A Review of Iterated Function System Theory for Image Compression

1. A Review of Iterated Function System Theory for Image Compression A presentation based on a publication by the same name by: Waite,J British Telecom Res. Labs., Ipswich Application of Fractal Techniques in Image Processing, IEE Colloquium Publication Date: 3 Dec 1990 12/03/1990 London

2. What is it? • A method of image compression based on the idea of fractals A fractal is a fractal is a fractal is a fractal is a fractal is a fractal is a fractal is a fractal is a fractal… • Exploits inherent self similarity in most images • Key landmarks: • Mandelbrot (1977) • Barnsley (1988) • Jacquin (1990)

3. D R|D R|D R The idea … • Consider a set of random points: [xd,yd] (called the Domain) • Map the points into the Range using a transform T xr=xd/1.2+1 yr=yd/1.4+.5 • Now set the obtained Range as the new domain and repeat the above steps • T(6,2) = ? The attractor • Attractor is independent of initial distribution • T(A) = A

4. Barnsley’s spleenwort fern Image sequence generated using matlab code from http://online.redwoods.cc.ca.us/instruct/darnold/ifs/ Extending … • Use a set of contractive mappings x' = 0x + 0y +.16, y' = 0x + 0y +0 x' = .85x + .04y + 0, y' = -.04x + .85y + 1.6 x' = .2x - .26y + 0, y' = .23 x + .22y +1.6 x' = -.15x + .28y + 0, y' = .26x + .24y + .44 • How would the attractor look? Would be a set of points • Scale independence Infinite resolution!! (algorithm) Suppose the transformation returned: (35.678, 68.3454) (35.193, 68.404) If we used a 100x100 support (35,68) (35,68) But on a 1000x1000 support (356,683) (351,684)

5. The inverse problem … Given an image I, can we find a set of transforms which will have I as its attractor? • Not practical to have the entire image as an attractor • How about dividing the image into a set of attractors? IFS: Iterated Function System

6. Domains Ranges The Collage Theorem “The inverse problem is solved approximately by the Collage Theorem. In the words of M. Barnsley, the theorem tells us that to find an IFS whose attractor is "close to" or "looks like" a given set, one must endeavor to find a set of transformations - contraction mappings on a suitable set within which the given set lies - such that the union, or collage, of the images of the given set under transformations is near to the given set.” http://www.cut-the-knot.org/ctk/ifs.shtml • Split I into non-overlapping Ranges • Try to find similarity within the image itself (Transform) • Domains > Ranges (Contractive!) • For each Range, Find the closest Domain-Transform pair • Domains can overlap Transform Pick T and D such that T(D) ~ R

7. Example approach • Split I into NxM Ranges of size 4x4 • For each Range r do: • For each Domain D 8x8 in I • Compute d = Shrink D by a factor of 2 • Compute a distance measure between d and r for all 8 orientations of d • Save the coordinates of d and the orientation o and the distance measure e • End For • Pick up the d for which e is the least • Transmit to the encoded stream: • Coordinates of d (Top left vertex suffices) • Value of o • The Mean Average Distance (MAD) between d and r • End For

8. Cons … • Computationally very expensive • Hunting around for a Domain • Computing the different orientations • Doing this for every range! • Quadtree, Clustering • Best match might not be good enough (exp. in small images) • Use a dictionary • Difficult to control the accuracy versus compression tradeoff • Control the Domain-Range sizes • Increase number of possible transforms • Smothered by JPEG • Sigh! ..more research needed

9. D R|D R|D R Pros … • Decompression is a breeze!(Did not even merit a separate slide) • Makes a lot of sense for use on the internet • Resolution independence You wont get blockiness, but it is a kind of extrapolation which does not represent the real world

10. Q’s?