BANDELET IMAGE APPROXIMATION AND COMPRESSION

Transcription

1 BANDELET IMAGE APPOXIMATION AND COMPESSION E. LE PENNEC AND S. MALLAT Abstract. Finding eicient geometric representations o images is a central issue to improve image compression and noise removal algorithms. We introduce bandelet orthogonal bases and rames that are adapted to the geometric regularity o an image. Images are approximated by inding a best bandelet basis or rame that produces a sparse representation. For unctions that are uniormly regular outside a set o edge curves that are geometrically regular, the main theorem proves that bandelet approximations satisy an optimal asymptotic error decay rate. A bandelet image compression scheme is derived. For computational applications, a ast discrete bandelet transorm algorithm is introduced, with a ast best basis search which preserves asymptotic approximation and coding error decay rates. Key words. Wavelets, Bandelets, Geometric epresentation, Non-Linear Approximation AMS subject classiications. 4A25, 42C40, 65T60. Introduction. When a unction deined over [0, ] 2 has singularities that belong to regular curves, one may take advantage o this geometrical regularity to optimize its approximation rom M parameters. Most current procedures such as M-term separable wavelet approximations are locally isotropic and thus can not take advantage o such geometric regularity. This paper introduces a new class o bases, with elongated multiscale bandelet vectors that ollow the geometry, to optimize the approximation. For unctions that are Hölderian o order α over [0, ] 2, M-term separable wavelet approximations M satisy M 2 C M α where stands or the L 2 norm. The decay exponent α is optimal in the sense that no other approximation scheme can improve it or all such unctions. I is Hölderian o order α over [0, ] 2 {C γ } γ G where the C γ are inite length curves along which is discontinuous then its M-term wavelet approximation satisies only M 2 C M. The existence o discontinuities drives entirely the decay o the approximation error. I the C γ are regular curves, several approaches [, 6, 8, 8] have already been proposed to improve the decay o this wavelet approximation error, or α 2. When α is unknown, the issue addressed by this paper is to ind an approximation scheme that is asymptotically as eicient as i was Hölderian o order α over its whole support, and to derive an image compression scheme. This is particularly important to approximate and compress images, where the contours o objects create edge transitions along piecewise regular curves. Section 2 reviews non-linear approximation results or piecewise regular images including edges. In the neighborhood o an edge, the image gray levels vary regularly in directions parallel to the edge, but they have sharp transitions across the edge. This anisotropic regularity is speciied by a geometric low that is a vector ield that indicates local direction o regularity. Section 3 constructs bandelets, which are anisotropic wavelets that are warped along this geometric low, and bandelet orthonormal bases in bands around edges. We study the approximation in bandelet bases o unctions including edges over such bands. Bandelets rames o L 2 [0, ] 2 are deined in Section 4. as a union o bandelet bases in dierent bands. A dictionary o ban- Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 7, 7525 Paris Cedex 05, France (lepennec@math.jussieu.r) Centre de Mathématiques Appliquées, Ecole Polytechnique, 928 Palaiseau Cedex, France

2 delet rames is constructed in Section 4.2 with dyadic square segmentations o [0, ] 2 and parameterized geometry lows. The main theorem o Section 4.2 proves that a best bandelet rame obtained by minimizing an appropriate Lagrangian cost unction in the bandelet dictionary yields asymptotically optimal approximations o piecewise regular unctions. I is Hölderian o order α over [0, ] 2 {C γ } γ G where the C γ are Hölderian o order α then the main theorem proves that an approximation rom M parameters in a best bandelet rame satisies M 2 C M α. To compress images in bits, an image transorm code is deined in Section 5. It is proved that a best bandelet rame yields a distortion rate that nearly reaches the Kolmogorov asymptotic lower bound, up to a logarithmic actor. For numerical implementations over digital images, Section 6. discretizes bandelet bases and rames. Section 6.2 describes a ast algorithm that inds a best discrete bandelet rame with an approximation error that decays like M α up to a logarithmic actor. 2. Geometric Image Model. We begin by establishing a mathematical model or geometrically regular images using the notion o edge. This model incorporates the act that the image intensity is not necessarily singular at edge locations, which is why edge detection is an ill-posed problem. We then review existing constructive procedures to approximate such geometrically regular unctions. Functions that are regular everywhere outside a set o regular edge curves deine a irst simple model o geometrically regular unctions. Let C α (Λ) be the space o Hölderian unctions o order α over Λ n deined or α > 0: C α (Λ) =, n : β = α, β sup (x,y) Λ 2 β x β xβn n (x) x β xβn n β x β xβn n exists and satisies (y) x y α α < (2.) with α = { α α i α / N i α N and α the integer just below α. For α integer, the space C α is slightly larger than the space o unction having bounded derivatives up to order α. The norm Cα (Λ) used through this paper is deined by: Cα (Λ) = max sup x Λ max β α x β sup max (x,y) Λ 2 β = α β (x), xβn n β x β xβn n (x) β x β xβn n (y) x y α α (2.2) We say that an edge curve is Hölderian o order α i the coordinates in 2 o the points along this curve has a parameterization by arc length which is Hölderian o order α. An image model with geometrically regular edges is obtained by imposing that C α (Λ) or Λ = [0, ] 2 {C γ } γ G, where the C γ are Hölderian o order α edge curves. For most images, this model is too simplistic because most oten the image intensity has a sharp variation but is not singular across an edge. In particular, discontinuities o the image intensity created by occlusions in the visual scene are blurred by optical diraction eects. This blurring eects along edges can be modeled 2

