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Pointwise Besov Space Smoothing of Images

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Pointwise Besov Space Smoothing of Images J. Math. Imaging and Vision manuscript No. (will be inserted by the editor) Pointwise Besov Space Smoothing of Images Gregery T. Buzzard · Antonin Chambolle · Jonathan D. Cohen · Stacey E. Levine · Bradley J. Lucier the date of receipt and acceptance should be inserted later 1 Abstract We formulate various variational problems uli of smoothness for the B1(L1(I)) Besov space semi- in which the smoothness of functions is measured using norm. We choose that particular space because it is Besov space semi-norms. Equivalent Besov space semi- closely related both to the space of functions of bounded norms can be defined in terms of moduli of smooth- variation, BV(I), that is used in Rudin{Osher{Fatemi 1 ness or sequence norms of coefficients in appropri- (ROF) image smoothing, and to the B1 (L1(I)) Besov ate wavelet expansions. Wavelet-based semi-norms have space, which is associated with wavelet shrinkage algo- been used before in variational problems, but existing rithms. It contains all functions in BV(I), which include algorithms do not preserve edges, and many result in functions with discontinuities along smooth curves, as blocky artifacts. Here we devise algorithms using mod- well as \fractal-like" rough regions; examples are given in an appendix. Furthermore, it prefers affine regions BJL was supported in part by the Office of Naval Research, to staircases, potentially making it a desirable regular- Contract N00014-91-J-1076, the Institute for Mathematics and its Applications, Minneapolis, USA, and the Simons izer for recovering piecewise affine data. While our mo- Foundation (Awards #209418 and #229816). SEL and JDC tivations and computational examples come from im- were supported in part by NSF-DMS 1320829. AC acknowl- age processing, we make no claim that our methods edges the support of the Isaac Newton Institute, Cambridge, \beat" the best current algorithms. The novelty in this and of a grant of the Simons Foundation. work is a new algorithm that incorporates a translation- G. T. Buzzard invariant Besov regularizer that does not depend on Department of Mathematics, Purdue University, 150 N. wavelets, thus improving on earlier results. Further- University St., West Lafayette, IN 47907, USA E-mail: [email protected] more, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the A. Chambolle CMAP, Ecole´ Polytechnique, CNRS, 91128 Palaiseau Cedex, smoothing parameter. We also analyze the norms of France smooth, textured, and random Gaussian noise data in E-mail: [email protected] 1 1 2 B1(L1(I)), B1 (L1(I)), BV(I) and L (I) and their dual J. D. Cohen spaces. Numerical results demonstrate properties of so- Department of Mathematics and Computer Science, lutions obtained from this moduli-of-smoothness{based Duquesne University, 440 College Hall, Pittsburgh, PA regularizer. 15282, USA Currently at Google Inc. E-mail: [email protected] S. E. Levine 1 Introduction Department of Mathematics and Computer Science, Duquesne University, 440 College Hall, Pittsburgh, PA This paper has several goals. 15282, USA E-mail: [email protected] We formulate various variational problems in which B. J. Lucier the smoothness of functions is measured using Besov Department of Mathematics, Purdue University, 150 N. space semi-norms. Equivalent Besov space semi-norms University St., West Lafayette, IN 47907, USA can be defined in terms of moduli of smoothness or se- E-mail: [email protected] quence norms of coefficients in appropriate wavelet ex- 2 Gregery T. Buzzard et al. pansions. Wavelet-based semi-norms have been used be- smoothing. Here one calculates an orthogonal wavelet fore in variational problems; here we devise algorithms decomposition of the function f, using moduli of smoothness. X f = c ; In particular, we build algorithms around the j;k; j;k 2 1 j≥0; k2Z ; 2Ψ B1(L1(I)) Besov space semi-norm. We chose that par- k k k k k ticular space because it is closely related both to the where j;k(x) = 2 (2 x−j) = 2 (2 (x−j=2 )) with space of functions of bounded variation (incorporated 2k the dyadic scale, j=2k a dyadic translation, and Ψ in Rudin{Osher{Fatemi (ROF) image smoothing) and a basic set of wavelet functions (often three functions 1 to the B1 (L1(I)) Besov space, which is associated with in two dimensions). Given a positive parameter λ, one wavelet shrinkage algorithms. then smooths f by shrinking the wavelet coefficients to While our motivations and computational examples calculate come from image processing, we make no claim that our ¯ X methods \beat" the best current algorithms