Vector Median Filters, Inf-Sup Operations, and Coupled PDE s: Theoretical Connections

Transcription

1 Journal of Mathematical Imaging and Vision 8, c 000 Kluwer Academic Publishers Manufactured in The Netherlands Vector Median Filters, Inf-Sup Operations, and Coupled PDE s: Theoretical Connections VICENT CASELLES Department of Informatics and Mathematics, University of the Illes Balears, Palma de Mallorca, Spain GUILLERMO SAPIRO AND DO HYUN CHUNG Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA guille@eceumnedu Abstract In this paper, we formally connect between vector median filters, inf-sup morphological operations, and geometric partial differential equations Considering a lexicographic order, which permits to define an order between vectors in IR N, we first show that the vector median filter of a vector-valued image is equivalent to a collection of infimum-supremum morphological operations We then proceed and study the asymptotic behavior of this filter We also provide an interpretation of the infinitesimal iteration of this vectorial median filter in terms of systems of coupled geometric partial differential equations The main component of the vector evolves according to curvature motion, while, intuitively, the others regularly deform their level-sets toward those of this main component These results extend to the vector case classical connections between scalar median filters, mathematical morphology, and mean curvature motion Keywords: vector median filtering, inf-sup operations, asymptotic behavior, anisotropic diffusion, curvature motion, coupled PDE s 1 Introduction Median filtering has been widely used in image processing as an edge preserving filter 1 The basic idea is that the pixel value is replaced by the median of the pixels contained in a window around it This idea has been extended to vector-valued images [6, 15, 16], based on the fact that the median is also the value that minimizes the L 1 norm between all the pixels in the window More precisely, the median of a finite series of numbers U =u i M i=1 is the number u j U such that M u i u j 1 i=1 M u i u z 1, u z U 1 i=1 Here, the pixels values u i i=1 N can be either a scalar or a vector In the case of vector-valued images, 1 has been used for directions as well [15, 16] the median is selected as the vector that minimizes the sum of angles between every pair in U When considering ux :IR IR as an image defined in the continuous plane, there is a close relationship between median filtering, inf-sup morphological operations, and partial differential equations [, 4], see Section The goal of this paper is to extend these theoretical results to the vectorial case, ux :IR IR N This is presented in Section 3 Other PDE s approaches for vector-valued images can be found for example in [7, 11, 17] Scalar Case The background material in this section is adapted from [4] We refer the interested reader to these notes and

2 110 Caselles, Sapiro and Chung references therein for details and proofs See also [9] for connections and references to connections between stack median filters and morphology Definition 1 Let ux :IR IR be the image, a map from the continuous plane to the continuous line The median of ux, with support given by a set B of bounded measure centered at x, is defined as: med B ux := inf λ IR : measurex B : ux λ measureb Definition The median filtering of the whole image u is the result of applying for all pixels x in the image From now on, we will consider and analyze only the local operation around a given point x, that is, Definition 1 It is understood that we refer to the application of this operation to the whole image Having these definitions in mind, it is easy to prove the following result: Proposition 1 med B u = inf B B sup x B ux, 3 where B := B : measureb measureb, B B This relation shows that the median is an inf-sup morphological operation This is the first relation we will extend to vector-valued images in the next Section We now report on the relation between median filtering and mean curvature motion, which constitutes the second result we will extend to vector-valued images a first version of the following theorem, connecting Gaussian weighted median filtering with mean curvature motion is due to Bence, Merriman, and Osher and to Barles and Evans; see [4] Theorem 1 Let u be three times differentiable,κu the curvature of the level-set of u, and let Dx 0, t be the disk centered at x 0 and with radius t Then, med Dx0,tu = ux κu u x 0t + Ot +1/3, 4 if ux 0 0,and meddx0,t ux 0 