3 through a convolution with an unknown kernel o compact support h(x). This means that we can write (x) = h(x) where C α (Ω) or Ω = [0, ] 2 {C γ } γ G. When, inding rom the exact locations o the edges C γ is an ill-posed problem, specially since h is unknown. The diiculty to locate blurred edges is well known in image processing [3]. The goal o this paper is to ind an approximation M rom M parameters which satisies M > 0, M 2 C M α, (2.3) with a constant C that does not depend upon the blurring kernel h. A wavelet approximation decomposes in an orthonormal wavelet basis and reconstructs M rom a partial sum o M wavelets corresponding to the largest amplitude coeicients. Over a class o unctions whose total variation are uniormly bounded then one can prove [5] that M 2 C M. (2.4) This result applies to unctions having discontinuities along regular edges. However, wavelet bases are unable to take advantage o existing geometric regularity in order to improve the asymptotic error decay M. Many beautiul ideas have already been studied to ind approximation schemes that take into account geometric image regularity. A surprising result by Candès and Donoho [] shows that it is not necessary to estimate the image geometry to obtain eicient approximations. A curvelet rame is composed o multiscale elongated and rotated wavelet type unctions. Candès and Donoho prove [2] that i α = 2 then an approximation M with M curvelets satisies M 2 C M 2 (log 2 M) 3. (2.5) Up to the (log 2 M) 3 actor, this approximation result is thus asymptotically optimal or α = 2. However, it is not adaptive in α in the sense that the optimal decay rate M α is not obtained when α < 2 or α > 2 [8]. Instead o choosing a priori a basis to approximate, one can rather adapt the approximation scheme to the image geometry. For example, one can construct an approximation M which is piecewise linear over an optimized triangulation including M triangles and satisies M 2 C M 2. This requires to adapt the triangulation to the edge geometry and to the blurring scale [2, 7]. However, there is no known polynomial complexity algorithm which computes such an approximation M with an error that always decays like M 2. Incorporating an unknown blurring kernel in the geometric model makes the problem much more diicult. Indeed, smooth edges are more diicult to detect than sharp singularities and the triangulation must be adapted to size o the blurring to approximate precisely the image transitions along the edges. Most adaptive approximation schemes that have been developed so ar can eiciently approximate geometrically regular images only i the edges are singularities, meaning that h = δ. In particular, many image processing algorithms have been developed to construct such approximations by detecting edges and constructing regular approximations between the edges where the image is uniormly regular [7]. To obtain ast polynomial time algorithms, Donoho introduced multiscale strategies to approximate the image geometry. Dictionaries constructed with wedgelets are used to compute approximations o unctions that are C 2 away rom C 2 edges[9]. The 3

4 approximation bound M 2 C M α holds only or α 2, and i there is no blurring (h = δ). Wakin et al.[8] propose a compression scheme that mixes the wedgelets and the wavelets to obtain better practical approximation results while ollowing a similar geometry optimization scheme. A dierent strategy developed by Cohen and Matei [6] uses a non-linear subdivision scheme to construct an approximation M that can reach a similar error decay bound, i the image has sharp discontinuities along C 2 edges. The goal o this paper is to ind a approximation M rom M parameters that satisies M 2 C M α or any α and any blurring kernel h. 3. Approximation in Orthonormal Bandelet Bases. 3.. Geometric Flow and Bandelet Bases. An image having geometrically regular edges, as in the model o Section 2, has sharp transitions when moving across edges but has regular variations when moving parallel to these edges. This displacement parallel to edges can be characterized by a geometric low which is a ield o parallel vectors that give the local direction in which has regular variations. Bandelet orthonormal bases are constructed by warping anisotropic wavelets bases with this geometric low. A geometric low is a vector ield τ(x, x 2 ) which gives directions in which has regular variations in the neighborhood o each (x, x 2 ). In the neighborhood o an edge, the low is typically parallel to the tangents o the edge curve. To construct an orthogonal basis with a geometric low, we shall impose that the low is locally either parallel in the vertical direction and hence constant in this direction or parallel in the horizontal direction. To simpliy the explanations, we shall irst consider horizontal or vertical horizon models [9] (or boundary ragments [3]), which are unctions that include a single edge C whose tangents have an angle with the horizontal or vertical that remains smaller than π/3, so that C can be parameterized horizontally or vertically by a unction g. Suppose that is a horizontal horizon model. We deine a vertically parallel low whose angle with the horizontal direction is smaller than π/3. Such a low can be written: τ(x, x 2 ) = τ(x ) = (, g (x )) with g (x ) 2. (3.) A low line is an integral curve o the low, whose tangent at (x, x 2 ) is collinear to τ(x, x 2 ). Let g(x) a primitive o g (x) deined by g(x) = x 0 g (x) dx, that we shall call low integral. Flow lines are set o points (x, x 2 ) Ω which satisy x 2 = g(x ) + cst. A band B parallel to this low is deined by: B = { } (x, x 2 ) : x [a, b ], x 2 [g(x ) + a 2, g(x ) + b 2 ]. (3.2) The band height a 2 and b 2 are chosen so that a neighborhood o C is included in B, as illustrated in Figure 3.. I the low directions are suiciently parallel to the edge directions then (x) has regular variations along each low line (x, g(x ) + cst). As a consequence, Figure 3.2 shows that the warped image W (x, x 2 ) = (x, x 2 + g(x )), (3.3) 4

5 Fig. 3.. Horizontal horizon model with a low induced by the edge and the corresponding band has regular variations along the horizontal lines (x 2 constant), in the rectangle obtained by warping the band B: W B = {(x, x 2 ) : (x, x 2 + g(x )) B} = {(x, x 2 ) : x [a, b ], x 2 [a 2, b 2 ]}. (3.4) W B W B Fig A band B and its warped band W B I Ψ(x, x 2 ) is a unction having vanishing moments along x or x 2 ixed, since W (x, x 2 ) is regular along x, the resulting inner product W, Ψ will be small. Note that W, Ψ =, W Ψ, (3.5) where W is the adjoint o the operator W deined in (3.3). This suggests decomposing over the warped image by W o a basis having vanishing moments along x. Observe then that W (x, x 2 ) = W (x, x 2 ) = (x, x 2 g(x )). (3.6) Since W is an orthogonal operator, an orthonormal amily warped with W = W remains orthonormal. By inverse warping, an orthogonal wavelet basis o the rectangle W B yields thus an orthogonal basis over the band B with basis unctions having vanishing moments along the low lines. A separable wavelet basis is deined rom one-dimensional wavelet ψ(t) and a scaling unction φ(t), that are here chosen compactly supported, which are dilated and translated [5, 6]: ψ j,m (t) = ( t 2 j ψ m ) 2 j 2 j and φ j,m (t) = ( t 2 j φ m ) 2 j 2 j 5. (3.7)