in these f = Sλ(cj;k; ) j;k; (2) 2 areas. Perhaps others can improve on our results. j2Z ; k≥0; 2Ψ We give further background and motivation in Sec- where tion 2. Section 3 introduces the modulus-of-smoothness 8 1 >t − λ, λ ≤ t; definition of the B1(L1(I)) semi-norm and discusses < its properties. Section 4 introduces discrete versions of Sλ(t) = 0; −λ ≤ t ≤ λ, (3) > Besov space and BV semi-norms based on moduli of :t + λ, t ≤ −λ. smoothness. The next Section 5 discusses previously- They derived similar results when using hard threshold- introduced algorithms for minimizing various varia- ing tional problems and how they can be adapted to our cir- 8t; λ ≤ t; cumstances. Section 6 gives an algorithm for a projec- <> tion needed in our algorithms. The very brief Section 7 Tλ(t) = 0; −λ ≤ t ≤ λ, > summarizes our main algorithm. We then examine the :t; t ≤ −λ, norms of smooth features, Gaussian noise, and periodic instead of Sλ(t) to calculate the wavelet coefficients of textures in various function spaces in Section 8. We pro- f~. vide computational examples in Section 9. Finally, an In 1992, DeVore and Lucier [11] related both varia- Appendix contains a one-parameter family of bounded, tional image smoothing as in (1) and wavelet shrinkage self-similar functions whose variation tends to infinity as in (2) to interpolation of function spaces using so- 1 while the B1(L(I)) norms are bounded. called (power) K-functionals: given two (quasi-)normed function spaces X and Y and any p; q > 0, one can con- sider 2 Background and motivation K(f; λ, X; Y ) = inf(kf − gkp + λkgkq ) (4) g X Y In this section we discuss some past image smoothing methods and motivations for the approach taken here. for any function f and smoothing parameter λ > 0. One Rudin, Osher, and Fatemi [20] introduced an image sees that (up to a constant) (1) is of the form (4) with smoothing method that is equivalent to the following: X = L2(I), p = 2, and Y = BV(I), q = 1. It is noted in given a noisy image f, represented as a function on the [11], however, that because Sobolev spaces and Besov unit square I = [0; 1]2, and a positive smoothing param- spaces have topologies that can be defined by norms of eter λ, find the function f~ that achieves the minimum the wavelet coefficients of functions in those spaces, we of the functional have that for appropriately chosen λk, k ≥ 0, ¯ X f = Tλ (cj;k; ) j;k (5) 1 2 k K(f; λ) = inf kf − gk + λkgkBV(I) ; (1) 2 g 2 L2(I) j2Z ; k≥0; 2Ψ is, to within a constant, a minimizer of K(f; λ, L (I);Y ) where kgk is the bounded-variation semi-norm of 2 BV(I) whenever Y is one of a family of Sobolev spaces g. (We will not give precise definitions here, see the W α(L (I)), α > 0, or Besov spaces Bα(L (I)), with references for details.) Around the same time, Sauer and 2 q q α > 0 and 1=q = α=2 + 1=2. Bouman [21] used a discrete version of the BV(I) semi- There was speculation in [11] that because BV(I) norm to regularize a tomography inversion problem. satisfies the strict inclusion Not much later, Donoho and Johnstone [15] in- 1 1 troduced the notion of wavelet shrinkage for image B1 (L1(I)) ⊂ BV(I) ⊂ B1(L1(I)) Pointwise Besov Space Smoothing of Images 3 1 1 1 between the Besov spaces B1 (L1(I)) and B1(L1(I)), functions whose norms in B1(L1(I)) are bounded but wavelet shrinkage or wavelet thresholding will likely whose norms in BV(I) tend to 1. 1 give a solution \similar" to the minimizer of (1). How- B1(L1(I)) smoothing should have another nice 1 ever, because the Besov space B1 (L1(I)) does not property. Minimizers of (1) applied to noisy images gen- include functions that are discontinuous across one- erally exhibit stair-casing or terrace-like effects because dimensional curves, what we shall call images with the BV(I) semi-norm does not distinguish between a edges, the minimizing image f~ of staircase or a plane that rises the same distance, while 1 1 the B1(L1(I)) semi-norm of a plane is zero, and the 2 1 kf − gk + λkgkB1(L (I)); B (L (I)) semi-norm of a staircase is nonzero. Thus, 2 L2(I) 1 1 1 1 the terracing effects should not appear in otherwise cannot have such discontinuities, i.e., it cannot have smooth regions of minimizers of (7). edges. In practice, however, if one applies the wavelet Later, Chambolle, DeVore, Lee, and Lucier [7] noted shrinkage algorithm for (7) to images, one finds that that because the minimizers do not generally look much different, X kfk2 = jc j2 qualitatively, than minimizers of (6) with q = 1. L2(I) j;k; j;k; This leads to the natural question about whether 1 other, equivalent, norms for B1(L1(I)) would give dif- for an orthonormal wavelet transform, and the topology 1 ferent results.
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