Fast Iterative Adaptive Multi-quadric Radial Basis Function Method for Edges Detection of Piecewise Functions—I: Uniform Mesh

Abstract

In Jung et al. (Appl Numer Math 61:77–91, 2011), an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy $O(n^3)$ computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman–Morrison–Woodbury formula. The $O(n^2)$ Levinson–Durbin recursive algorithm that employed Yule–Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Several classical benchmark examples show that the IAMQ-RBF-Fast based edges detection method can be at least three times faster than the original IAMQ-RBF based one. And it can capture an edge with fewer grid points than the multi-resolution analysis (Harten in J Comput Phys 49:357–393, 1983) and just as good as if not better than the L1PA method (Denker and Gelb in SIAM J Sci Comput 39(2):A559–A592, 2017). Preliminary results in the density solution of the 1D Mach 3 extended shock–density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection method demonstrating an excellent performance in term of speed and accuracy, are also shown.

Appendices

Appendix A: Brief Description of the L1PA Method

The L1PA method is used to determine edges from regularized reconstruction using the variance signature. Given the (noisy) Fourier coefficients of a piecewise smooth function from the set $\hat{\mathbf {T}}_q = \hat{\mathbf {F}}\cup \hat{\mathbf {R}}_q$, where $\hat{\mathbf {F}} = \{\hat{f}_k:~-\beta N\le k \le \beta N\}$ with $0 \le \beta \le 1$, and $\hat{\mathbf {R}}_q$ is $\zeta (\gamma -\beta )(2N+1)$ randomly selected coefficients from $\hat{f}_k, |k|>\beta N$, $q = 1,\ldots ,Q$. Here, we set $\beta = 0.3, \gamma = 0.95, \zeta = 0.5, Q = 30$ for each test.

For each subsampled set $\hat{\mathbf {T}}_q$, we reconstruct f on a set of uniform grid points $x_j, j = -N,\ldots ,N$ as

$$\begin{aligned} \vec {f}_q = {\mathrm {argmin}}_{\vec {u}} \left| \left| F_q\vec {u} - \vec {\hat{f}}_q\right| \right| _2^2 + \lambda ||{\mathbf {P}}^{p}\vec {u}||_1, \end{aligned}$$

which can be solved efficiently by the split Bregman algorithm [17]. Here $\vec {\hat{f}}_q$ is vector of Fourier coefficients formed from the subsampled set $\hat{\mathbf {T}}_q$, and $F_q$ is the discrete Fourier transform generating the coefficients to match $\vec {\hat{f}}_q$. The polynomial annihilation transform ${\mathbf {P}}^{p}$, is used as a sparsifying operator in the penalty term. Here we choose $p = 2$. The variance is calculated as

$$\begin{aligned} \vec {v}({\mathbf {Q}})_j = \frac{1}{Q}\sum _{q=1}^{Q}\left( \vec {f_{q}}_{j} - \frac{1}{Q}\sum _{q=1}^{Q}\vec {f_{q}}_{j}\right) ^2, \quad j = 1,\ldots , 2N+1. \end{aligned}$$

Each element in $\vec {v}({\mathbf {Q}})$ differs in convergence properties only within each jump region, that is, the convergence is similar in smooth regions. Using this variance signature, edges detection algorithm is developed for piecewise smooth functions (For details, see [1, 13] and references therein).

Appendix B: List of Parameters

Here, we list what we considered to be optimized settings of parameters that are used in the IAMQ-RBF-Fast method, MR analysis and L1PA method in Table 3 for 1D problems, Table 4 for 2D images and Table 5 for noisy image.

Table 3 The corresponding parameters for the IAMQ-RBF-Fast method, MR analysis and L1PA method for piecewise linear function and 1D Euler equations

Table 4 The corresponding parameters for the IAMQ-RBF-Fast method, MR analysis and L1PA method for 2D images

Table 5 The corresponding parameters for noisy images with different $\omega$