Approximation of Weighted Local ean Operators Jianzhong Wang Department of athematics and Statistics Sam Houston State University Huntsville, TX 7734 Abstract Recently, weighted local mean operators are widely used in image processing, compressive sensing, and other areas. A weighted local mean operator changes its characteristics depending on a function content within a local area in order to preserve the function features. The directional diffusion filter and Yaroslavsky neighborhood filter (also called the sigma filter) are discrete versions of such operators. Although these operators are not convolution ones, due to their sparsity, the corresponding numerical algorithms have simple structure and fast performance. In this paper, we study the approximate properties of the weighted local mean operators, particularly focus on their asymptotic expansions, which are related to nonlinear diffusion equations. Keywords: Weighted local mean, anisotropic diffusion, approximation, asymptotic expansion, neighborhood filters. Introduction Using local integral mean to approximate functions is well-known in the classic approximation. Let f be a function defined on the d-dimensional (d-d) Euclidean space R d and write tx P R d, }x} ď hu, h ą, where }x} denotes the norm of the d-d vector x. The local integral mean of f is defined by A h pfqpxq Volp q fpx ` zq dz, @x P R d, () where Volp q hd π d Γp d `q is the volume of. The local mean operator in () is a convolution one, which produces an isotropic mean for f in its domain. Hence, the operator does not preserve the function features that are characterized by the gradient of f. In many applications, like image enhancement, for example, edge is an important feature that should be preserved. However, the local mean in () blurs edge. In order to preserve such features of functions, recently, the weighted local mean operators are introduced. These operators change their characteristics depending on the contents of functions such that small weights are assigned to feature areas while large weights are assigned to harmonic areas, so that a slower diffusion is created across region boundaries while a faster diffusion acts on harmonic areas. Suppose f is a differentiable function whose gradient is denoted by f. The first type of weighted local mean operators is defined as follows: W h pfqpxq ă fpxq, z ą exp Spxq h fpx ` zq dz, @x P R d, () where ă, ą denotes the inner product in R d, hpą q is a time-like parameter, and ă fpxq, z ą Spxq exp h dz (3) The research was supported by SHSU-ERG-99
is the normalization factor. The operator W h is nonlinear and has the smallest weight in the gradient direction of f and the largest weight in the level-space directions of f so that W h preserves the region boundaries of f. In this paper, we call W h the nonlinear directionally local mean (DL) operator. A similar but slightly modified version of W h for d was introduced for removing image noise in [6] and also for the study of diffusion maps in [3]. The second type of weighted local mean operators is defined as follows: Y h pfqpxq Dpxq fpx ` zq fpxq exp h fpx ` zq dz, @x P R d, (4) where h ą is a parameter and fpx ` zq fpxq Dpxq exp h dz (5) is the normalization factor. The operator Y h is also nonlinear and has the