flooding. The image is examined iteratively n times, eah iteration step the threshold is raised and pixels of the gradient image falling below the new threshold are examined to be labeled with a orresponding region. Initially eah region will ontain the soure of its athment basin when the flooding level reahes it. Eah flooded region is also haraterized by a hromatiity value, that orresponds to the soure pixel hromatiity. This hromatiity value is used to perform region merging simultaneously with the flooding proess. A pixel whose neighboring pixels belong to different regions is a watershed pixel. When a watershed pixel is deteted, the adjaent regions may be merged into one if the hromati distane between the region hromati values is below a hromati threshold. The merged region hromati value is the average of that of the merged regions. The final labeling of the image regions is performed taking into aount the equivalenes established by the merging proess. Watershed pixels whose adjaent regions do not merge into one are labeled as region boundary pixels and retain their hromatiity. 4 Experimental results The general watershed-merge method is parametrized by: The number of iterations n, whih determines the resolution of the flooding proess going over the gradient magnitude image level sets. The gradient operator used to ompute the gradient magnitude image, whih an be either the intensity gradient G (x) of equation or the fuzzy gradient F G (x) of equation 4. The olor representation of the image. Assuming the RGB spae, it an be either the Cartesian representation I (x) or the zenithal and azimuthal angles of the spherial representation P (x). This seletion determines the seletion of the hromati distane. The hromati distane, whih an be either the Eulidean distane in the RGB Cartesian spae, or the hromati distane of equation 1. The hromati distane threshold δ, whih determines the hromati resolution of the region merging proess. We will use a well known benhmark image [12] to ompare our proposed segmentation proess with variations of method obtained with other parameter settings. The dark regions are ritial to the pereptually orret gradient omputation, while the bright spots may indue false edge detetion. The method does not ompute any speular free image to remove this latter problem. The operational parameter settings are n = 100 and δ = 0.1. In figure2 we show the segmentation results on this image for all ombinations of the remaining parameter settings. The olumn of images labeled Gradient has the gradient magnitude images. From top to bottom, figures 2(a), 2(e), 2(i) show, respetively the result of the intensity gradient, the hromati gradient of equation (2), and the fuzzy gradient of equation (4). The olumn of images labeled Watershed orrespond to the image region partition performing only to the flooding proess, without any region merging, on the orresponding gradient magnitude images. 3
Original image Gradient Watershed Segmentation (a) (b) () (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 2: Image segmentation results with different parametrizations 4
It an be appreiated that the fuzzy gradient watershed removes most of the dark miroregions originated by the hromati gradient. There are, however, some regions with different olors in this rough dark region whih are not fully identified by the intensity gradient watershed of Fig. 2 (b) and are better deteted by the fuzzy gradient watershed in Fig. 2(j). The two image olumns with the heading segmentation show the results of the region merging from the orresponding gradient watershed in the same row. The left olumn shows the results of using of the Eulidean distane on the RGB Cartesian oordinates. The right segmentation olumn show the results of the using the hromati distane of equation (1). If we want to asertain the effet of the olor representation and the hromati distane we must ompare the rightmost olumns in Fig. 2. We find that the general effet is that the hromati distane on polar oordinates is better identifying the subtle olor regions in the darkest areas of the image, it detets better the shape of the objets, has better olor onstany properties, and it is muh less sensitive to bright spots or shining areas. Comparing the gradient operators attending to the final segmentation we observe that the fuzzy gradient is better than the others in removing noise from the dark regions and maintain the objet integrity. Overall the best result is obtained with our proposal as shown in Fig. 2(l), where we an easily identify the subtle regions in the upper dark area, the shadow of the lowermost objet, and we an learly identify objet with the same olor unaffeted by shading and bright spots. 5 Conlusions The paper introdues a fuzzy watershed and region merging segmentation based on the zenithal and azimuthal angles of the spherial representation of olors in the RGB spae. These definitions allows the onstrution of a robust fuzzy hromati gradient that we use to realize a robust hromati watershed segmentation. This gradient operator has good olor edge detetion in lightened areas and does not suffer from the noise in the dark areas. The fuzzy watershed is omplemented by a region merging based on the defined hromati distane. We give a general shema of the algorithm performing both watershed and region merging. Out proposal an be stated by this algorithm fixing the olor representation, gradient operator, and region merging distane. We ompare our approah with other algorithms obtained with different setting of the general shema, obtaining the best qualitative segmentation. Referenes [1] J. Angulo and J. Serra, Modelling and segmentation of olour images in polar representations, Image and Vision Computing, vol. 25, pp. 475 495, Apr. 2007. [2] A. Hanbury and J. Serra, Mathematial Morphology in the HLS Colour Spae. Proeedings of the 12th British Mahine Vision Conferene, 2001. [3] R. Moreno, M. Graña, and E. Zulueta, Rgb olour gradient following olour onstany preservation, Eletronis Letters, vol. 46, pp. 908 910, jun. 2010. [4] R. Moreno, M. Graña, and A. d Anjou, A olor transformation for robust detetion of olor landmarks in roboti ontexts, in Trends in Pratial Appliations of Agents and Multiagent Systems, pp. 665 672, 2010. [5] R. Moreno, J. Lopez-Guede, and A. d njou, Hybrid olor spae transformation to visualize olor onstany, in Hybrid Artifiial Intelligene Systems, pp. 241 247, 2010. [6] S. A. Shafer, Using olor to separate refletion omponents, Color Researh and Apliations, vol. 10, pp. 43 51, april 1984. [7] M. G. R. Moreno and A. d Anjou, A image olor gradient preserving olor onstany, in FUZZ-IEEE 2010, pp. 710 714, WCCI, July 2010. [8] Y. Mileva, A. Bruhn, and J. Weikert, Illumination-Robust variational optial flow with photometri invariants, in Pattern Reognition, pp. 152 162, 2007. [9] J. van de Weijer and T. Gevers, Robust optial flow from photometri invariants, in ICIP: 2004 INTERNATIONAL CON- FERENCE ON IMAGE PROCESSING, VOLS 1-5, pp. 1835 1838, IEEE, 2004. [10] B. JAMES CLERK MAXWELL, Experiments on olour, as pereivedd by the eye, with remarks on olour blindness, TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH, VOL. XXI., PART II., vol. 21, 2, pp. 274 299, 1885. [11] S. Beuher and C. Lantuejoul, Use of Watersheds in Contour Detetion, in International Workshop on Image Proessing: Real-time Edge and Motion Detetion/Estimation, Rennes, Frane., September 1979. [12] R. T. Tan, K. Nishino, and K. Ikeuhi, Separating refletion omponents based on hromatiity and noise analysis., IEEE Trans Pattern Anal Mah Intell, vol. 26, pp. 1373 1379, Ot 2004. 5