Pointwise Besov Space Smoothing of Images

Abstract

We formulate various variational problems in which the smoothness of functions is measured using Besov space semi-norms. Equivalent Besov space semi-norms can be defined in terms of moduli of smoothness or sequence norms of coefficients in appropriate wavelet expansions. Wavelet-based semi-norms have been used before in variational problems, but existing algorithms do not preserve edges, and many result in blocky artifacts. Here, we devise algorithms using moduli of smoothness for the $B^1_\infty (L_1(I))$ Besov space semi-norm. We choose that particular space because it is closely related both to the space of functions of bounded variation, ${\text {BV}}(I)$, that is used in Rudin–Osher–Fatemi image smoothing, and to the $B^1_1(L_1(I))$ Besov space, which is associated with wavelet shrinkage algorithms. It contains all functions in ${\text {BV}}(I)$, which include functions with discontinuities along smooth curves, as well as “fractal-like” rough regions; examples are given in an appendix. Furthermore, it prefers affine regions to staircases, potentially making it a desirable regularizer for recovering piecewise affine data. While our motivations and computational examples come from image processing, we make no claim that our methods “beat” the best current algorithms. The novelty in this work is a new algorithm that incorporates a translation-invariant Besov regularizer that does not depend on wavelets, thus improving on earlier results. Furthermore, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the smoothing parameter. We also analyze the norms of smooth, textured, and random Gaussian noise data in $B^1_\infty (L_1(I))$, $B^1_1(L_1(I))$, ${\text {BV}}(I)$ and $L^2(I)$ and their dual spaces. Numerical results demonstrate properties of solutions obtained from this moduli of smoothness-based regularizer.

Appendix

Here we describe a family of self-similar functions with fractal-like properties that illustrate some of the differences between membership in the Besov space $B^1_\infty (L_1(I))$ and the space of functions of bounded variation ${\text {BV}}(I)$.

We begin with the function $\varPhi$ that is continuous on $\mathbb R^2$, zero outside of I, with $\varPhi ((\frac{1}{2},\frac{1}{2}))=1$, and a linear polynomial on the triangles in I delineated by the boundary of I, the line $x_2=x_1$, and the line $x_1+x_2=1$. The graph of $\varPhi$ is a pyramid with base I and height 1.

We define $\phi (x)=\frac{1}{4}\varPhi (2x)$, so $\phi$ is a scaled dyadic dilate of $\varPhi$ with support $[0,1/2]^2$.

We construct the sequence of functions $f_0(x)=\phi (x)$ and

$$\begin{aligned} f_{k+1}(x)= & {} \phi (x) +\frac{2}{3}[f_k(2x-(1,0))\nonumber \\&+f_k(2x-(0,1))+f_k(2x-(1,1))]. \end{aligned}$$

(27)

We see that $f_k$ consists of $f_{k-1}$ plus $3^k$ dyadically dilated translates $\phi _{j,k}(x)=\phi (2^kx-j)$ for some multi-indices $j=(j_1,j_2)$ with coefficients $(2/3)^k$. The supports of all the $\phi _{j,k}$ are essentially disjoint.

Finally, we let $f=\lim _{k\rightarrow \infty }f_k$. We can write

$$\begin{aligned} f=\sum _{k=0}^\infty \sum _jc_{j,k}\phi _{j,k}(x),\quad c_{j,k}=\Bigl (\frac{2}{3}\Bigr )^k. \end{aligned}$$

(28)

Thus, f is an infinite sum of scaled, dilated, and translated versions of the single pyramid $\varPhi$; Fig. 7 illustrates the graph of f.

Fig. 7

Graph of the function f in (28) that is in $B^1_\infty (L_1(I))$ but not in ${\text {BV}}(I)$

The arguments in DeVore and Popov [12]¹ show that for any $0<p,q\le \infty$ and $\alpha < \min (2,1+1/p)$ we have

$$\begin{aligned} \Vert f\Vert _{B^\alpha _q(L_p(I))}\asymp \biggl (\sum _k\biggl (\sum _j \bigl [2^{\alpha k}\Vert c_{j,k}\phi _{j,k}\Vert _{L_p(I)}\bigr ]^p\biggr )^{\frac{q}{p}}\biggr )^{\frac{1}{q}}, \end{aligned}$$

with the usual changes when p or q are $\infty$. In our case, we have

$$\begin{aligned} \Vert f\Vert _{B^1_\infty (L_1(I))}\asymp \sup _k\sum _j2^k\Bigl (\frac{2}{3}\Bigr )^k\Vert \phi _{j,k}\Vert _{L_1(I)}, \end{aligned}$$

where for each k there are $3^k$ different offsets j.

We note that $\Vert \phi _{j,k}\Vert _{L_1(I)}=4^{-k}\Vert \phi \Vert _{L_1(I)}$; because there are $3^k$ terms in the sum for each k,

$$\begin{aligned} \sum _j2^k\Bigl (\frac{2}{3}\Bigr )^k\Vert \phi _{j,k}\Vert _{L_1(I)}=\Vert \phi \Vert _{L_1(I)}, \end{aligned}$$

and we have $\Vert f\Vert _{B^1_\infty (L_1(I))}\asymp \Vert \phi \Vert _{L_1(I)}<\infty$.

We’ll now see that f is not in ${\text {BV}}(I)$. Denoting the variation of f by V(f), a simple scaling argument shows that $V(\phi _{j,k})=2^{-k}V(\phi )$. Since the supports of all the $\phi _{j,k}$ in the definition of f are essentially disjoint, the co-area formula shows that

$$\begin{aligned} \begin{aligned} V(f)&=\sum _k\sum _j\Bigl (\frac{2}{3}\Bigr )^kV(\phi _{j,k}) \\&=\sum _k\Bigl (\frac{2}{3}\Bigr )^k\times 3^k\times 2^{-k} V(\phi ) \\&=\sum _kV(\phi )=\infty . \end{aligned} \end{aligned}$$

In other words, there is a constant C such that for all $h\in \mathbb R^2$,

$$\begin{aligned} \Vert f({\cdot }+2h)-2f({\cdot }+h)+f\Vert _{L_1(I_{2h})}\le C|h|, \end{aligned}$$

but there is no constant C such that for all $h\in \mathbb R^2$,

$$\begin{aligned} \Vert f({\cdot }+h)-f\Vert _{L_1(I_{h})}\le C|h|. \end{aligned}$$

Note that by replacing 2 / 3 in (27) with any $0<r<2/3$, the resulting limit f is in both $B^1_\infty (L_1(I))$ and ${\text {BV}}(I)$ (indeed, it’s in $B^1_1(L_1(I))\subset {\text {BV}}(I)$). In this case, we have

$$\begin{aligned} V(f)=\frac{1}{1-(3/2)r}V(\phi ), \end{aligned}$$

so the variation of f tends to $\infty$ as $r\rightarrow 2/3$ (as one might expect), while $\Vert f\Vert _{B^1_\infty (L_1(I))}$ remains bounded.

And if $r>2/3$, the function f is in neither ${\text {BV}}(I)$ nor ${B^1_\infty (L_1(I))}$.

Thus both ${\text {BV}}(I)$ and ${B^1_\infty (L_1(I))}$ contain fractal-like functions, but their norms in ${B^1_\infty (L_1(I))}$ can be arbitrarily smaller than their norms in ${\text {BV}}(I)$.