similar behavior as W h. When d, Y h was known as Yaroslavsky neighborhood filter in [7] and the sigma filter in [5]. Therefore, we will call Y h the nonlinear Yaroslavsky local mean (YL) operator. The approximate properties of W h and Y h describe their important characters. In this paper, we study the approximation of W h and Y h, particularly put emphasis on their asymptotic expansions as h Ñ. In Section, we introduce the notations and preliminaries. In Section 3, we study the approximation of W h. In Section 4, we study the approximation of Y h. Preliminaries and notations In this section, we assemble the main notions and notations, which will be used throughout the paper. Assume that X Ă R d is a locally compact set. Let CpXq denote the linear space of all real-valued continuous functions defined on X. The sup-norm of f P CpXq is defined by }f} X sup fpxq. (6) xpx If the set X need not to be stressed, we simply write the norm as }f}. We by C k pxq denote the space of the functions that have all k th order continuous partial derivatives. The space C k pr d q is often abbreviated to C k. Let X x tz P R d, z ` x P Xu and X x tz P R d, z ` x P Xu. The (point-wise) first modulus and the (point-wise) second modulus of continuity of f P CpXq at x P XpĂ R d q are defined by and ωpf, x, δq ω pf, x, δq sup fpx ` zq fpxq, δ ą, zpb δ XpX xq sup fpx ` zq ` fpx zq fpxq, δ ą, zpb δ XX x respectively. The first (global) modulus and the second (global) modulus of continuity on X are defined by ωpf, X, δq sup xpx ωpf, x, δq and ω pf, X, δq sup xpx ω pf, x, δq respectively. As usual, Gamma function is defined by Γpsq 8 t s e t dt, s ą, and the (real-valued) modified Bessel function of the first kind is defined by I n psq π π e s cos t cospntq dt, n ą {, s P R,
which can also be represented by I n psq The Kummer s function is defined by pa; b; sq s n n Γ n ` Γpbq Γpb aqγpaq Γ π e s cos t sin n t dt. e st t a p tq b a dt, s P R, (7) which is also called the confluent hypergeometric function of the first kind with the notation F pa; b; sq. ore details of Bessel functions and the Kummer s functions can be found in [] and [4]. For n P Z d` and k P Z d`, we write n! n!n! n d! and n n ` n ` ` n d. For s P p, 8q d, we write Γpsq Γps q Γps d q. For x P R d, we define the k-power of x by x k x k xk xk d d and say that x k is an odd power if there is j such that k j is odd. The k-moment of a function f P CpXq is defined by m X pf, x k q x k fpxq dx. When X, we denote it by m h pf, x k q. 3 Approximation of the directionally local mean operators X To study the approximation of the nonlinear DL operator W h, we first generalize it to the following linear DL operator (associated with g P C ): ă gpxq, z ą W g,h pfqpxq exp S g,h pxq h fpx ` zq dz, x P R d, (8) where ă S g,h pxq exp gpxq, z ą h dz, x P R d. (9) It is obvious that W f,h pfqpxq W h pfqpxq. We define the non-normalized kernel of W g,h by ă gpxq, z ą W g,h px, zq exp h and the normalized kernel by w g,h px, zq () S g,h pxq W g,hpx, zq. () Later, we will simply denote them by S g pxq, w g px, zq, and W g px, zq if the parameter h is not stressed. It is easy to verify that the kernel w g px, zq satisfies () w g px, zq ě ; () w g px, zq w g px, zq; and (3) ş w g px, zq dz. By these properties, we immediately get the following: Theorem Suppose g P C. Then and, for each compact set X Ă R d, W g,h pfqpxq fpxq ď ω pf, x, hq, f P C, W g,h pfqpxq fpxq ď ω pf, X, hq, x P X. Furthermore, if f P C, then W g,h pfqpxq fpxq Oph q. 3