ux0 t +Ot 3 5 otherwise This result also leads to the following very interesting relation: If t 0, that is, the region of support shrinks to a point, iterating the median filter is equivalent to solving the geometric partial differential equation u = κ u 6 t This relation is basically obtained by moving the curvature term to the left side and dividing by t, with a rescaling of time t t See [4] for the formal derivation This is one of the most popular image flows used for image enhancement via anisotropic diffusion [1], see also [10] This flow is diffusing the image in the direction of the level-sets of u, that is, perpendicular to u Therefore, iterated median filtering is basically anisotropic diffusion From this point of view, median filtering, which is classically used in image processing in its discrete form and with finite support B, and was of course derived long before curvature motion, can be considered a-posteriori as a non-consistent numerical implementation of anisotropic diffusion This explains for example the observed instabilities of discrete median filters, which do not occur in correct numerical implementations of 6 3 Vectorial Case As pointed out in the introduction, median filtering was extended to vector-valued images using the L 1 minimization property In order to use the definition analogous to the one given by Eq, we need to impose an order in the vector space This order is necessary to obtain the vectorial extensions of the results presented in the previous Section In Section 4 we discuss the use of other definitions to obtain median-type operations for vector-valued images

3 Vector Median Filters 111 In order to be able to compare between two vectors, we assume a lexicographic order 3 Given two N dimensional vectors u and v, we say that u v if and only if u = v, oru 1 >v 1,oru i =v i, for all 1 i < j N and u j >v j This order means that we compare the coordinates in order, until we find the first component that is different between the vectors It is well known that the lexicographic order is a total order, and any two vectors in IR N can be compared Given a set of vectors IR N we define its supremum infimum as the least of the upper bounds respectively, the greatest of the lower bounds That is, u = sup means that u u for all u and if u v for all u then also u v Analogous relations hold for inf with instead of The supremum and infimum always exist in the lexicographic order See also Lemma 1 below With this order in mind, the definition given by Eq is consistent for the vectorial case as well, with minor modifications: Definition 3 Let ux :IR IR N be the image, a map from the continuous plane to the continuous space IR N The vector median filter of u, with support given by a set B of bounded measure centered at x, is defined as over lines stands for set closure: med B u := inf λ IR N : measurex B : ux λ measureb 7 Remark It is necessary to define the vector median filter over the closure of a set to guarantee that if a series of vectors u ɛ is greater or equal than a fix vector v, the limit of the series is also greater or equal than v This property does not hold in the general case if the closure is omitted eg, v = [0, 1000] and u = [ɛ, 0], ɛ 0 Note also that as in the scalar case, we consider only the local behavior of the filter, while the median of the whole image is obtained shifting B and applying 7 to all the image pixels We proceed below to develop the main results of this paper concerning the relations between vector median filtering with a lexicographic order, inf-sup morphological operators, and geometric PDE s We should note that for the developments below, it is enough to consider N =, that is, a two dimensional vector The operations we show for N = will hold as well for N >, where we relate the component i + 1tothe component i in the same way we will relate the second component i = to the first one i = 1 in the developments below In other words, in order to process the component i + 1 we simulate the behavior of the component i as if i = 1, and then the behavior of the component i + 1 is obtained as if it were the second component of the vector The process is then performed in pairs two-dimensional vectors, i = 1, i =, i =, i = 3, i = 3, i = 4,,i = N 1,i = N, and the processed vector is given by the two components of the first pair and all the second components for the rest We should also note that in many cases there is no natural lexicographic order between all the vector components, but there is just a relation of importance between