Image coding based on a fractal theory of iterated contractive image transformations

Abstract

The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that (i) it relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis, and (ii) it approximates an original image by a fractal image. The author refers to the approach as fractal block coding. The coding-decoding system is based on the construction, for an original image to encode, of a specific image transformation-a fractal code-which, when iterated on any initial image, produces a sequence of images that converges to a fractal approximation of the original. It is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0.5-1.0 b/pixel. The fractal block coder has performance comparable to state-of-the-art vector quantizers.

Presentation-Oriented Key-Frames Coding Based on Fractals

“…The fractal code for the original image is a collection of such extracted local codes. This approach proposed by Jacquin in [6], gives a representation of the image as composed by copies of parts of the image itself.…”

“…The fractal image can be seen as a collage composed by copies of parts of an original image that have been transformed through opportune geometric and massive transformations (that is, luminance or contrast shift). The mathematical foundation of this technique is the general theory of contractive iterated transformations, based on the works of Barnsley [5] and Jaquin [6] …”

“…To face this problem, a fractal decoding with adaptive averaging is proposed in [6]. It is based on the following consideration: In the case of perfect fractal coding, the four independent codes (in C I ) provide quite similar results.…”

“…With respect to these, fractal coding of an image and video adds the feature of expansion of the dataset during decoding; this characteristic is not usually provided in standard codecs and has to be considered as an external separate issue if standard codecs are used for key-frames coding. In the past, fractals have been proposed for image data compression by exploiting pseudo-self-similarity inside natural images [5], [6]. The resulting algorithms have the advantage of allowing the expansion of the signal along its dimensions during decoding.…”

“…The fractal code for the original image is a collection of such extracted local codes. This approach proposed by Jacquin in [6], gives a representation of the image as composed by copies of parts of the image itself.The classical fractal decoding stage consists in an iterated process starting from an arbitrary initial image, 0 µ . In fact, if τ is a contractive transformation, the τ 's attractor…”