6 Following [5], the index j goes to when the wavelet scale 2 j decreases. In this paper, j is thus typically a negative integer. The amily o separable wavelets { } φj,m (x ) ψ j,m2 (x 2 ), ψ j,m (x ) φ j,m2 (x 2 ) (3.8), ψ j,m (x ) ψ j,m2 (x 2 ) (j,m,m 2) I W B deines an orthonormal basis over the rectangle W B, i one modiies appropriately the wavelets whose support intersect the boundary o W B [4]. The index set I W B depends upon the width and length o the rectangle W B. The wavelet construction is slightly modiied to cope with really anisotropic rectangle : it is started with scaling unction o size o order the largest dimension instead o the smallest dimension. This basis could thus already include some anisotropic unctions. This ensures that a polynomial on a rectangle could always be reproduced with a ixed number o coeicients. Since W is orthogonal, applying its inverse to each o the wavelets (3.8) yields an orthonormal basis o L 2 (B), that is called a warped wavelet basis: { } φj,m (x ) ψ j,m2 (x 2 g(x )), ψ j,m (x ) φ j,m2 (x 2 g(x ))., ψ j,m (x ) ψ j,m2 (x 2 g(x )) (j,m,m 2) I W B (3.9) Warped wavelets are separable along the x variable and along the x 2 = x 2 g(x ) variable, so that or a given x 2 one ollows the geometric low lines within the band B. The wavelet ψ(t) has p α vanishing moments but φ(t) has no vanishing moments. As a consequence, the separable wavelets ψ j,m (x ) φ j,m2 (x 2 ) and ψ j,m (x ) ψ j,m2 (x 2 ) have vanishing moments along x but not φ j,m (x ) ψ j,m2 (x 2 ). However, {φ j,m (x )} m is an orthonormal basis o a multiresolution space which also admits an orthonormal basis o wavelets {ψ l,m (x )} l>j,m that have vanishing moments, besides a constant number o scaling unctions. This suggests replacing the orthogonal amily {φ j,m (x ) ψ j,m2 (x 2 )} j,m,m 2 by the amily {ψ l,m (x ) ψ j,m2 (x 2 )} j,l>j,m,m 2 which generates the same space. This is called a bandeletization. Ater applying the inverse warping W the resulting unctions ψ l,m (x ) ψ j,m2 (x 2 g(x )) are called bandelets because their support is parallel to the low lines and is more elongated (2 l > 2 j ) in the direction o the geometric low. Inserting these bandelets in the warped wavelet basis (3.9) yields a bandelet orthonormal basis o L 2 (B): { } ψl,m (x ) ψ j,m2 (x 2 g(x )), ψ j,m (x ) φ j,m2 (x 2 g(x )), ψ j,m (x ) ψ j,m2 (x 2 g(x )) j,l>j,m,m 2. (3.0) Suppose now that is a vertical horizon model, with an edge along a curve C whose tangents have an angle smaller than π/3 with the vertical direction. We then deine a geometric low τ(x, x 2 ) that is parallel in the horizontal direction and which has an angle smaller than π/3 with the vertical direction: τ(x, x 2 ) = τ(x 2 ) = (g (x 2 ), ) with g (x 2 ) 2. (3.) Flow lines are points (x, x 2 ) with x = g(x 2 ) + cst. We deine a band which is parallel to the geometric low: { } B = (x, x 2 ) : x [g(x 2 ) + a, g(x 2 ) + b ], x 2 [a 2, b 2 ]. (3.2) 6

7 The width l = b a o the band is adjusted so that a neighborhood o C is included in B. Similarly to the previous case, the warped wavelet basis o L 2 (B) is then deined by: { φj,m (x g(x 2 )) ψ j,m2 (x 2 ), ψ j,m (x g(x 2 )) φ j,m2 (x 2 ) }, ψ j,m (x g(x 2 )) ψ j,m2 (x 2 ) (j,m,m 2) I W B. (3.3) The bandeletization replaces each amily o scaling unctions {φ j,m2 (x 2 )} m2 by a amily o orthonormal wavelets that generates the same approximation space. The resulting bandelet orthonormal basis o L 2 (B) is: { φj,m (x g(x 2 )) ψ j,m2 (x 2 )), ψ j,m (x g(x 2 )) ψ l,m2 (x 2 )) }, ψ j,m (x g(x 2 )) ψ j,m2 (x 2 )) j,l>j,m,m 2. (3.4) 3.2. Bandelet Approximation in a Band. The bandelet approximation o geometrically regular horizon models is studied in bands around their edge. The ollowing deinition introduces such geometrical regular unctions according to the model o Section 2. Deinition 3.. A unction is a C α horizon model over [0, ] 2 i = or = h, with C α (Λ) or Λ = [0, ] 2 {C}, the blurring kernel h has a support included in [ s, s] 2 and is C α with h C α s (2+α), the edge curve C is Hölderian o order α and its tangents have an angle with the horizontal or vertical direction that remains smaller than π/3, with a distance larger than s rom the horizontal (respectively vertical) boundary. Observe that the kernel h can be rewritten as a dilation by s: h(x) = s 2 h (s x) where h has a support in [, ] and h = h. Normalizing the amplitude o h by setting h C α = s (2+α) is equivalent to setting h C α =. The convolution with h diuses the edge C over a tube C s deined as the set o points within a distance s o C. Outside this tube is uniormly C α but within this tube its regularity depends upon s which may be an arbitrarily small variable. We study the bandelet approximation o within a band B that includes the tube C s. A bandelet basis is constructed rom a geometric low. To optimize the approximation o with M parameters in a bandelet basis, it is necessary to speciy this geometric low with as ew parameters as possible. A vertically parallel low is speciied in a horizontal band B o length l = b a by parameterizing the low integral g(x ) over a amily o orthogonal scaling unctions {θ k,m (t)} at a scale m l2 k 2k. The scaling unctions {θ k,m (t)} m l2 whose support do not intersect the border k o [a, b ] can be written θ k,m (t) = θ(2 k t m), and t [a, b ], g(t) = l 2 k m= α m θ k,m (t) with α m = g, θ k,m θ k,m 2. (3.5) We suppose that the space V k generated by the orthogonal amily {θ k,n (t)} n l2 k includes polynomials o degree p over [a, b ], and that θ(t) is compactly supported and p times dierentiable. The decomposition (3.5) deines a parameterized low that depends upon the scale 2 k and the (b a )2 k coeicients α n. Since g (t) 2 and g is deined up to a constant, one can set g(a ) = 0 so that g(t) 2 l. As a 7

