Optimally Sparse Representations of Cartoon-Like Cylindrical Data

Abstract

Sparse representations of multidimensional data have received a significant attention in the literature due to their applications in problems of data restoration and feature extraction. In this paper, we consider an idealized class ${\mathcal {C}}^2(Z) \subset L^2({\mathbb {R}}^3)$ of 3-dimensional data dominated by surface singularities that are orthogonal to the xy plane. To deal with this type of data, we introduce a new multiscale directional representation called cylindrical shearlets and prove that this new approach achieves superior approximation properties not only with respect to conventional multiscale representations but also with respect to 3-dimensional shearlets and curvelets. Specifically, the N-term approximation $f_N^S$ obtained by selecting the N largest coefficients of the cylindrical shearlet expansion of a function $f \in {\mathcal {C}}(Z)$ satisfies the asymptotic estimate $\Vert f - f_N^S\Vert _2^2 \le c \, N^{-2} \, (\ln N)^3, \quad \text {as } N \rightarrow \infty .$ This is the optimal decay rate, up the logarithmic factor, outperforming 3d wavelet and 3d shearlet approximations which only yield approximation rates of order $N^{-1/2}$ and $N^{-1}$ (ignoring logarithmic factors), respectively, on the same type of data.

A: Optimal Approximation Rates

Our analysis of the optimal sparsity rate adapts the method in [5] where, for functions f in a function class ${\mathcal {F}}$, one considers adaptive representations in an overcomplete dictionary ${\varPhi } =\{\phi _i: i \in I\} \subset L^2({\mathbb {R}}^2)$ of the form

$$\begin{aligned} f = \sum _{i \in I_f} c_i(f) \, \phi _i, \end{aligned}$$

(24)

and the selection of $I_f$ in I is required to satisfy a polynomial depth search constraint to avoid situations which are computationally unfeasible. The sparsity of the expansion (24) is measured in terms of the quasi-norm $\Vert c(f)\Vert _{w \ell ^p}$, where $c(f)=(c_i(f))$, with the optimal degree of sparsity being defined as the smallest p such that $\Vert c(f)\Vert _{w \ell ^p}$ is bounded. Hence, denoting by $|c(f)|_m$ the m-th largest entry in the coefficient sequence (|c(f)|), there is a constant $C>0$ such that

$$\begin{aligned} \sup _{f \in {\mathcal {F}}}|c(f)|_m \le C \, m^{-\frac{1}{p}}, \end{aligned}$$

(25)

and no decay rate faster than $m^{-\frac{1}{p}}$ is possible. It follows that, if ${\varPhi }$ is also a Parseval frame, then, from (25) we have

$$\begin{aligned} \Vert f-f_N\Vert ^2 \le \sum _{m>N} |c(f)|^2_m \le C \sum _{m>N} m^{-2/p} \le C \, N^{-2/p+1} \end{aligned}$$

and $O(N^{-2/p+1})$ is the optimal decay rate as no better approximation can be achieved under the procedure described above.

The argument in [5] to determine the optimal p for which $\Vert c(f)\Vert _{w \ell ^p}$ is bounded for $f \in {\mathcal {F}}$ requires to assess the value p such that ${\mathcal {F}}$ contains a copy of $\ell ^p$. Recall that ${\mathcal {F}}$ contains a copy of $\ell ^p$ if it contains embedded orthogonal hypercubes of dimension $M({\varDelta })$ and side ${\varDelta }$ such that, for some sequence $({\varDelta }_k) \rightarrow 0$, there is a constant $C>0$ such that

$$\begin{aligned} M({\varDelta }_k) \ge C \, {\varDelta }_k^{-p}, \quad k=k_0, k_0+1, \dots \end{aligned}$$

Theorem 4

The class ${\mathcal {C}}^2(Z) \subset L^2({\mathbb {R}}^3)$ contains a copy of $\ell ^p$ for $p=\frac{2}{3}$.

Theorem 4 implies that no representation system satisfying polynomial depth search constraint can provide approximations for ${\mathcal {C}}^2(Z)$ with the coefficients $\Vert c(f)\Vert _{w\ell ^p} < \infty$, for $p < \frac{2}{3}$, i.e., $p=\frac{2}{3}$ is the optimal value. It follows that, if $\Vert c(f)\Vert _{w \ell ^{\frac{2}{3}}} < \infty$, then there is a constant $C>0$ such that

$$\begin{aligned} \sup _{f \in {\mathcal {C}}^2}|c(f)|_m \le C \, m^{-\frac{3}{2}}, \end{aligned}$$

and no decay rate faster than $m^{-\frac{3}{2}}$ is possible. As seen above, the last inequality implies that, if $f_N$ is the best N term approximation to $f \in {\mathcal {C}}^2(Z)$ using a Parseval frame, then

$$\begin{aligned} \Vert f-f_N\Vert ^2 \le C \sum _{m>N} m^{-3} \le C \, N^{-2}. \end{aligned}$$

Hence Theorem 3 shows that the approximation rate of cylindrical shearlets is nearly optimal. Note the rate $O(N^{-2})$ is also the optimal approximation rate for functions in the class of 2d cartoon-like images [5].