one vector and the rest For example, in color representations like Lab and Yuv, the first component is usually more important than the other two, but there is no natural order between the last two by themself In this case, the system is treated as a collection of two dimensional vectors of the form u 1, u i, 1 < i N For the Lab space for example, two pairs of two-dimensional vectors are processed, L, a and L, b, and the processed Lab vector if given by the two components of the first pair and the second component of the second pair 31 Vector Median Filtering as an Inf-Sup Operator Lemma 1 contains implicitly the fact that, if u is a scalar function, we may restrict the sets in B to be level sets of u This is an obvious fact, if u :IR IR is a measurable function and B a subset of IR of finite measure we always have where S := med B u = inf B S sup x B ux, 8 [u b] B : b IR, measure[v b] B measureb A formula similar to 8 also holds in the vectorial case Proposition Let u = u 1, u :IR IR be a bounded measurable function such that measure[u 1 = α] = 0 for all α IR and B a subset of IR of finite

4 11 Caselles, Sapiro and Chung measure Then where G := med B u = inf B G sup x B ux, 9 [u λ] B : λ IR, measure[u λ] B measureb Proof: For simplicity, for any function v from IR to IR N, N = 1, we shall denote by [v λ] the set x B : vx λ, λ IR N To prove this proposition, we first observe that the infimum in the right hand side of 9 is indeed attained We have that ϕ := inf sup = inf ux : B G x B sup ux : B G x B 10 Since the infimum in the lexicographic order of a closed set is attained then ϕ = ϕ 1,ϕ sup x B ux : B G Let B n := [u λ n ] G be such that λ n := sup B n u ϕ Observe that B n = [u λ n ] Let λ n = λ n,1,λ n, Let λ n,1 = supλ n,1, λ n+1,1,, λ n, = supλ n,,λ n+1,, Then λ n,1 ϕ 1, λ n, ϕ Thus, we have λ n = λ n,1,λ n, λ n, λ n is decreasing and λ n ϕ It follows that [u 1 ϕ 1 ] = n [u 1 λ n,1 ] Since measure [u 1 = α] = 0 for all α IR and [u λ] = [u 1 <λ 1 ] [u 1 =λ 1,u λ ] for all λ = λ 1,λ IR, then measure [u λ] = measure[u 1 λ 1 ] for all λ IR As a consequence, we have that measure[u ϕ] = measure[u 1 ϕ 1 ] = lim measure[u 1 λ n n,1 ] = lim measure[u λ n n ] lim sup measure[u λ n ] n measureb Now, by the definition of ϕ,wehave ϕ sup u ϕ [u ϕ] This justifies 10 In a similar way, one can show that med B u F B := λ: measurex B : ux λ measureb 11 and med B u = infλ : λ F B Now, since measure[u ϕ] measureb,wehave med B u ϕ Since also measure[u med B u] measureb then ϕ sup [u medb u] u med Bu Both inequalities prove the Proposition For simplicity, we shall assume in what follows that u = u 1, u :IR IR is a continuous function and measure[u 1 = α] = 0 for all α IR As above we shall write [v λ] to mean [v λ] B Proposition 3 Assume that u = u 1, u :IR IR is a continuous function, measure[u 1 = α] = 0 for all α IR and B is a compact subset of IR Then med med B u = B u 1 1 inf [x B:u1 x=med B u 1 ] u x Proof: Let med µ := B u 1 inf [x B:u1 x=med B u 1 ] u x 13 Since we assumed that measure[u 1 = α] = 0 for all α IR we have that measure[u µ] measureb Then, by definition of med B u we have that med B u µ Now, let λ = λ 1,λ F B Since measure[u 1 λ 1 ] = measure[u λ] measureb then med B u 1 λ 1 Ifmed B u 1 <λ 1, then µ λ Thus we may assume that med B u 1 = λ 1 Since [u λ] = [u 1 < λ 1 ] [u 1 = λ 1,u λ ], if inf [x B:u1 x = med B u 1 ] u x>λ, then [u λ] = [u 1 <λ 1 ] Since u 1 is continuous then med B u 1 sup [u λ] u 1 <λ 1 This contradiction proves that inf [x B:u1 x=med B u 1 ] u x λ Thus µ λ We conclude that µ med B u Therefore µ = med B u This Proposition means that for the first component of the vector, the median is as in the scalar case, while

5 Vector Median Filters 113 for the second one, the result is obtained looking for the infimum over all the pixels of u corresponding to the positions on the image plane where the median of the first component is obtained This result is expected, since the first component already selects the whole possible vectors, and then the positions in the plane where the second component can select from are determined Note also that new vectors, and then new pixel values, are not created, as expected from a median filter These properties will also hold for the alternative definition we present now: Definition 4 Let u = u 1, u :IR IR be a measurable function and B IR be of finite measure Define where H := med B u = inf sup ux, 14 B H x B [u 1 b] B : b IR, measure[u 1 b] B measureb Proposition 4 Let u = u 1, u :IR IR be a continuous function and B be