For further study, we introduce following notations. For g P C, we define a local orthonormal (o.n.) basis at x P R d associated with g by " te,, e B tb,, b d u d u, } gpxq}, tη,, η d u, } gpxq}, where te,, e d u is the standard o.n. basis of R d and tη, η,, η d u is an o.n. basis with η g } g}. Under the basis B, the partial derivative Bgpxq Bb j (in the direction b j ) is denoted by D j gpxq or g j pxq and the second-order partial derivative B gpxq Bb ibb j denoted by D ij gpxq or g ij pxq. Later, for z P R d, we always assume that z pz, z,, z d q is the B-coordinate of z: z ř d j z jb j. Under the B-coordinate system, we have ă gpxq, z ą z } gpxq}. () Definition Let F be an operator on C. We say that F has the linearly-reproducing property if it reproduces all linear functions, that is, Fpfq f for each linear function f P C. In many applications, the linearly-reproducing property is important. For instance, in image denoising, the linearly-reproducing filters can reduce the staircase effect [, 6]. We now prove the following: Theorem The DL operator W g,h has the linearly-reproducing property. Proof. For a linear function L on R d, there is a P R d and c P R such that, for each y P R d, Lpyq c ` řd j a jy j. Hence, Lpx ` zq Lpxq ř d j a jz j. By w g px, zq w g px, zq, we have w g px, zqz j dz, ď j ď d, which yields W g plqpxq w g px, zqlpx ` zqdz w g px, zqlpxqdz Lpxq. The theorem is proved. The problem of associating integral operators with PDEs is a very natural and fruitful research area. Note that the corresponding PDEs can be derived from the asymptotic expansions of these operators. For example, the Gauss-Weierstrass integral has the asymptotic expansion G t pvqpxq pπtq d{ R d e }x y} t vpyqdy, @v P C, G t pvqpxq vpxq ` t vpxq ` ptq, t Ñ `, where ř d j B is the spatial Laplacian operator. It is known that tg Bx t u tą is a semigroup and j upx, tq G t pvqpxq gives the solution of the initial-value heat diffusion equation problem " B Btupx, tq upx, tq, t ą, upx, q vpxq, x P R d. If we set t h and consider t as a time variable, then the operator W g,h also produces a diffusion process. Although the family tw g,h u hą is no longer a semigroup, its asymptotic expansion describes the diffusion process in terms of PDEs. We now give the asymptotic expansion of W g,h in the following: 4
Theorem 3 Let f, g P C and B tb, b,, b d u be a local o.n. basis at x associated with g. Write p } gpxq} and define # pd`q App q B ` Bpp q ř d L g,x Bb j B, if p, Bb j pd`q, if p, (3) where Apuq ` 3 ; ` d ; u ` ; ` d Bpuq ; u, Then, W g,h has the following asymptotic expansion. Furthermore, if f P C 4, then ` ; ` d ; u ` ; ` d (4) ; u. W g,h fpxq fpxq ` h pl g,x fq pxq ` ph q, h Ñ. (5) W g,h fpxq fpxq ` h pl g,x fq pxq ` Oph 4 q, h Ñ. (6) In applications, the case of d is particularly useful. We present the special form of (3) for d in the following: Corollary 4 Let f, g P C and B tb, b u be the local o.n. basis at x associated with g. Write p } gpxq}. Then the operator L g,x in Theorem 3 has the following form. $ & L g,x % I p p I p `I p