Proof of Theorem 4

Since our proof follows closely the Proof of Theorem 3 in [5], we will mainly emphasize the modifications needed for our case.

Let g be a smooth and non-negative function with compact support in $[0,2 \pi ]$. For scalars A and $m(A,\delta )$ to be determined, let $g_{i,m}(t)= A \, m^{-2} g(m t- 2 \pi i)$ for $i =0,1, \ldots , m-1.$ Notice that $\Vert g_{i,m}\Vert _{C^2} = A \, \Vert g\Vert _{C^2}$ and $\Vert g_{i,m}\Vert _{L^1}= A \, m^{-3} \Vert g\Vert _{L^1}$. We introduce polar coordinates $(\rho ,\theta )$ with origin in $\big (\frac{1}{2},\frac{1}{2}\big )$. For $\rho _0=\frac{1}{4}$, using cylindrical coordinates in ${\mathbb {R}}^3$, we set

$$\begin{aligned} \psi _{i,m}(\rho ,\theta ,z) = \left( \chi _{\{\rho \le \rho _0\}}(\rho ,\theta ) - \chi _{\{\rho \le g_{i,m}+\rho _0\}}(\rho ,\theta )\right) w(z), \quad i =0,1, \dots , m-1, \end{aligned}$$

where w is a $C^2$ function with compact support. Hence, we define the radius functions $r_\xi =\frac{1}{4}+\sum _{i=1}^m \xi _{i} \, g_{i,m},$ where $\xi _{i} \in \{0,1\}$ and the corresponding functions

$$\begin{aligned} f_\xi (\rho ,\theta ,z)=\chi _{\{\rho \le \rho _0\}}(\rho ,\theta ) \, w(z)+\sum _{i=1}^m \xi _{i} \, \psi _{i,m}(\rho ,\theta ,z), \quad \xi _{i} \in \{0,1\}. \end{aligned}$$

The functions $\psi _{i,m}$ are bulges around a cylinder of radius $\rho _0$ and have disjoint support; each $f_\xi$ is the indicator function of the cylinder of radius $\rho _0$ plus some addition bulges. Since g is bounded and non-negative, a direct calculation shows that

$$\begin{aligned} \Vert \psi _{i,m}\Vert _{L^2}^2 \simeq \Vert g_{i,m}\Vert _{L^1}= A \, m^{-3} \Vert g\Vert _{L^1}, \end{aligned}$$

and, for each radius function $r_\xi$, $\Vert r_\xi \Vert _{C^2} \le \Vert g_{i,m}\Vert _{C^2} = A \, \Vert g\Vert _{C^2}.$ Hence, the hypercube embedding is achieved whenever $A \le Z/\Vert g\Vert _{C^2}$.

Whenever $A \le Z/\Vert g\Vert _{C^2}$, the sidelength ${\varDelta } = \Vert \psi _{i,m}\Vert _{L^2}$ of the hypercubes satisfies:

$$\begin{aligned} \Vert \psi _{i,m}\Vert _{L^2}^2 = {\varDelta }^2 \simeq \Vert g_{i,m}\Vert _{L^1}= A \, m^{-3} \Vert g\Vert _{L^1} \le Z \, m^{-3} \frac{\Vert g\Vert _{L^1}}{\Vert g\Vert _{C^2}}. \end{aligned}$$

Hence, setting

$$\begin{aligned} m(\delta ) = \left\lfloor \left( \frac{\delta ^2}{Z}\frac{\Vert g\Vert _{C^2}}{\Vert g\Vert _{L^1}} \right) ^{-\frac{1}{3}}\right\rfloor , \quad A(\delta ,Z) = \delta ^2 m^{3}/\Vert g\Vert _{L^1}, \end{aligned}$$

it follows that $A \le Z/\Vert g\Vert _{C^2}$ and ${\varDelta } \simeq \delta$, which shows that the hypercube embedding is satisfied with sidelength ${\varDelta } \simeq \delta$ and the dimension of the hypercube obeys $m(\delta ) \ge K \, Z^{\frac{1}{3}} \, \delta ^{-\frac{2}{3}},$ for all $0< \delta < \delta _0,$ where $\delta _0$ is the solution of

$$\begin{aligned} 2=\left( \frac{\delta _0^2}{Z } \frac{\Vert g\Vert _{C^2}}{\Vert g\Vert _{L^1}} \right) ^{-1/3}, \quad K = \frac{1}{2} \, \left( \frac{\Vert g\Vert _{C^2}}{\Vert g\Vert _{L^1}} \right) ^{-1/3}. \end{aligned}$$

$\square$