a compact subset of IR Then med B u = med B u 1 sup [x B:u1 x=med B u 1 ] u 15 x To prove this proposition we need the following simple Lemma: Lemma 1 Let π i :IR IR be the projection of IR onto the i coordinate Let IR Then i π 1 sup = sup π 1 ii If π 1 ū = π 1 sup for some ū then π sup = supπ u : u, π 1 u = π 1 sup 16 In particular, if is compact we always have 16 Similar statement holds for the infimum Proof of Proposition 4: q = Let med B u 1 sup [x B:u1 x=med B u 1 ] u x Since measure [u 1 med B u 1 ] measureb,wehave that med B u sup [x B:u 1 x med B u 1 ] u Now, observe that if u 1 x = med B u 1 then u x sup [u1 =med B u 1 ] u From this relation it follows that sup [x B:u1 x med B u 1 ] u q Hence med B u q To prove the opposite inequality, observe that if b IR is such that measure[u 1 b] measureb then b med B u 1 and, in consequence, [u 1 med B u 1 ] [u 1 b] Since by Lemma 1, q = sup [u1 med B u 1 ] u, we have q sup [u1 b] u for all b IR such that measure[u 1 b] measureb Hence q med B u Observe that the infimum in 15 is attained 3 Asymptotic Behavior and Coupled Geometric PDE s Since π 1 med B u = π 1 med B u = med Bu 1, then Theorem 1 describes the asymptotic behavior of the first coordinate of the vector median med Dx0,tu for a three times differentiable function u :IR IR, Dx 0, t being the disk of radius t > 0 centered at x 0 IR Let us describe the asymptotic behavior of the second coordinate of med Dx0,tu The formula will be written explicitly only when Du x 0 Du 1 x If Du x 0 Du 1 x 0 =0, the formula can be written using the Taylor expansion of u given in the next Proposition Proposition 5 Let u :IR IR be three times differentiable, u = u 1, u Assume that Du 1 x 0 0 Then u x = u x 0 + x x 0,e 1 Du x 0 e t κu 1 x 0 Du x 0 e 1 κu 1x 0 x x 0, e 1 Du x 0 e + 1 D u x 0 e 1,e 1 x x 0, e 1 + ot, 17 for x [u 1 = med Dx0,tu 1 ] Dx 0, t, where e 1 = Du 1 x 0 Du 1 x 0, e = Du 1x 0 Du 1 x 0, such that e 1, e is positive oriented In particular, if Du x 0 Du 1 x 0 0, then π meddx0,tu = sup [u1 =med Dx0,tu 1 ] Dx 0,t u = u x 0 t Du Du 1 x 0 x 0 Du 1 x 0 + Ot 18

6 114 Caselles, Sapiro and Chung Similarly, π med Dx0,t u = inf [u1 =med Dx0,tu 1 ] Dx 0,tu = u x 0 + t Du Du 1 x 0 x 0 Du 1 x 0 + Ot 19 Before giving the proof of this proposition let us observe that the above formula for u coincides with the asymptotic expansion 4 if u = u 1 Indeed, if x [u 1 = med Dx0,tu 1 ] Dx 0,t, using that Du x 0 e 1 = 0 and Du x 0 e = Du 1 x 0, we have u x = u x t κu 1 x 0 Du 1 x 0 1 κu 1x 0 x x 0, e 1 Du 1 x D u x 0 e 1, e 1 x x 0, e 1 + ot Since u = u 1 and Du 1 x 0 κu 1 x 0 = D u x 0 e 1,e 1, the last two terms in the above expression cancel each other and we have u x = u 1 x = med Dx0,tu 1 = u x t κu 1 x 0 Du 1 x 0 +ot, expression consistent with 4 More generally, if Du x 0 e 1 = 0, we may write u x = u x t κu 1 x 0 Du 1 x 0 1 κu 1x 0 x x 0, e 1 Du x 0 e + 1 D u x 0 e 1, e 1 x x 0, e 1 + ot for x [u 1 = med Dx0,tu 1 ] Dx 0,t Now, observe that Du x 0 e =± Du x 0 In particular, if Du x 0 e = Du x 0 0, since Du x 0 Du x 0 is colinear to e 1, the expression for u x can be reduced to u x = u x t κu 1 x 0 Du 1 x Du x 0 κu x 0 κu 1 x 0 x x 0,e 1 +ot The value of the corresponding infimum or supremum depend on the sign of the terms containing x x 0, e 1 and we shall not write them explicitly Let x [u 1 = med Dx0,tu 1 ] Dx 0, t Obvi- Proof: ously x x 0 = x x 0,e 1 e 1 + x x 0,e e To compute x x 0, e we expand u 1 in Taylor series up to the second order and write the identity u 1 x = med Dx0,tu 1 as u 1 x 0 + Du 1 x 0, x x D u 1 x 0 x x 0, x x 0 +ot =med Dx0,tu 1 Using 4, we have Du 1 x 0, x x 0 = 1 6 t κu 1 x 0 Du 1 x 0 1 D u 1 x 0 x x 0, x x 0 +ot Thus, we may write x x 0 = x x 0,e 1 e t κu 1 x 0 e D u 1 x 0 x x 0, x x 0 e + ot Du 1 x 0 0 Introducing this expression for x x 0 in the right hand side of 0 we obtain x x 0 = x x 0,e 1 e t κu 1 x 0 e D u 1 x 0 e 1, e 1 x x 0,e 1 e + ot, Du 1 x 0 expression which can be written as x x 0 = x x 0, e 1 e t κu 1 x 0 e 1 κu 1x 0 x x 0, e 1 e + ot, 1