8 result, one can veriy that there exists C θ > 0 that only depend upon θ such that α m C θ (b a ). There are many possible approaches to compute a geometric low or a horizon model. Section 4.2 describes an algorithm that computes the low by minimizing the approximation error o the resulting bandelet representation. One can also use a simpler and computationally more eicient approach that minimizes a Sobolev norm in the direction o the low [4]. I the horizon model is horizontal, the edge C can be parameterized horizontally by (x, c(x )). For each x, because o the blurring eect, the position o the edge can only be estimated up to an error bounded by the size s o the blurring kernel. The resulting estimate g(x ) o the edge position c(x ) satisies g c C d s, where C d depends upon precision o the algorithm that is used. Let P Vk be the orthogonal projection on the approximation space V k. To represent the geometry with ew coeicients, a regularized edge is considered: g(t) = P Vk g(t) = l 2 k m= α m θ k,m (t) with α m = g, θ k,m θ k,m 2. (3.6) We shall prove that i the scale 2 k is small enough then the distance between C and the resulting regularized edge remains o the order o s. Let us now construct a bandelet orthonormal basis over a band B parallel to the geometric low deined by the low integral g, with a one-dimensional wavelet ψ which has p α vanishing moments. To simpliy notations, in the ollowing we write B = {b m } m the orthonormal bandelet basis deined in (3.0). An approximation o in B is obtained by keeping only the bandelet coeicients above a threshold T : M =, b m b m. (3.7) m,b m T The total number o parameters is M = M G + M B where M G is the number o parameters α n in (3.6) that deine the geometric low in B, and M B is the number o bandelet coeicients above T. The ollowing theorem computes the resulting approximation error. Theorem 3.2. Let be a C α horizon model with an edge parameterized by c and α < p. Let g be an edge estimation such that g c C d s. There exists C such that or any threshold T, the approximation error in a bandelet basis deined by the the low integral g = P Vk ( g) with 2 k = max( c /α C, ) max(s, T 2α/(α+) ) /α α satisies M 2 C C 2 l α+ M α (3.8) with C = max( C α (Λ) max( c α C α, ) max( c α C α, C d, ), min( c (α+)/(2α) α+ Cα, ), C (Λ) ). The constant C is deined as the maximum o 3 quantities : Cα (Λ) max( c α C α, ) max( c α C α, C d, ) that controls the regularity o W, min( c (α+)/(2α) C, ) that appears in the geometry approximation, α α+ C (Λ) that controls the error away rom the singularities. This does not correspond to any ine optimization in the relationship between T, s and the regularity o and C γ but on the natural quantities that arises in the proo. Proo. [Theorem 3.2] 8

9 We irst prove the ollowing proposition that considers the case T s (α+)/(2α) where the geometric approximation scale is 2 k = max( c /α C, ) α s/α. Proposition 3.3. Under the hypotheses o Theorem 3.2, there exists C that only depends upon the edge geometry such that or any threshold T s (α+)/(2α) the bandelet approximation error satisies and M 2 C C 2/(α+) l T 2α/(α+) (3.9) M max(c C 2/(α+) l T 2/(α+), C log 2 T ). (3.20) Proo. [Proposition 3.3] Let g be the projection o g on the space V k with 2 k = max( c /α C, ) α s/α, B the associated band and B the associated bandelet basis. By construction, the number o geometric parameters M G satisies and, as s T 2α/(α+), M G max((b a ) max( c /α, ) s /α, C) (3.2) M G max(c C 2/(α+) l T 2/(α+), C log 2 T ) (3.22) and we should now ocus on the number M B o bandelet coeicients above T. The bandelet basis B o B is separated in two amilies, B the bandelets whose support does not intersect C s and B 2 the bandelets whose support does intersect C s. Using the orthogonality o the basis B, we veriy that M 2 = m, b m 2 (3.23) M 2 =,b m <T, b m 2 +, b m 2 (3.24) b m B,b m <T b m B 2,b m <T and M B = M B, + M B,2 where M B, and M B,2 are respectively the number o bandelets coeicients above the threshold T in B and B 2. The proo is then divided in two lemmas. The Lemma 3.4 takes care o the outer bandelets, B, that do not intersect the singularities and relies on the regularity o. Lemma 3.4. Under the hypotheses o Theorem 3.2, there exists a constant C such that and b m B,b m <T, b m 2 C C 2/(α+) l T 2α/(α+) (3.25) M B, max(c C 2/(α+) l T 2/(α+), C). (3.26) Lemma 3.5 considers the inner bandelets o B 2, that capture the sharp transitions o. Lemma 3.5. Under the hypotheses o Theorem 3.2, there exists a constant C such that, b m 2 C C 2/(α+) l T 2α/(α+) (3.27) and b m B 2,b m <T M B,2 max(c C 2/(α+) l T 2/(α+), C log 2 T ). (3.28) 9

10 Combining (3.22), (3.25), (3.26) (3.27) and (3.28) allows to conclude. To prove Lemma 3.4 and Lemma 3.5, we go back to the deinition o the bandelets. With a slight abuse o notation, we note b l,j,m (x, x 2 ) the bandelets deined by ψ l,m (x ) ψ j,m2 (x 2 g(x )) i j < l and either by ψ j,m (x ) ψ j,m2 (x 2 g(x )) or ψ j,m (x ) φ j,m2 (x 2 g(x )) i j = l. In the last case, both ψ j,m (x ) ψ j,m2 (x 2 g(x )) and ψ j,m (x ) φ j,m2 (x 2 g(x )) have vanishing moments along x and, as this is the only direction in which the moments are used, could be handled in the exact same way. From now on, we suppose that b l,j,m (x, x 2 ) = ψ l,m (x ) ψ j,m2 (x 2 g(x )). As seen in (3.5) and (3.0),, b l,j,m = (x, x 2 + g(x )), ψ l,m (x ) ψ j,m2 (x 2 ) = W (x, x 2 ), ψ l,m (x ) ψ j,m2 (x 2 ). (3.29) The regularity o W is the key to control the bandelet coeicient. I is C k in a the neighborhood o a point (x, x 2 + g(x )), a straightorward calculation shows that along the x 2 axis α W x α 2 (x, x 2 ) α W x α 2 (x, x 2 ) C α x 2 x 2 α α. Along the x axis, a control on the regularity o g is required and ater some calculation one derives that i g is C α then W is also C α and that α W (x x α, x 2) α W (x x α, x 2 ) C α max( g, g α ) x x α α. The deinition o g implies that the dierence between g and the true curve c, g c, is even C α as stated by the ollowing lemma proved in Appendix A.3 : Lemma 3.6. There exists a constant C such that i c is C α over [a, b ] and g satisies g c C d s then g = P Vk g with 2 k = max( c /α C, ) α s/α is C α and satisies and β α, (g c) (β) C max( c C α, C d, ) s β/α (3.30) x x x 0 K2 k, (g c) ( α ) (x) (g c) ( α ) (x 0 ) C max( c C α, C d, ) x x 0 α α. (3.3) An immediate consequence o this lemma is that g is C α with a norm bounded by C max( c C α, C d, ) + c C α. The proo o Lemma 3.4 in Appendix A. combines this lemma with the regularity o outside C s to obtain a regularity o W along x and thus obtained a decay o the bandelets coeicients with their vanishing moments in this direction. In Appendix A.2 to obtain Lemma 3.5, some regularity along x or W in the neighborhood o the smoothed singularity is required. This is given by the ollowing lemma proved in Appendix A.4: 0

