[Paper Review] Beltrami Representation and its applications to texture map and video compression

Lok Ming Lui, Ka Chun Lam+2 more|arXiv (Cornell University)|Oct 18, 2012

Computer Graphics and Visualization Techniques40 references3 citations

TL;DR

This paper proposes a novel compression method for bijective surface homeomorphisms using Fourier approximation of the Beltrami representation—a complex-valued function on triangular faces with sup norm < 1. By leveraging the 1-1 correspondence between surface maps and their Beltrami representations, and using a new Linear Beltrami Solver for exact reconstruction, the method reduces storage by up to 2/3 while preserving bijectivity. The approach enables significant compression in texture mapping and video compression, achieving up to 56:1 compression ratio on motion vector fields with negligible quality loss.

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 3D coordinates each vertex is mapped to, which often requires lots of storage 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 using the Linear Beltrami Solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage memory can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we proposed to apply the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improve the compression ratio.

Motivation & Objective

• To address the high storage and bandwidth demands of storing bijective surface maps as 3D coordinates in computer graphics and imaging.
• To develop a compression scheme that preserves the bijective (1-1 and onto) nature of surface homeomorphisms during data reduction.
• To enable efficient compression of texture maps and motion vector fields in video by representing them via Beltrami representations.
• To introduce a Linear Beltrami Solver for exact reconstruction of surface maps from their Beltrami representations.
• To demonstrate the effectiveness of Fourier approximation on Beltrami representations for achieving high compression ratios without loss of bijectivity.

Proposed method

• Represent surface homeomorphisms using the Beltrami representation, a complex-valued function on mesh faces with sup norm strictly less than 1, which encodes local conformal distortion.
• Leverage the 1-1 correspondence between surface homeomorphisms and Beltrami representations under suitable normalization to ensure injective and surjective reconstruction.
• Introduce the Linear Beltrami Solver to exactly reconstruct a bijective surface map from its Beltrami representation using a PDE-based numerical scheme.
• Apply Fourier approximation to the Beltrami representation to compress the data, exploiting its low-constraint structure (only sup norm < 1) to maintain bijectivity.
• Use the compressed Beltrami representation to reconstruct texture maps and motion vector fields in video, replacing direct storage of 3D coordinates.
• Validate the method through experiments on textured surfaces and video sequences, measuring PSNR and compression ratio of reconstructed data.

Experimental results

Research questions

• RQ1 Can the Beltrami representation be used to compress bijective surface maps while preserving their bijective property?
• RQ2 Can Fourier approximation of the Beltrami representation achieve high compression ratios without compromising map bijectivity?
• RQ3 How effective is the Linear Beltrami Solver in reconstructing surface maps from their Beltrami representations with minimal error?
• RQ4 To what extent can Beltrami-based compression reduce storage for texture maps and motion vector fields in video?
• RQ5 What is the impact of compressed motion vector fields on the overall video compression ratio and visual quality?

Key findings

• The proposed method reduces storage for surface homeomorphisms by up to 66% (1/3 of original size) using Beltrami representation and Fourier approximation.
• The Linear Beltrami Solver enables exact reconstruction of bijective surface maps from their Beltrami representations with high numerical accuracy.
• Motion vector fields in video can be compressed at a ratio of up to 56:1 using 0.5% of Fourier coefficients, with PSNR above 90 for reconstructed P-frames.
• The PSNR of reconstructed P-frames remains close to 90 even after multiple cycles, indicating negligible error accumulation in motion vector fields.
• When motion vector fields account for 1/3 of video file size, compressing them with the proposed method reduces total video storage by approximately 32%.
• No visible differences are observed between original and compressed P-frames, demonstrating high visual fidelity at low bit rates.