7 Vector Median Filters 115 since κu 1 x 0 = D u 1 x 0 e 1,e 1 Du 1 x 0 Introducing this in the Taylor expansion of u, u x = u x 0 + Du x 0, x x D u x 0 x x 0, x x 0 +ot we obtain 17, the first part of the proposition Now, from 1 we have for x [u 1 = med Dx0,tu 1 ] Dx 0,t, x x 0 = x x 0,e 1 e 1 +Ot In particular, the curve [u 1 = med Dx0,tu 1 ] Dx 0,t intersects the axis e at some point x such that x x 0 = Ot We also deduce that sup x x 0, e 1 =sup x x 0 +Ot = t+ot where the sup s are taken in [u 1 = med Dx0,tu 1 ] Dx 0,t Taking the supremum of the last expression for u on [u 1 = med Dx0,tu 1 ] Dx 0, t we obtain 19 In a similar way we deduce 18 In analogy to the scalar case, this result can also lead to deduce the following result: Modulo the different scales of the two coordinates, when the median filter is iterated, and t 0, the second component of the vector, u x, is moving its level-sets to follow those of the first component u 1 x, 5 which are by themself moving with curvature motion This is expressed with the equation u x, t t =± Du x, t Du x, t Du 1 x, t Du 1 x, t Du x,t, where the sign depends on the exact definition of the median being used Deriving this equation from the asymptotic result presented above is much more complicated than in the scalar case, and this is beyond the scope of this paper In spite of the very attractive notation, this is not a well-defined partial differential equation since the right hand side is not defined when Du 1 x=0 see remark below, and certainly this happens in images and in those which are solutions of mean curvature motion Note that Du 1 x = 0 means that there is no level-set direction at that place, and then the level-sets of u have nothing to follow This equation clarifies the meaning of the vector median and gives it a very intuitive interpretation The next terms in the Taylor expansion of u depend on curvatures of u 1 and u These terms play a role when the previous one is zero, and, in particular, this will happen when the level sets of both components of the vector are equal The precise form of med Dx0,tu and meddx 0,t u can be deduced from the asymptotic expansion for u previous to the proof of the last proposition and we shall not write them explicitly Let us only mention that if u = u 1 on a neighborhood of a point, then u moves with curvature motion, as expected To illustrate the case when Du 1 x 0 = 0 and x 0 is non-degenerate, we first assume that x 0 = 0, 0 and u 1 x 1, x = Ax1 +Bx, x =x 1,x IR, A, B > 0 It is immediate to compute med Dx0,tu 1 = inf α IR : measure[u 1 α] = t AB measuredx 0, t The set X [u 1 = t AB ] Dx 0,t = [x 1, x A Dx 0, t : B x 1 + B A x = t ] Again, it is straightforward to obtain sup X u x 1, x = u 0, 0 + t A B u x + B A u y 1/ + ot Consider now the case where u 1 is constant in a neighborhood of x 0 Suppose that u 1 = α in Dx 0, t Then either using or 3 we conclude that In this case, med Dx0,tu = α sup Dx0,t u x sup u x = u x 0 + t Du x 0 +ot Dx 0,t We conclude that there is no common simple expression for all cases Therefore, in contrast with the scalar case, the asymptotic behavior of the median filter when the gradient of the first component, u 1 x, is zero is not uniquely defined and decisions need to be taken when the equation is implemented see below

8 116 Caselles, Sapiro and Chung 31 Projected Mean Curvature Motion If we set aside for a moment the requirement for an inf-sup morphological operator, and start directly from the definitions in 1 and 15 instead of 9 and 14, we obtain an interesting alternative to the median filtering of vector-valued images we once again consider only two dimensional vectors: Definition 5 Let u = u 1, u :IR IR be a continuous function and B IR a compact subset Define med B u := med B u 1 med [x B:u1 x=med B u 1 ]u x 3 In contrast with the previous definitions, we here considered also the median of the second component, restricted to the positions where the first component achieved its own median value The asymptotic expansion 17 in Proposition 5 is of course general Replacing sup by median at the end of the proof we obtain that the expression analogous to 18 and 19 for med is π med Dx0,t u = u x 0 + t Du x, t κ u1 x, t Du 1x,t Du 1 x,t +ot 4 Note that the time scale of this expression is t,asin the scalar case Theorem 1, and then as in the asymptotic expansion of the first component of the vector This is in contrast with a time scale of t for the expressions having inf or sup in the second component equations 18 and 19 The PDE corresponding to the expression above, and therefore to the second component of the vector, is u x, t t = κ u1 x, t Du 1 x, t Du 1 x, t Du x,t Du x,t Du x, t 5 This equation shows that the level-sets of the second component u are moving with the same geometric velocity as those of the first one, u 1, meaning mean curvature motion the projection reflects the well known fact that tangential velocities do not affect the geometry of the motion Under certain smoothness assumptions, short term existence of this flow can be derived from the