11 Lemma 3.7. Suppose is a C α horizon model with s > 0. I g satisies β α, (g c) (β) C max( c C α, C d, ) s β/α (3.32) and x x x 0 Ks /α, (g c) ( α ) (x) (g c) ( α ) (x 0 ) then α x α W (x, x 2 ) α W (x, x 2 ) x α C max( c C α, C d, ) x x 0 α α. (3.33) C C α (Λ) max( c α C α, ) max( c α C α, Cα d, ) s x x α α. (3.34) This bound and the condition on the threshold T s (α+)/(2α) are suicient to prove Lemma 3.5 The hypotheses o Proposition 3.3 requires a small threshold, T s α+/(2α), but a similar results also holds or larger T. Proposition 3.8. Under the hypotheses o Theorem 3.2, there exists a constant C such that or all threshold T the resulting bandelet approximation error satisies and M 2 C C 2/(α+) l T 2α/(α+) (3.35) M max(c C 2/(α+) l T 2/(α+), C log 2 T ). (3.36) Proo. [Proposition 3.8] I g c C d s and T 2α/(α+) s, Proposition 3.3 applies. Otherwise, T 2α/(α+) > s and we prove that the condition on the edge estimation o Theorem 3.2, g c C d s, can even be relaxed to g c C d max(s, T 2α/(α+) ). Indeed, let mod = h mod where ( s ) 2 ( s ) h mod = h T 2α/(α+) T x (3.37) 2α/(α+) is a dilation o the smoothing kernel h supported in [ T 2α/(α+), T 2α/(α+) ] 2, which satisies h mod C α (T 2α/(α+) ) (2+α). The ollowing lemma proved in Appendix A.5 implies mod 2 C 2 C (Λ) l T 2α/(α+) : (3.38) Lemma 3.9. I is as C α horizon model, there exists a constant C such that 2 h C 2 C (Λ) l s (3.39) Furthermore Proposition 3.3 applies to mod so and mod mod,mmod 2 C.C 2/(α+) l T 2α/(α+) (3.40) M mod max(c C 2/(α+) l T 2/(α+), log 2 T ). (3.4)

12 Let J be the set o bandelets such that mod, b m T, as mod 2 C C 2/(α+) l T 2α/(α+), (3.42) one can veriy that, b m 2 2( mod, b m 2 + mod, b m 2 ) (3.43) m/ J m/ J m/ J 2( mod mod,mmod 2 + mod 2 ) (3.44) m/ J, b m 2 C C 2/(α+) l T 2α/(α+) (3.45) while Card (J ) C C 2/(α+) l T 2/(α+). To conclude, we rely on the optimality o the thresholding strategy in the ollowing sense: Lemma 3.0. Let B = {u m } m J be a amily o unctions and J T = {m :, u m > T }. I J J then, u m 2 + T 2 Card (J T ), u m 2 + T 2 Card (J ) m/ J T m/ J. (3.46) Proo. [Lemma 3.0] For every J J,, u m 2 + T 2 Card (J ) = (( m J ), u n 2 + m J T 2 ) m/ J m J. (3.47) This implies that each term o the sum is minimized i m J only when, u m 2 T 2 i.e. the sum is minimal or J = J T. Indeed this implies that the bounds obtained or the subset J remains valid or the subset J T obtained with the thresholding. As proved in Lemma 3.0,, u m 2 + T 2 Card (J T ), u m 2 + T 2 Card (J ) (3.48) m/ J T m/ J C C 2/(α+) l T 2α/(α+) + T 2 C C 2/(α+) l T 2/(α+) (3.49) m/ J T, u m 2 + T 2 Card (J T ) C C 2/(α+) l T 2α/(α+) (3.50) which immediately implies (3.35) and (3.36). Theorem 3.2 proves that or a horizon model the bandelet approximation error has an optimal asymptotic decay i the low integral g is computed at an appropriate scale 2 k that depends upon, rom an estimation g that is suiciently precise. Section 4.2 explains how to compute such a low and ind the approximation scale 2 k, in a more general setting, with a best basis strategy that minimizes the approximation error. 4. Bandelet Frames and Approximations. 4.. Bandelet Frames. To approximate piecewise regular images having several edges over [0, ] 2, the image support is partitioned into regions [0, ] 2 = i Ω i, inside each o which the restriction o is either uniormly regular or is a horizon model or has two edges that meet. This means that in each Ω i has either no edge 2

13 or a single edge whose tangents have an smaller angle than π/3 with the horizontal or with the vertical, or has an edge junction. This is illustrated by Figure 4. with square regions. The size o squares become smaller in the neighborhood o junctions. Bandelet bases are deined over bands that include the regions Ω i and it is shown that their union deines a rame o L 2 [0, ] 2. Fig. 4.. Partition o an image I is uniormly regular in a region Ω i then there is no need to deine a geometric low. Similarly, i has an edge junction in Ω i then there is no geometric regularity and no appropriate geometric low can be deined. In both cases, is decomposed in a separable wavelet basis B i = {b i,m } m, that is constructed over the smallest rectangle B i that includes Ω i. I is a horizon model over Ω i then a vertically or horizontally parallel geometric low is deined over Ω i. Let B i be the most narrow band parallel to the low in Ω i and that includes Ω i. Figure 4.2 gives an example. Section 3. explains how to construct a bandelet orthonormal basis B i = {b i,m } m o L 2 (B i ). The ollowing proposition proves that the union o such orthonormal bases over a segmentation o [0, ] 2 deines a rame o L 2 ([0, ] 2 ), that is called a bandelet rame. We write P Ωi the orthogonal projector deined by P Ωi (x) = { (x) i x Ωi 0 i x / Ω i. (4.) B i Ω i Fig A square Ω i with a low and its associated minimum band B i. 3