B Bb ` 8 p I p 3p `pp qi p B I p `I p Bb if p,, if p. By the relation W f,h fpxq W h fpxq, applying Theorem 3, we immediately obtain the following: Theorem 5 Let f P C pr d q and B tb, b,, b d u be a local o.n. basis at x associated with f. Then, the nonlinear DL operator W h has the following asymptotic expansion. Furthermore, if f P C 4 pr d q, then W h fpxq fpxq ` h pl f,x fq pxq ` ph q, h Ñ. (7) W h fpxq fpxq ` h pl f,x fq pxq ` Oph 4 q, h Ñ. (8) To prove Theorem 3 and Corollary 4, we establish two lemmas. Lemma Assume p } gpxq} and d ě. Let B tb, b,, b d u be a local o.n. basis at x associated with g. Then we have m h pw g,h, z k q, z k is odd, (9) h d π d S g,h pxq Γp ` d q ; ` d ; p, () m h pw g,h, zq h `d π d Γp ` d q ; ` d ; p, () m h pw g,h, z j q h `d π d Γ ` ` d ; ` d ; p, ď j ď d. () 5
Proof. Since the kernel W g,h px, zq is symmetric with respect to z, (9) is obtained. It is obvious that S g,h pxq m h pw g,h, q. By () and (), we have m h pw g,h, q exp p z h dz h π exp p r cos pθq r d sin d pθqdrdθ pd qπd{ { Γ ` d` hd π d Γp ` d q ; ` d ; p Equation () is obtained. Similarly, m h pw g,h, zq z exp p z h dz pd qπd{ { Γ ` d` h`d π d Γp ` d q h π ; ` d ; p. h exp p r cos pθq, which yields (). Finally, for ď j ď d, m h pw g,h, zj q zj exp p z h dz πd{ { h π Γ ` exp p r cos pθq d` h h`d π d Γp ` d q The lemma is proved. By Lemma, we have the following: h ; ` d ; p r d` cos pθq sin d pθqdθdr r d` sin d pθqdrdθ, ď j ď d. Lemma Assume p } gpxq} and d ě. Let B tb, b,, b d u be a local o.n. basis at x associated with g. Then Proof. For any k P Z d`, m h pw g,h, z k q, z k is odd, (3) m h pw g,h, zq h ` 3 ; ` d ; p d ` ` ; ` d (4) ; p, m h pw g,h, zj h ` q ; ` d ; p d ` ` ; ` d ď j ď d. (5) ; p, m h pw g,h, z k q m hpw g,h, z k q. S g,h pxq Then, (3), (4), and (5) are derived from Lemma. We now complete the proofs of Theorem 3 and Corollary 4. Proof of Theorem 3. Let B be the local o.n. basis at x associated with g. Set z ř d j z jb j. We have, for f P C, fpx ` zq fpxq ` j f j pxqz j ` i j f ij pxqz i z j ` `ˇˇz ˇˇ, (6) 6
which yields pfpx ` zq ` fpx zqq fpxq ` When p, by the direct computation, we have W g,h fpxq fpxq ` When p, by Lemma and (7), we have h W g,h fpxq fpxq ` App q B fpxq pd ` q Bb i j f ij pxqz i z j ` `ˇˇzˇˇ. (7) h pd ` q fpxq ` ph q. (8) ` Bpp q j B fpxq Bb ` ph q, (9) j where Apuq and Bpuq are given by (3). The proof of (5) is completed. If g P f 4 pr 4 q, then in (7) the term `ˇˇzˇˇ can be replaced by O `ˇˇz4ˇˇ, which yields (6). Proof of Corollary 4. When d, Γ ` ` d Γpq and ; ` d p p ; p ; ; p e p I ` I. (3) By the identities we have Applying the recurrent formula I pxq x e x p 4e ; 3; p p I ; 3; x, I p xq I pxq, p p 4e p I p. (3) pb qpa; b ; xq apa ` ; b; xq pb a qpa; b; xq, (3) we have ; 3; p 4 3 ; ; p 3 ; 3; p 4e p p p I ` I 3 4e p p p p 3p I ` pp qi 4e p 3p I p. (33) The identities (3), (3), and (33) lead us to the corollary. To reveal the anisotropic diffusion behavior of the directionally local mean operator, we give the properties of the functions Apuq and Bpuq in the following: Theorem 6 The function Apuq is decreasing on p, 8q and the function Bpuq is increasing on p, 8q. Furthermore, Apq Bpq, (34) lim Apuq, uñ8 (35) lim uñ8 d `. (36) 7
