8/20/2019 1996 Lossless Acceleration of Fractal Image Compression by Fast Convolution http://slidepdf.com/reader/full/1996-lossless-acceleration-of-fractal-image-compression-by-fast-convolution 1/4 LOSSLESS ACCELERATION OF FRACTAL IMAGE COMPRESSION BY FAST CONVOLUTION Dietmar Saupe, Hannes Hartenstein Universitat F’reiburg, Institut fiir Informatik, Am Flughafen 17, 791 10 Freiburg, Germany ABSTRACT In fractal image compression the encoding step is com- putationally expensive. We present a new technique for reducing the computational complexity. It is lossless, i.e., it does not sacrifice any image quality for the sake of the speedup. It is based on a codebook coherence characteristic to fractal image compression and leads to a novel application of the fast Fourier transform- based convolution. The method provides a new con- ceptual view of fractal image compression. This paper focuses on the implementation issues and presents the first empirical experiments analyzing the performance benefits of the convolution approach to fractal image compression depending on image size, range size, and codebook size. T he results show acceleration factors for large ranges up to 23 (larger factors possible), outper- forming all other currently known lossless acceleration methods for such range sizes. 1. INTRODUCTION In fractal image compression [l, 1 image blocks (ranges) have to be compared against a large codebook of blocks similar to vector quantiza tion. For each such com- parison a computationally expensive least-squares op timization is required. Typical codebooks consist of many thousands of blocks and the straightforward im- plementation of fractal image compression by “b rute force” suffers from long encoding times. The existing methods to reduce the computational complexity are: discrete methods (classification and adaptive cluster- ing), continuous methods (functionals or feature vec- tors), and dimensionality reduction methods. For a survey of these see [3]. In these methods suitable sub- sets of the codebook are eliminated from the search. Most of th e techniques are lossy in the sense th at they trade in a speedup for some loss in image fidelity. In contrast, with a lossless method the codebook block with the minimal (collage) error is obtained rathe r th an an acceptable but suboptimal one. Three lossless meth- ods are known: (1) Rademacher labelings [4]. Speedup factor 1.5. 0-7803-3258-X/96/ 5.00 996 IEEE 2) Dimension reduction by partial distortion elimina- 3) Dimension reduction by image pyramids [6] Speedup tion [5]. Speedup factor < 4. factor 4. Due to the moderate speedup factors these tech- niques are used in conjunction with other (lossy) meth- ods. An almost lossless acceleration method based on image pyramids with impressive results is given in [7]. Our new solution offered in [8] is the first one that takes advantage of the fact that the codebook blocks, taken from the image, are usually overlapping. The fast convolution ased on the convolution theorem and carried out in the frequency domain s ideally suited to exploit this sort of codebook coherence. This pa- per focuses on implementation issues and presents the first computer experiments analyzing the performance benefits of the convolution approach to fractal image compression depending on image size, range size, and codebook size. 2. FRACTAL IMAGE COMPRESSION VIA CONVOLUTION Let us consider the generic type of fractal image com- pression and remark later about generalizations. A partitioning of the image into disjoint image blocks (called ranges is defined. A pool of (larger) image blocks (called domains) serves as a source of blocks from which ranges can be approximated s the sum of a DC component and a scaled copy of a domain block (collage). For a range block R the domain blocks are twice the linear size. The domain blocks are shrunken by pixel averaging to match the range block size. This gives a pool of codebook blocks D1,. . , DN~. or range R and codebook block D we let where 1 is the constant block with unit intensity at ev- ery pixel. The scaling coefficient s s clamped to [--1,1] to ensure convergence in the decoding and then both s and o are uniformly quantized. The collage eror for 185
4
Embed
1996 Lossless Acceleration of Fractal Image Compression by Fast Convolution
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/20/2019 1996 Lossless Acceleration of Fractal Image Compression by Fast Convolution