14 Proposition 4.. For any segmentation [0, ] 2 = i Ω i and L 2 [0, ] 2 = i,m, b i,m b i,m with bi,m = P Ωi b i,m. (4.2) and 2 i,m, b i,m 2. (4.3) I the sup over all x [0, ] 2 o the number o bands B i that includes x is a inite number A then 2 A, b i,m 2 (4.4) i,m and the union o bandelet bases F = i B i is a rame o L 2 ([0, ] 2 ). Proo. [Proposition 4.] Since B i = {b i,m } m is an orthonormal basis o L 2 (B i ), P Bi = m, b i,m b i,m. Moreover Ω i B i and [0, ] 2 = i Ω i so = i P Ωi = i P Ωi P Bi = i,m, b i,m P Ωi b i,m. (4.5) which proves (4.2). To prove (4.3) we shall veriy a slightly more general property that will be useul later: I = α i,m 2. (4.6) i,m α i,m bi,m then 2 i,m Indeed P Ωi = PΩi m α i,m b i,m so P Ωi 2 m α i,m 2 and since 2 = i P 2 Ω i we get (4.6). Applying (4.6) to (4.2) yields (4.3). Since the sup over all x [0, ] 2 o the number o regions B i that includes x is A, one veriies that i 2 B i A 2 rom which we derive (4.4) by inserting (4.5). This scheme provides a direct reconstruction o an image rom its bandelet coeicients. The discontinuous nature o the border b i,m leads to blocking eects which have to be avoided in image processing, but which do not degrade the error decay. To suppress these discontinuities, the reconstruction can be computed with the classical iterative rame algorithm on the ull bandelet rame. This would yield a smaller approximation error and may avoid the blocking eect but requires an iterative reconstruction algorithm. Another solution is proposed in [4], where the bandelets themselves are modiied in order to cross the boundaries, which removes the blocking arteacts. The bandelet liting scheme removes the blocking eects but we then have no proo that the resulting best basis algorithm described in the next section yields an approximation whose error decay is optimal or geometrically regular unctions Approximation in a Dictionary o Bandelet Frames. A bandelet rame is deined by geometric parameters that speciy the segmentation o the image support into subregions Ω i and by the geometric low in each Ω i. A dictionary o bandelet rames is constructed with segmentations in dyadic square regions and a parameterization o the geometric low in each region. Within this dictionary, a best bandelet rame is deined by minimizing a Lagrangian cost unction. The main 4

15 theorem computes the error when approximating a geometrically regular unction in a best bandelet rame. Like in the wedgelet bases o Donoho [9], the image is segmented in dyadic square regions obtained by successive subdivisions o square regions into our squares o twice smaller width. For a square image support o width L, a square region o width L 2 λ is represented by a node at the depth λ o a quad-tree. A square subdivided into our smaller squares corresponds to a node having our children in the quadtree. Figure 4.3 gives an example o a dyadic square image segmentation with the corresponding quad-tree. This tree will be coded by the list o the splitting decision at each inner node Fig A dyadic square segmentation and its corresponding quadtree In each dyadic square Ω i o size 2 λi a variable indicates i the basis is a wavelet or bandelet basis and in this last case whether the geometric low is constant vertically or horizontally. I it exists, the low is characterized by the decomposition (3.5) o an integral curve over a amily o scaling unctions at a scale 2 ki. Let F = i B i be the bandelet rame resulting rom such a dyadic segmentation o [0, ] 2 and such a parameterized low. An approximation o in F is obtained by keeping only the bandelet coeicients above a threshold T : M =, b i,m b i,m, (4.7) i,m,b i,m T with b i.m = P Ωi b i,m. The total number o parameters is M = M S + M G + M B where M S is the number o geometric parameters that deine the dyadic image segmentation, M G is the number o parameters that deine the geometric lows in all the dyadic squares and M B is the number o bandelet coeicients above T. Optimizing the rame means inding F, that depends upon and M, in a dictionary D o bandelet rame so that M 2 C M β (4.8) or the largest possible exponent β. Since M =,b i,m <T, b i,m b i,m we derive rom (4.6) that M 2 i,m,b i,m <T 5, b i,m 2. (4.9)

16 To control M 2 or a ixed number o parameters M = M S + M G + M B, we thus need to minimize,b, b i,m <T i,m 2. This is done indirectly by minimizing a Lagrangian, similarly to the strategy used by Donoho [9] to optimize the geometry o wedgelet approximations: L(, T, F) =, b i,m 2 + T 2 M with M = M S + M G + M B. (4.0) b i,m F,b i,m <T The Lagrangian multiplier is T 2 because at the threshold level the squared amplitude T 2 o a bandelet coeicient should increase the Lagrangian by the same amount as an increase by o the number M B o bandelet coeicients. The best bandelet rame F is the one that minimizes the Lagrangian (4.0) over a dictionary o bandelets. To reduce the dictionary D to a inite size, the resolution o the image geometry is limited to T 2, which we shall see is suicient to approximate eiciently. This means irst that the width o the square image regions remain larger than T 2 and hence that the maximum depth o the quadtree representing the dyadic square image segmentation is log 2 T 2. It also means that the decomposition coeicients α n o the geometry deined in (3.5) are quantized: Q T 2(α n ) = q T 2 i (q /2) T 2 α n < (q + /2) T 2 with q Z. In a dyadic square o size 2 λ, since α n C θ 2 λ, we necessarily have q C θ 2 λ T 2 C θ T 2. The dictionary D T 2 o bandelet rames constructed over this geometry o resolution T 2 thus has a inite size and we can ind a best basis that minimizes the Lagrangian (4.0) over this dictionary. We now concentrate on the approximation capabilities o a best bandelet rame obtained by minimizing L(, T, F) over the dictionary o rames D T 2, when has some geometric regularity. The ollowing deinition speciies the geometric regularity conditions on that will used in the remaining o the paper. Deinition 4.2. A unction is C α geometrically regular over [0, ] 2 i = or = h with C α (Λ) or Λ = [0, ] 2 {C γ } γ G, the blurring kernel h is C α, compactly supported in [ s, s] 2 and h C α s (2+α), the edge curves C γ are Hölderian o order α and do not intersect tangentially i α >. The ollowing theorem gives an upper bound o the approximation error o in a best bandelet rame. Theorem 4.3. Let be a C α geometrically regular unction and α < p. There exists C that only depends upon the edges geometry such that or any T > 0 the thresholding approximation o in a best bandelet rame o D T 2 yields an approximation M that satisies with M 2 C C 2 max(, l C) α+ M α (4.) with M C max(, l C ) C 2/(α+) T 2/(α+) (4.2) where C = max( Cα (Λ) max( c α C α, )2, c (α+)/(2α) C α length o the curves. 6, α+ C (Λ) ) and l C is the total