Proof. We first prove that Apuq is decreasing on p, 8q. By (3), we have The identity yields d du Apuq ; ` d ` 3 ; ` d ; u ` ; ` d ; u ` 3 ; ` d ; u ` ; ` d ; u ; u ; ` d 3 5 4 ` d ; 3 ` d ; u C e ut t 3 d p tq dt ď, @u P p, 8q, where C Γ ` d d ` Γ ` ` 3 Γ re-write Bpuq as the following: ` ; ` d ; u. pa; b; uq a pa `, b `, uq (37) b ` d ; u ; ` d ; ` d ; u e ut t p tq d ; u ` ` d dt {; ` d ; ` d ; u ; u e ut t p tq d dt. Hence Apuq is decreasing. By the recurrent formula (3), we Bpuq pd ` q ` ; d ` ; u ` 3 ; d ` ; u pd ` q ` ; d ` ; u d ` d ` Apuq, (38) d ` which yields the increase of Bpuq. Note that the equation (34) is a trivial consequence of pa; b; q. To prove (35), applying Kummer s transform pa; b; xq e x pb a; b; xq, we have ; ` d d ` ; u e u ; ` d ; u, ; ` d d ` ` ; u e u ; ` d ; u. Applying the Poincaré-Type expansions of ` d` ; ` d ; u and ` d` ; ` d ; u, we have ; ` d ; u Γ ` ` d nÿ Γ ` p q 3 u k Γ ` 3 3 ` k k!γ `, (39) d` which yield ; ` d ; u Γ ` ` d Γ ` u k nÿ k k u k ` O `u n p q k Γ ` ` k k!γ ` d` k u k ` O `u n, (4) Apuq d ` u p ` pqq, (4) so that (35) holds. Finally, by (38) and (35), we obtain (36). Let us set t h as a time variable. Then, the operator W g,h describes a diffusion processing. Theorem 6 illustrates that, as the magnitude of the gradient of g enlarges, the diffusion speed in the gradient direction reduces and tends to, while the diffusion speed along the tangent space of g increases up to the ceiling d` d`. Thus, in the area where } gpxq} is small, the operator produces a 8
nearly isotropic diffusion, while in the area where } gpxq} is large, it almost stops the diffusion in the gradient direction of g. In Figures and, the graphs of the functions App q and Bpp q are presented for d and d 4 respectively. They show that App q is decreasing and approximates, while Bpp q is increasing and approximates d` d`. Figure : The graphs of the functions App q (left) and Bpp q (right) for d. Figure : The graphs of the functions App q (left) and Bpp q (right) for d 4. 4 Approximation of Yaroslovsky local mean operators We now discuss YL operator Y h defined in (4). Suppose g P C. Again, we first generalize Y h to the following: gpx ` zq gpxq Y g,h pfqpxq exp D g,h pxq h fpx ` zq dz, @x P R d, (4) where D g,h pxq is the normalization factor defined by gpx ` zq gpxq D g,h pfqpxq exp h dz, @x P R d. (43) We denote the non-normalized kernel of Y g,h by gpx ` zq gpxq K g,h px, zq exp 9 h (44)
and denote the normalized kernel by k g,h px, zq D g,h pxq K g,hpx, zq. (45) We will simply denote them by D g pxq, K g px, zq, and k g px, zq if the parameter h is not stressed. It is easy to verify that the kernel k g px, zq is nonnegative and satisfies the normalization condition ş k g px, zq dz. Therefore, we immediately get the following: Theorem 7 Let f P C. Then Therefore, for each compact set X Ă R d, Y g,h pfqpxq fpxq ď ωpf, x, hq. Y g,h pfqpxq fpxq ď ωpx, f, hq, x P X. Unlike the kernel of the DL operator, the kernel k g,h px, zq is not symmetric with respect to z. Therefore, it has no linearly-reproducing property. We now present the asymptotic expansion of Y g,h in the following: Theorem 8 Let f, g P C pr d q and B tb, b,, b d u be a local o.n. basis at x associated with g. Let L g,x be the operator in (3). Define $ & 3Epp qg pxq ` F pp B q g jj pxq ` F pp q g j pxq B, p, Q g,x Bb j Bb j j %, p, where Epuq ` 5 ; 3 ` d ; u ` ; ` d F puq ; u, Then, Y g,h has the following asymptotic expansion: Y g,h fpxq fpxq ` h L g,x ` 3 ; 3 ` d ; u ; ` d ` (46) ; u. (47) p pd ` qpd ` 4q Q g,x fpxq ` ph q, h Ñ. (48) The following corollary presents the special form of Q g,x for d. Corollary 9 When d, the functions Epuq and F puq in (47) have the following forms. 