Figure 1: Red rock: A 19588×4457 (83-megapixel) panorama (top row) from 9 photos produced by our curl-free wavelet projection in 26.11s on CPU and 0.45s on GPU. The bottom row is the extracted wavelet coefficients by our method. (Data is courtesy of Aseem Agarwala.) AbstractGradient-domain compositing has been widely used to create a seamless composite with gradient close to a composite gradient field generated from one or more registered images. The key to this problem is to solve a Poisson equation, whose unknown variables can reach the size of the composite if no region of interest is drawn explicitly, thus making both the time and memory cost expensive in processing multi-megapixel images. In this paper, we propose an approximate projection method based on biorthogonal Multiresolution Analyses (MRA) to solve the Poisson equation. Unlike previous Poisson equation solvers which try to converge to the accurate solution with iterative algorithms, we use biorthogonal compactly supported curl-free wavelets as the fundamental bases to approximately project the composite gradient field onto a curl-free vector space. Then, the composite can be efficiently recovered by applying a fast inverse wavelet transform. Considering an n-pixel composite, our method only requires 2n of memory for all vector fields and is more efficient than state-of-the-art methods while achieving almost identical results. Specifically, experiments show that our method gains a 5x speedup over the streaming multigrid in certain cases.

We introduce a new biorthogonal wavelet approach to creating a water‐tight surface defined by an implicit function, from a finite set of oriented points. Our approach aims at addressing problems with previous wavelet methods which are not resilient to missing or nonuniformly sampled data. To address the problems, our approach has two key elements. First, by applying a three‐dimensional partial integration, we derive a new integral formula to compute the wavelet coefficients without requiring the implicit function to be an indicator function. It can be shown that the previously used formula is a special case of our formula when the integrated function is an indicator function. Second, a simple yet general method is proposed to construct smooth wavelets with small support. With our method, a family of wavelets can be constructed with the same support size as previously used wavelets while having one more degree of continuity. Experiments show that our approach can robustly produce results comparable to those produced by the Fourier and Poisson methods, regardless of the input data being noisy, missing or nonuniform. Moreover, our approach does not need to compute global integrals or solve large linear systems.

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Figure 1: Tear layer cake. We simulate it as an anisotropic and heterogenous material in different fracture patterns with horizontal fiber orientation. From left to right: (a) Global orthotropic. (b) Only the upper black layer is orthotropic. (c) Only the lower black layer is orthotropic. (d) Both the upper and lower black layers is orthotropic.