results in [3] Using Lemma 1 it is possible to show that med,as defined in 3, is also a morphological inf-sup operation of the type of 9 and 14 This time, the set over which the inf-sup operations are taken is given by R := λ = λ 1,λ : measure[u 1 λ 1 ] measureb, [u 1 λ 1 ], measure[u 1 = λ 1, u λ ] measure[u 1 = λ 1 ] B Recapping, the second component of the vector can be obtained via inf, sup, or median operations over a restricted set In all the cases, the filter is a morphological inf-sup operation, computed over different structuring elements sets, and in all the cases a corresponding asymptotic behavior and PDE interpretation can be given In the case of the median operation, the asymptotic expansion of the vector components have all the same scale, and the second component levelsets are just moving with the geometric velocity of the first component ones In the other cases, the level-sets of the second component move toward those of the first component In all the cases then, the level-sets of the second component follow those of the first one, as expected from a lexicographic order Figure 1 shows an example of the theoretical results presented in this paper 4 Concluding Remarks In this paper, we have extended to vector-valued images the relation between median filtering, inf-sup morphological operators, and PDE s based interpretations In order to obtain the results here reported, we have assumed a lexicographic order that permits to compare between vectors in addition to assuming a coupling of the channels, which is common in the literature If we do not want to use this assumption, we will not have an order, and then an infimum-supremum type of operation Therefore, both the positive and negative results reported in this paper are a direct consequence of imposing and order in IR N In order to avoid this, we need to follow a different approach to compute the median filter, for example, Eq 1 We should note that for continuous signals, minimizing the L 1 norm of a vector is equivalent to the independent minimization of

9 Vector Median Filters 117 Figure 1 Examples of the theoretical results presented in this paper The original image is on the top left The top right shows the result of alternating 1 and 15 for 1 step with a 3 3 discrete support since these equations correspond to erosion and dilation respectively, alternating them constitutes an opening filter The bottom figures show results of the vectorial PDE derived from the mean curvature motion for the first component and projected mean curvature motion for the rest after and 0 iterations respectively All computations were performed on the Lab color space Images reproduced here without color each one of its components, reducing then the problem to the scalar case, where, for example, each plane is independently enhanced via mean curvature motion, see Section 6 Therefore, in order to have equations that are coupled, we need to look for a different approach, like the one presented in this paper Inspired by the work on median filtering of angles and directions, in [1 14] we propose a different alternative based on minimizing the norms of the gradient of the chromaticity vectors, following the theory of harmonic maps In addition to the study of direction diffusion, the theory introduced in this paper leads to another interesting flow: ux, t t = u vx, t u, u where u :IR IR is the deforming image and vx, t is a given vector field This flow is inspired on Eq, but, since there is no absolute value, when the regularity of the vector field can be controlled, the equation can be well-defined This PDE is basically deforming the level-sets to follow certain direction The theoretical and practical results regarding this flow will be reported elsewhere Acknowledgments GS thanks Prof R Kohn from the Courant Institute, NYU, for motivating him to think again about filtering vectorial images Part of this work was performed while GS was visiting the University of Illes Balears This work was partially supported by the Spanish