17 The remarks on the non optimality o the constant C o Theorem 3.2 apply here. In addition, the dependency o C on the edges geometry is not explicitly controlled but involves the number o curves G as well as the geometric coniguration (distance, angle o the crossing,... ). Proo. [Theorem 4.3] The theorem proo is based on the ollowing lemma which exhibits a bandelet rame in D T 2 having suitable approximation properties when α > and thus the edges have tangents. The remaining case, α =, is obtained with the classical wavelet basis that is included in the dictionary. Lemma 4.4. Under the hypotheses o Theorem 4.3 and i α >, or any T > 0 there exists a bandelet rame F D T 2 such that L(, T, F) C C 2/(α+) max(, l C ) T 2α/(α+), (4.3) or a constant C that only depends upon the edges geometry. We irst prove that this lemma implies Theorem 4.3 and will then prove the lemma. Let F = {b i,m } i,m be a best bandelet rame that minimizes L(, T, F) in D T 2. Clearly, L(, T, F ) L(, T, F), where F is the rame o Lemma 4.4, and hence L(, T, F ) = As a result and b i,m F,b i,m <T b i,m F,b i,m <T, b i,m 2 + T 2 M C C 2/(α+) max(, l C ) T 2α/(α+). (4.4), b i,m 2 C C 2/(α+) max(, l C ) T 2α/(α+) (4.5) M C C 2/(α+) max(, l C ) T 2α/(α+) T 2 (4.6) M C C 2/(α+) max(, l C ) T 2/(α+). (4.7) Inserting (4.7) in (4.5) yields, b i,m 2 C C 2 max(, l C ) M α, (4.8) b i,m F,b i,m <T and since we saw in (4.9) that M 2 b i,m F,b i,m <T, b i,m 2, this proves the theorem result (4.). The core o Theorem 4.3 is thus the Lemma 4.4 whose proo is constructive. Proo. [Lemma 4.4] We irst design a dyadic square segmentation associated to and T and a low in each square so that the resulting bandelet rame F satisies (4.3). This segmentation is constructed by separating squares that are close to an edge rom the others. egular squares i I are squares which are distant by more than s rom all edges C γ. In such squares, is uniormly regular. No geometric low is deined in regular squares, which means that the bandelet basis is a separable wavelet basis. 7

18 Let us now consider squares that include or are adjacent to a single edge. We say that a region Ω includes a single horizontal edge component i it is crossed horizontally by a single edge C γ that remains at distance smaller than s and whose tangents have angles with the horizontal smaller than π/3 and i there is no other edge at a distance smaller than s. It is tempting to use this deinition directly on the Ω i. However, to eiciently approximate with such bandelets, one must also veriy that the larger band B i includes no other edge component. To enorce this property, i the size o Ω i is 2 λ, a larger vertical rectangle Ω i o height 7 2 λ and width 2 λ, centered in Ω i is considered. We shall veriy that B i Ω i. A square Ω i is said to be a horizontal edge square i Ω i is at a distance smaller than s rom an edge curve and i Ω i includes a single horizontal edge component C γ and remains at a distance larger than s rom the two straight horizontal segment o the boundary o [0, ] 2, and we write i I H. In this case, a vertically parallel geometric low integral is deined in Ω i by approximating c i at a scale 2 k = max( c /α C, ) α η/α, where η = max(s, T 2α/(α+) ) plays the role o a geometric resolution, and by quantizing the resulting coeicients: with g i (x) = 2 λ k n= Q T 2 ( ) ci, θ k,n θ k,n 2 θ k,n (x) (4.9) Q T 2(x) = qt 2 i (q /2)T 2 x < (q + /2)T 2 with q Z. Lemma 3.6 proves that choosing 2 k = max( c /α C, ) α η/α implies that the error between g i and c i satisies g i c i C max( c C α, ) η. Vertical edge squares are deined similarly by inverting the roles o the horizontal and vertical directions and hence o x and x 2. I Ω i is a square o size 2 λ then the Ω i is a horizontal rectangle o height 2 λ and width 7 2 λ. A square Ω i is said to be a vertical edge square i Ω i is at a distance smaller than s rom an edge curve and i Ω i includes a single vertical edge component C γ and remains at a distance larger than s rom the two straight vertical segment o the boundary o [0, ] 2. We then write i I V. A horizontally parallel low is then deined in Ω i by approximating the edge parameterization c i as in (4.9). A square that is both a horizontal edge square and a vertical edge square is considered as a horizontal edge square by deault. The ollowing algorithm constructs a dyadic image segmentation by labeling regular, vertical edge and horizontal edge squares, and introduces a third residual class called junction squares because we shall see that they are close to junction points. No geometric low is deined in junction squares because they are close to several edges, and the corresponding bandelet basis is thus a separable wavelet basis. Initialization: label the square [0, ] 2 a temporary square. Step : Split in our every temporary square and remove it. Step 2: Label, in the ollowing order, each new subdivided square Ω i as a: regular square i I i it is at a distance larger than s rom all edges junction square i I J i its size is smaller than η horizontal edge square i I H i Ω i includes a single horizontal edge component vertical edge square i I V i Ω i includes a single vertical edge component. temporary square otherwise. 8