4 p q I p p q ` pp ` 4qI p p 4 p q I p p q ` pp 4qI p p Epuq p 4 pi p p q ` I p p qq, F puq p 4 pi p p q ` I p p qq. (49) Since in application we are more interested in the nonlinear YL operator Y h, we give the asymptotic expansion of Y h in the following: Theorem Let f P C and B tb, b,, b d u be a local o.n. basis at x associated with f. Write p } fpxq} and let Apuq and Bpuq be the functions in (3). Then, the nonlinear YL operator Y h has the following asymptotic expansion: Y h fpxq fpxq ` h pd ` q Gpp q B fpxq Bb ` App q j B fpxq Bb ` ph q, h Ñ, (5) j
where Particularly, when d, Apuq and Gpuq are given by Gpuq pd ` qbpuq pu ` dqapuq. (5) Apuq 4I p u q p pi p u q ` I p u qq, Gpuq 4uI p u q 4pu ` 3qI p u q p pi p u q ` I p u qq. (5) By Theorem, when fpxq, the expansion of Y h has the simple form: Y h fpxq fpxq ` h pd ` q fpxq ` ph q. We now prove Theorem 8, Corollary 9, and Theorem. lemmas. We need to establish the following two Lemma 3 Assume g P C, d ě, and p } gpxq}. Let B tb, b,, b d u be a local o.n. basis at x associated with g. Then we have and, for ď j ď d, m h pw g,h, zq 4 3h4`d π d 5 4Γp3 ` d q ; 3 ` d ; p, (53) m h pw g,h, zz j h4`d π d q 4Γ `3 ` d Proof. We have m h pw g,h, zq 4 z 4 exp p z h dz pd qπd{ { Γ ` d` 3h4`d π d 4Γp3 ` d q and, for ď j ď d, m h pw g,h, zz j q zz j exp The lemma is proved. h π πd{ { h π Γ ` d` h4`d π d 4Γ `3 ` d 5 ; 3 ` d ; p p exp p r cos pθq h dz ; 3 ` d ; p. (54) h exp p r cos pθq h ; 3 ` d ; p Lemma 4 Assume g P C and p } gpxq}. Then. r d`3 cos 4 pθq sin d pθqdθdr r d`3 cos pθq sin d pθqdrdθ D g,h pxq S g,h pxq ` ph 3 q, h Ñ. (55)
Proof. Let B be a local o.n. basis at x associated with g. We have K g,h px, zq exp g ij pxqz i z j h pz ` ` i j e p z h pz h which yields i j D g,h pxq m h pk g,h, q m h pw g,h, q p h By (9), m h pw g,h, z z i z j q, which yields (55). ˇˇ `ˇˇz g ij pxqz i z j ` `ˇˇz 3ˇˇ, (56) i j g ij pxqm h pw g,h, z z i z j q ` ph 3 q. Proof of Theorem 8. When p, by simple computation, we obtain Y g,h fpxq fpxq ` When p, by (56) and (6), we have h pd ` q fpxq ` ph q. K g,h pxqpfpx ` zq fpxqq W g,h px, zq f j pxqz j ` j j j f ij pxqz i z j pz h f j pxqz j j j j g ij pxqz i z j ` p z q, which yields Y g,h pfqpxq fpxq ` p h By Lemmas, 3, and 4, we have j m h pw g,h, zj q f jj pxq p D g,h pxq h j m h pw g,h, z 4 q D g,h pxq m h pw g,h, z 4 q D g,h pxq g pxq ` m h pw g,h, zz j q g j pxqf j pxq D j g,h pxq m h pw g,h, zz j q g jj pxq f pxq ` ph q. (57) D g,h pxq j m h pw g,h, zj q f jj pxq D g,h pxq L g,h fpxq ` ph 3 q, (58) ` 5 ; 3 ` d ; p m h pw g,h, z z j q D g,h pxq Hence, (48) holds. The proof is completed. 3h 4 pd ` 4qpd ` q h 4 pd ` 4qpd ` q Proof of Corollary 9. By the identities (3), (33), and ` ; ` d ; p ` ph3 q, (59) ` 3 ; 3 ` d ; p ` ; ` d ; p ` ph3 q. (6) bpa; b; zq zpa; b ` ; zq bpa ; b; zq, (6)
