基于选择初始图像的快速分形解码方法

给出推广的拼贴定理，并对迭代函数系统进行有效变形，记录下变形后的参数，形成不动点图像。迭代时选择不动点图像为初始图像，并证明不动点图像也是迭代函数系统的吸引子的一个较好的近似，实验结果说明，在保持同样的压缩比情况下，只须少数迭代就可得到理想的恢复图像。

A Fast Fractal Decoding Algorithm Based on the Selection of a Special Initial Image

Real-time reconstruction of images demands speed in fractal image compression(FIC). Our fast fractal decoding algorithm is based on the selection of a special initial image that we call fixed-point image tiled by all the fixed points, where each fixed point corresponds to one of the contraction mappings. The selection of this special initial image is based on our theoretical research. The collage theorem, used in fractal decoding, has to be extended. Our Theorem 1 is the extended collage theorem, which, when the conditions stipulated in Theorem 1 are satisfied, can be mathematically expressed by Eq. (1). Starting from Theorem 1,we prove Theorem 2, which, when the conditions stipulated in Theorem 2 are satisfied, can be mathematically expressed by inequality(2). In the past, fractal decoding neglects the second term on the right-hand side of inequality (2). Applying Theorem 2, we select the fixed-point image as the special initial image. Our proof that the fixed-point image is a good approximation of the attractor of the IFS (iterated function system) includes Eqs. (3) through (6), where Eqs. (5) and (6) are particularly important. We reconstruct the image in Fig. 1 by the traditional method and by our algorithm respectively and obtain Figs. 2 and 3 respectively. The quality of Fig. 3 is somewhat better, as PSNR is 39. 34 for Fig. 3 and 38. 77 for Fig. 2. Fig. 3 requires only 5 iterations while Fig. 2 requires 11 iterations with CR (compression ratio) kept at the high value of 21. 3 for both Figs. 2 and 3.