10 118 Caselles, Sapiro and Chung DGICYT, Project PB , European Network PAVR FMRXCT960036, the Office of Naval Research ONR-N , the Office of Naval Research Young Investigator Award to GS, the Presidential Early Career Awards for Scientists and Engineers PECASE to GS, the National Science Foundation CAREER Award to GS, by the National Science Foundation Learning and Intelligent Systems Program LIS, and NSF-IRI Geometry Driven Diffusion Notes 1 Although edges are not completely preserved with a median filter, they are indeed much better preserved than with ordinary linear filters κ = div u u 3 Lexicographic order has recently been used in vector-valued morphology as well; see [5] for the most recent published results 4 Du := uand Du, Du =0, while Du = Du 5 The Beltrami flow [7] also has the property that the level-sets tend to follow each other [8] 6 In the classical discrete case, since the median belongs to the finite set of vectors in the window, the vectorial case is not reduced to a collection of scalar cases 11 G Sapiro and D Ringach, Anisotropic diffusion of multivalued images with applications to color filtering, IEEE Trans Image Processing Vol 5, pp , B Tang, G Sapiro, and V Caselles, Direction diffusion, ECE Department Technical Report, University of Minnesota, Feb B Tang, G Sapiro, and V Caselles, Direction diffusion, in Proc Int Conference Comp Vision, Greece, Sept B Tang, G Sapiro, and V Caselles, Color image enhancement via chromaticity diffusion, ECE Department Technical Report, University of Minnesota, March PE Trahanias and AN Venetsanopoulos, Vector directional filters A new class of multichannel image processing filters, IEEE Trans Image Processing, Vol, pp , PE Trahanias, D Karakos, and AN Venetsanopoulos, Directional processing of color images: Theory and experimental results, IEEE Trans Image Processing, Vol 5, pp , RT Whitaker and G Gerig, Vector-valued diffusion, in Geometry Driven Diffusion in Computer Vision, B ter Haar Romeny Ed, Kluwer: Boston, MA, 1994 References 1 L Alvarez, PL Lions, and JM Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J Numer Anal, Vol 9, pp , 199 V Caselles, JM Morel, G Sapiro, and A Tannenbaum Eds, Special issue on partial differential equations and geometrydriven diffusion in image processing and analysis, IEEE Trans Image Processing, Vol 7, pp 69 73, LC Evans and J Spruck, Motion of level-sets by mean curvature II, in Trans American Mathematical Society, Vol 30, No 1, pp 31 33, F Guichard and JM Morel, Introduction to Partial Differential Equations in Image Processing Tutorial Notes, IEEE Int Conf Image Proc, Washington, DC, Oct HJAM Heijmans and JBTM Roerdink Eds, Mathematical Morphology and Its Applications to Image and Signal Processing, Kluwer: Dordrecht, The Netherlands, DG Karakos and PE Trahanias, Generalized multichannel image-filtering structures, IEEE Trans Image Processing, Vol 6, pp , R Kimmel, R Malladi, and N Sochen, Image processing via the Beltrami operator, in Proc of 3rd Asian Conf on Computer Vision, Hong Kong, Jan 8 11, R Kimmel, Personal communication 9 P Maragos and RW Schafer, Morphological systems for multidimensional image processing, Proc IEEE, Vol 78, pp , P Perona and J Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans Pattern Anal Machine Intell, Vol 1, pp , 1990 Vicent Caselles received the Licenciatura and PhD degrees in mathematics from Valencia University, Spain, in 198 and 1985, respectively Currently, he is an associate professor at the University of Illes Balears in Spain He is an associate member of IEEE His research interests include image processing, computer vision, and the applications of geometry and partial differential equations to both previous fields Guillermo Sapiro was born in Montevideo, Uruguay, on April 3, 1966 He received his BSc summa cum laude, MSc, and PhD from the Department of Electrical Engineering at the Technion, Israel Institute of Technology, in 1989, 1991, and 1993 respectively After post-doctoral research at MIT, Dr Sapiro became Member of Technical Staff at the research facilities of HP Labs in Palo Alto, California He is currently with the Department of Electrical and Computer Engineering at the University of Minnesota G Sapiro works on differential geometry and geometric partial differential equations, both in theory and applications in computer

11 Vector Median Filters 119 vision and image analysis He recently co-edited a special issue of IEEE Image Processing in this topic G Sapiro 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 199, 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 1999 G Sapiro is a member of IEEE National University, Seoul, Korea in 1994 and 1996 respectively, Currently, he is a PhD candidate at the Department of Electrical & Computer Engineering, University of Minnesota, Minneapolis, MN His research interests include 3-D computer vision and partial differential equation based image processing Do Hyun Chung received his BS in Eng and MS in Eng from the Department of Control & Instrumentation Engineering, Seoul