we have p 3 ; 4; p q 3 p p ; 3; p q p 3 ; 3; p q 4e p p 4e p p 4 p I p p I ` p p I ` pp 4qI p 3 p p I. (6) Then, by (3) and (6), we have p 5 ; 4; p q p 3 ; 3; p q p 3 ; 4; p q 4e p p 4 Finally, by (63), (6), and (3), we obtain (49). p I p Proof of Theorem. By Y h fpxq Y f,h fpxq, we have Bf Bb j ď d. Then, the operator Q f,x is reduced to By (48), we obtain Y h fpxq fpxq ` pd ` q ` Q f,x 3pEpp qd fpxq ` pf pp q h pd ` q By (6), we have ; 3 ` d ; p h p ` pp ` 4qI. (63) } fpxq} p and Bf Bb j D jj fpxq. j App q 6p d ` 4 Epp q D fpxq Bpp q p d ` 4 F pp q 4 ` d p and, by (3) and (65), we have 5 ; 3 ` d ; p 4 ` d 3 pp ` d ` q 4 ` d 6p j, ď D jj fpxq ` ph q, h Ñ. (64) ; ` d ; p ; ` d ; p, (65) ; ` d ; p ; ` d ; p d ` 3 pd ` q ; 3 ` d ; ` d ; p Thus, (5) is yielded from (65) and (66). Finally, (5) is derived from (49) and (5). ; p. (66) The anisotropic diffusion behavior of the nonlinear YL operator is characterized by Gpp q and App q, where the properties of App q is already given in Theorem 6. We present the properties of Gpp q in the following: Theorem Let the function Gpuq be given by (5). Then Gpq, lim Gpuq and there is a uñ8 u P p, 8q such that Gpuq ă for u ą u. Hence, Gpuq has a zero in p, 8q. 3
Proof. By Apq and Bpq, we get Gpq. The limit lim Gpuq is derived from (4) uñ8 and (36). To prove that there is u ą such that Gpuq ă for u ą u, we write Gpuq pd ` qbpuq pu ` dqapuq pd ` q pu ` d ` qapuq ` ; ` d ; u pd ` q ; ` d ; u pu ` d ` q ; ` d ; u. By (39) and (4), Γ ` 3 Γ ` ` d Γp q Γ ` d` Γ ` 3 pd ` q Γ ` 3 u Γ ` d 4 d ; ` d Γ ` d pd ` q Γ ` 3 Γ ` 3 Γ ` d ; u pu ` d ` q ; ` d ` Γ 3 u 3 ` O u 5 pu ` d ` q Γ ` d` u 3 ` O u 5 ` Γ ` 5. Γ ` d Γ ` d` u 3 ` O u 5 ; u Γ ` 5 u 3 Γ ` d u 5 ` O u 7 Hence, when u is large enough, Gpuq ă. The theorem is proved. It can be seen that the diffusion behavior of Y h is different from W h when they act on a same function f. In the gradient direction of f, when the magnitude of the gradient is large, Y h causes a backward diffusion, while W h still produces a forward one although it becomes very weak. In the directions on the level space of f, Y h produces the diffusion that tapers off when the magnitude of the gradient becomes large, while W h produces a nearly isotropic one. We show the graphs of Gpp q and App q in Figure 3 for d. Figure 3: The graphs of the functions Gpp q (left) and App q (right) for d. Acknowledgment. The author would like to thank the reviewers of this paper for their valuable comments. References []. Abramowitz and I.A. Stegun, (eds.) Handbook of athematical Functions with Formulas, Graphs, and athematical Tables, 9th printing. New York: Dover, 97. [] A. Buades, B. Coll, and J.. orel, The staircasing effect in neighborhood filters and its solution, IEEE Trans. Image Process., 5(6), pp. 499-55, 6. 4
[3] R.R. Coifman and. aggioni, Diffusion wavelets, in Special Issue on Diffusion aps and Wavelets, Appl. Comput. Harmo. Anal.,, pp. 53-94, 6. [4] N.N. Lebedev, and Richard R. Silverman (Translator), Special Functions and Their Applications, Prentice-HAll, 965. [5] J.S. Lee, Digital image smoothing and the sigma filter, Comput. Vis., Graph., Image Process., 4, pp. 53-69, 983. [6] J.Z. Wang, Construction of local nonlinear filter without staircase effect in image restoration, Applicable Analysis, 9(8), pp. 57-73, 9. [7] L.P. Yaroslavsky, Digital Picture Processing An Introduction, New York, Springer-Verlag, 985. 5