by Hector Chahuara, Anna Krokhine and Elshadai Tegegn

## Introduction

Controlling caustics is a difficult task as any change to specular surface can have large effects on the caustic image. In this post, we address a building block of the optimization framework for computing the shape of refractive objects presented in Schwartzburg et al (2014) and propose to improve a building block on its formulation that performs a mesh reconstruction. The proposed improvement was tested on flatland meshes and performs reasonably well in the presence of noise.

## Theory

Regularization is a technique often needed in optimization for enforcing certain characteristics such as sparsity or stability in the solution. In this particular case, Schwartzburg et al (2014) apply a generalized Tikhonov regularization to achieve a stable solution. Given the optimization-based formulation of Schwartzburg et al (2014), it is possible to isolate this problem and to see that this is in fact mesh denoising. In addition, it is important to mention that Tikhonov regularization is usually outperformed by other methods, among which total variation (TV) denoising distinguishes itself by its reasonable computational cost and good results.

In the following, we apply the framework described in Zhang et al (2015) that applies TV to the noisy normalized normals N_{0} of a mesh, then the optimization problem to solve becomes

min_{N} ||N-N_{0}||^{2} + λ.WVTV(N),

where WVTV is a weighted vectorial TV version, adapted from the one described in Zhang et al (2015) for flatland, defined as

WVTV(N) = Σ_{e}ω_{e} (l_{e}(D_{x}N)^{2}+(D_{y}N)^{2})^{0.5}

l_{e} is the length of the edge “e”, and the weights ω_{e} , dependent on the difference of two consecutive normals N_{i} and N_{i+1} i.e. normals that correspond to adjacent edges are defined as

ω_{e}=exp(-||N_{i}-N_{i+1}||^{4})

to penalize less sharp features (high difference between consecutive normals) than the smooth ones (similar consecutive normals). It is important to mention that TV denoising is a non-smooth optimization problem, so a solver that rely solely on gradient or Hessian information could not reach the optimum. By using the iteratively reweighted least squares (IRLS) Wolke et al (1988), a well-known method for optimization, it is straightforward to build an algorithmto solve this problem .

## Results

The described approach was implemented in MATLAB R2021b. Visual results can be observed in the following animations that show the denoising process of two figures corrupted by noise: a square and a circle.

## Conclusion

The implemented method yields reasonable results for the presented cases. While this exploratory experiments indicate that the method has the potential to improve results if embedded in the general caustics framework. Nonetheless, more experiments are needed to confirm this and to assess the impact on the quality of the result.

## References

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design, “ ACM Trans. Graph. 33, 4, 2014

H. Zhang, C. Wu, J. Zhang and J. Deng, “Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space,” in *IEEE Transactions on Visualization and Computer Graphics*, vol. 21, no. 7, pp. 873-886, 1 July 2015

Wolke, R., Schwetlick, H., “Iteratively Reweighted Least Squares: Algorithms, Convergence Analysis, “ and Numerical Comparisons. *SIAM Journal on Scientific and Statistical Computing, 9*, 907-921, 1988

L. Condat, “Discrete total variation: New definition and minimization,” *SIAM Journal on Imaging Sciences*, vol. 10, no. 3, pp. 1258-1290, 2017