19 J H J J J J J J J J J J H H H V V J J H H H V V V V V V H H V V V V V H H V V V V V V V V V V V V V V H H V V H H H V V V V V V V H V V V V V V H V J J J H H H H H J J J H H H H J J H H H H H H H H H H H H H H H H H H H H H H H H H H H J J H H H H H H H H H H H H V H J J J J H H Fig Partition with dyadic square labeled J,, H and V. H H H V H H V V V V H H V V V V V V H H H H H H H V V V V V V V V H H V V H H H H H H H H H H H H H V H H V V V Fig Close up on a junction zone with an intermediate partition with dyadic square labeled, H, V and temporary square. Step 3: Go to step i there remains temporary squares. Figure 4.4 illustrates such a labeled segmentation. From such a dyadic image partition, we associate a bandelet rame F as the union o the bandelet bases deined with the geometric low constructed (or not) in each square depending upon their label. We now prove that the resulting Lagrangian satisies the property (4.3) o Lemma 4.4. To evaluate the Lagrangian, L(, T, F) =, b i,m 2 + T 2 (M S + M G + M G ) (4.20) b i,m F,b i,m <T we separate the geometrical cost M S + M G and decompose it into: L(, T, F) = T 2 (M S + M G ) + L(, T, Bi ) + L(, T, Bi ) i I i I J + L(, T, Bi ) + L(, T, Bi ), i I H i I V with L(, T, B i ) = b i,m B i,b i,m <T 9 (4.2), b i,m 2 + T 2 M B,i (4.22)

20 where M B,i is the number o inner-products, b i,m T or a ixed i. To prove that L(, T, F) C C 2/(α+) max(, l C ) T 2α/(α+), a similar bound or T 2 (M S + M G ) and each o the our partial sums corresponding to dierent class o squares in (4.2) shall be proved. The irst lemma characterizes the dyadic segmentation obtained by our splitting algorithm and computes the resulting number o geometric parameters. Its proo can be ound in Appendix B.. Lemma 4.5. There exists a constant C that depends upon the edges geometry such that the resulting dyadic image segmentation deined recursively includes at most C log 2 max(s, T 2α/(α+) ) squares with at most C junction squares. Furthermore T 2 (M S + M G ) C l C max( c /α C α, ) T 2α/(α+). (4.23) The next lemma computes the Lagrangian value on regular squares by using standard wavelet approximation properties or uniormly regular unctions. Its proo is the appendix B.2. Lemma 4.6. There exists a constant C that depends upon the edges geometry such that the sum o the partial Lagrangian L i over each regular square Ω i (i I ) satisies Li (, T, B i ) C C 2/(α+) T 2α/(α+). (4.24) i I For junction squares the Lagrangian value is calculated using the act that there are ew such squares and they have a small size bounded by η. The lemma s proo is in Appendix B.3. Lemma 4.7. There exists a constant C that depends upon the edges geometry such that the sum o the partial Lagrangian L i over all junction square Ω i (i I J ) satisies Li (, T, B i ) C C 2/(α+) T 2α/(α+). (4.25) i I J The inal lemma gives an upper bound on the Lagrangian o bandelets constructed over vertical edge squares. This proo in appendix B.4. It is at the core o our construction and relies on the precision o the geometric low used in each edge square. Lemma 4.8. There exists a constant C that depends upon the edges geometry such that the sum o the partial Lagrangian L i over all horizontal edge square Ω i (i I H ) satisies L(, T, Bi ) C C 2/(α+) max(, l C ) T 2α/(α+). (4.26) i I H Transposing this result to vertical edge squares proves that L(, T, Bi ) C C 2/(α+) max(, l C ) T 2α/(α+). (4.27) i I V Inserting (4.23), (4.24), (4.25), (4.26) and (4.27) in equation (4.2) proves the result (4.3) o Lemma

21 This theorem provides a constructive approximation scheme with an optimal approximation bound. The decay rate M α o the error is optimal as it is the same as the optimal one or uniormly C α unctions. It is much better than the decay rate M or the wavelets and improves the decay rate M 2 (log 2 M) 3 o the curvelets even i α = 2. Furthermore, the Lagrangian minimization does not require any inormation on the regularity parameter α or on the smoothing kernel h. However, an exhaustive search to minimize the Lagrangian in the dictionary D T 2 requires an exponential number o operations which prohibits its practical use. Section 6.2 introduces a modiied dictionary and a ast algorithm that inds a best bandelet basis with a polynomial complexity, at the cost o adding a logarithmic actor in the resulting approximation error. 5. Image Compression. For image compression, we must minimize the total number o bits required to encode the approximation as opposed to the number o parameters M. An image is compressed in a bandelet rame by irst coding the segmentation o the image support and a geometric low in each region o the segmentation. The decomposition coeicients o the image in the resulting bandelet rame are then quantized and stored with a binary code. This very simple algorithm does not provide a scalable scheme but does give an almost optimal distortion-rate. We denote by the resulting total number o bits to encode a bandelet rame and the bandelet coeicients o in this rame. It can be decomposed into = S + G + B (5.) where S is the number o bits to encode the dyadic square segmentation, G is the number o bits to encode the low in each square region and B is the number o bits to encode the quantized bandelet coeicients. We saw in Section 4.2 that a dyadic square segmentation o [0, ] 2 is represented by a quadtree whose leaves are the square regions o the partition. Each interior node o the tree corresponds to a square that is split in our subsquares. This splitting decision is encoded with a bit equal to. The leaves o the tree correspond to squares which are not split, which is encoded with a bit equal to 0. With this code, the number o bits S that speciies the segmentation quadtree is thus equal to the number o nodes o this quad-tree. Over a square o size 2 λ, the geometric low is parameterized at a scale 2 k in (3.5) by 2 λ k quantized coeicients α m = q T 2 with q T 2 C θ. We thus need 2 λ k log 2 (C θ T 2 ) bits to encode these α m. The number o bits G to encode the lows is the sum o these values over all squares where a low is deined plus the number o bits required to speciy the scale. In a bandelet rame F = {b i,m } i,m, all bandelet coeicients, b i,m are uniormly quantized with a uniorm quantizer Q T o step T : Q T (x) = qt i (q /2)T x < (q + /2)T with q Z. The indices i, m o the M B non-zero quantized coeicients are encoded together with the value Q T (, b i,m ) 0. The proo o Theorem 4.3 shows that the M B non-zero quantized bandelet coeicients whose amplitude is larger than T appear at a scale larger than C ψ T. Since there are C2 ψ 2 T 2 such coeicients, to encode an index i, m is equivalent to encode an integer in [, Cψ 2 2 T 2 ] which requires log 2 (Cψ 2 2 T 2 ) bits. Since, b i,m, each non-zero quantized coeicient is 2