Texture Map and Video Compression Using Beltrami Representation
08 Oct 2013-Siam Journal on Imaging Sciences (Society for Industrial and Applied Mathematics)-Vol. 6, Iss: 4, pp 1880-1902
TL;DR: This work proposes an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation, a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1.
Abstract: Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as three-dimensional coordinates each vertex is mapped to, which often requires lots of memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed usin...
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TL;DR: FLASH is proposed, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment for genus-0 closed brain surfaces with limited bijectivity under large deformations and slow computation.
Abstract: Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark-matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by f...
121 citations
Cites background or methods from "Texture Map and Video Compression U..."
...We fix the landmark positions and also the region in C that corresponds to a small region enclosing the south pole, and then apply LBS [3] to reconstruct the quasi-conformal map using the same Beltrami differential μ....
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...[3] proposed a linear algorithm, called the linear Beltrami solver (LBS), to reconstruct a quasi-conformal map gi from its Beltrami coefficient μ = ρ+ i τ on rectangular domains in C....
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...In our algorithms, we apply the LBS proposed in [3] to reconstruct a quasi-conformal map from a given Beltrami differential with fixed vertex locations of the big triangle....
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...For the implementation of LBS, please refer to [3]....
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...It is noteworthy that the linear algorithm introduced by [3] works only for rectangular domains....
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TL;DR: In this paper, the conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories.
Abstract: Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.
44 citations
TL;DR: In this article, the conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories.
Abstract: Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.
38 citations
TL;DR: In this article, a double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus 0 closed surfaces can be applied.
Abstract: Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a Mobius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm.
31 citations
References
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TL;DR: DARTEL has been applied to intersubject registration of 471 whole brain images, and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects.
6,999 citations
TL;DR: A review of recent as well as classic image registration methods to provide a comprehensive reference source for the researchers involved in image registration, regardless of particular application areas.
6,842 citations
"Texture Map and Video Compression U..." refers methods in this paper
...A commonly used method is to find surface maps satisfying certain constraints, such as matching landmarks, matching intensity or minimizing distortions [7, 54]....
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TL;DR: The decomposition of deformations by principal warps is demonstrated and the method is extended to deal with curving edges between landmarks to aid the extraction of features for analysis, comparison, and diagnosis of biological and medical images.
Abstract: The decomposition of deformations by principal warps is demonstrated. The method is extended to deal with curving edges between landmarks. This formulation is related to other applications of splines current in computer vision. How they might aid in the extraction of features for analysis, comparison, and diagnosis of biological and medical images in indicated. >
5,065 citations
"Texture Map and Video Compression U..." refers background in this paper
...Landmark-based registration approaches have also been studied [4, 6, 26, 49], which compute registrations that match landmark features consistently....
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TL;DR: This work has developed a means for generating an average folding pattern across a large number of individual subjects as a function on the unit sphere and of nonrigidly aligning each individual with the average, establishing a spherical surface‐based coordinate system that is adapted to the folding pattern of each individual subject, allowing for much higher localization accuracy of structural and functional features of the human brain.
Abstract: The neurons of the human cerebral cortex are arranged in a highly folded sheet, with the majority of the cortical surface area buried in folds. Cortical maps are typically arranged with a topography oriented parallel to the cortical surface. Despite this unambiguous sheetlike geometry, the most commonly used coordinate systems for localizing cortical features are based on 3-D stereotaxic coordinates rather than on position relative to the 2-D cortical sheet. In order to address the need for a more natural surface-based coordinate system for the cortex, we have developed a means for generating an average folding pattern across a large number of individual subjects as a function on the unit sphere and of nonrigidly aligning each individual with the average. This establishes a spherical surface-based coordinate system that is adapted to the folding pattern of each individual subject, allowing for much higher localization accuracy of structural and functional features of the human brain.
3,024 citations
"Texture Map and Video Compression U..." refers background in this paper
...For image registration, different intensitybased registration algorithms have been developed [1, 2, 9, 12, 46], which aim to obtain 1-1 correspondences between images based on image intensity....
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TL;DR: Design of the MPEG algorithm presents a difficult challenge since quality requirements demand high compression that cannot be achieved with only intraframe coding, and the algorithm’s random access requirement is best satisfied with pure intraframes coding.
Abstract: The Moving Picture Experts Group (MPEG) standard addresses compression of video signals at approximately 1.5M-bits. MPEG is a generic standard and is independent of any particular applications. Applications of compressed video on digital storage media include asymmetric applications such as electronic publishing, games and entertainment. Symmetric applications of digital video include video mail, video conferencing, videotelephone and production of electronic publishing. Design of the MPEG algorithm presents a difficult challenge since quality requirements demand high compression that cannot be achieved with only intraframe coding. The algorithm’s random access requirement, however, is best satisfied with pure intraframe coding. MPEG uses predictive and interpolative coding techniques to answer this challenge. Extensive details are presented.
2,447 citations