Fractal Image Compression

Size: px

Start display at page:

Download "Fractal Image Compression"

Bennett Cain
5 years ago
Views:

1 Ball State University January 24, 2018

2 We discuss the works of Hutchinson, Vrscay, Kominek, Barnsley, Jacquin. Mandelbrot s Thesis 1977 Traditional geometry with its straight lines and smooth surfaces does not resemble the geometry of trees and clouds and mountains. Fractal geometry, with its convoluted coastlines and detail ad infinitum, does.

3 Go to http: // to see examples of fractals.

5 It turns out that Sierpinski s Triangle is the limit of the process depicted above. Note how Sierpinski s Triangle is the union of 3 shrunken copies of itself.

6 What is the underlying mathematics behind the Multiple Reduction Copy Machine Each B+W picture is a set in R 2. Each photocopy is the union of the shrunken images of this set under 3 different functions.

7 W is the action of the photocopy machine. The functions g 1, g 2, g 3 help make up the action W.

8 (IFS s) Let g 1,..., g n : R 2 R 2 be contractions, i.e., for some α < 1 for all x, y R 2. g i (x) g i (y) α x y Let S = {closed and bounded subsets of R 2 }. Define photocopy function W : S S by n W (E) = g i (E). i=1

9 By the CMT there is a unique set A S s.t. 1) for any B S we have W k (B) A n 2) W (A) = A i.e. g i (A) = A i=1 (A is a union of shrunken copies of itself). Note: W k denotes the k-th iterate, that is, running the copy machine k times.

10 von Koch curve (different program for MRCM) This image can be fully described by the 16 parameters which define the 4 functions, yet it is in some sense infinitely complex.

11 Given an IFS (set of contraction functions) how do we draw the corresponding attractor set? Let B = {x 0 }, a single point, be the seed. W (B) contains n points W 2 (B) contains n 2 points W k (B) contains n k points

12 The Spleenwort fern can be described by only 28 IFS (copy machine) parameters (4 functions). Note that this fern is the union of FOUR shrunken copies of itself.

13 What is compression? Take a picture of the above Spleenwort fern with, say, a digital camera. Instead of saving the picture pixel by pixel which uses a lot of memory we could just save the 28 parameters needed to fully describe the picture. In fact, our 28 parameters will give a better description since it is a mathematical description and we can then zoom in as far as we want without any distortion (up to the capabilities of the the computer). What is Fractal image compression? Fractal Image compression looks to use the power of IFS s to store complicated images encoded in the parameters of a few functions in such a way that the picture can easily be decoded and rapidly reproduced.

14 Given a picture that we want to store, not pixel by pixel, but with an IFS, how do we go about finding an appropriate IFS?

15 Given a B+W target image T (subset of R 2 ) we want to find g 1,..., g n whose attractor A is (close to) T. Collage Theorem Given a B+W target image T can we find g 1,..., g n such that ( ) T n g i (T ) = W (T )? i If so, then since A = these notes) implies T A. n g i (A) the CMT (found on last page of i Note that ( ) says that T is approximately the union of shrunken copies of itself.

16 Give the target image T as the leaf below (imagine it is colored in completely) we try to find a simple way of covering it with 4 shrunken copies of itself.

18 Drawbacks 1) Not all pictures are (approximately) unions of shrunken copies of themselves. 2) What about gray scale and color pictures? 3) How could we automate this procedure so that a computer could find an appropriate IFS (copy machine) given some target picture?

19 -Jacquin Self similarity between part of the picture and the whole picture is unrealistic, but we can often find some similarity between smaller parts and larger parts.

20 Grayscale target image is a function z = T (x, y) defined on [0, 1] [0, 1] where T (x, y) {0,..., 255}. T (x 0, y 0 ) = 0 means color (x 0, y 0 ) white. T (x 0, y 0 ) = 255 means color (x 0, y 0 ) black.

21 Given a Target fcn (pic) T (x, y) we will create a new copy machine F such that 1) We input a fcn (pic) and output a fcn (pic), i.e., U(x, y) F V (x, y). 2) Iterating (feeding a picture output back into the machine) produces a limit picture that is (approximately) T (x, y). For (2) to occur we must have F (T ) T, i.e., putting T (x, y) through copy machine F produces (approximately) the same pic T (x, y).

24 Fix a child block in target image T (x, y) and select parent block also in target image T (x, y). Determine how well parent block matches child block when we 1) Shrink parent block 2) Apply symmetry (4 rotations, 4 flips) 3) Adjust contrast and brightness of parent block, i.e., Z (x, y ) = sz(x, y) + σ. s= contrast σ=brightness

25 Letting (e, f ) denote the center of parent block and m denote the symmetry one could show that (1)-(3) can be performed by using very specific 3 3 matrices with just 5 parameters (e, f, m, σ, s). x y z = s a b 0 c d x y z + For fixed child we find best parent block and parameters to match. (large compression time) Total (1,024 child blocks) 5 parameters = 1, bits =32,768 bits. Pixel by pixel representation=524,288 bits. Compression ratio 16:1 e f σ

26 Determine operator F (based on target function T ) as follows: For ANY fcn (pic) Z : [0, 1] [0, 1] {0,..., 255} define Z(x, y) F Z (x, y ) where Z : [0, 1] [0, 1] {0,..., 255}. It turns out that F is a contraction operator (on L 1 ) and so has an attractor, i.e., some fcn (pic) A(x, y) s.t. A(x, y) F A(x, y). 1) Since F (T ) T we have A T. 2) For any (pic) fcn U(x, y) we have F n (U) A T. Thus the limit pic from iterating the copy machine is (approx) T.

27 Decompression is Fast: Feed midgray pic U(x, y) 100 into copy machine F and iterate 6 times and F 6 (U(x, y)) T (x, y).

29 Tweak procedure for better results: Large variation use smaller blocks better pic quality Small variation use larger blocks faster compression, higher ratio Use only parent block containing child block faster compression

Method 1 (left): Searching over ALL parent blocks, 3

Method 3 (right): Uses place-dependent grey level maps

30 Method 1 (left): Searching over ALL parent blocks, 3 hours of computing time, Error = Method 2 (middle): Uses parent block containing child block, 34 seconds of computing time, Error = Method 3 (right): Uses place-dependent grey level maps and parent block containing child block, 27 seconds of computing time, Error =

31 Drawbacks: Text and diagonal lines

32 Contraction Mapping Theorem (CMT) Let (X, ρ) be a nonempty complete metric space and let f : X X be a contraction map, i.e., ρ(f (x), f (y)) αρ(x, y) for all x, y X where α < 1. Then there exists a unique ATTRACTOR x 0 X such that 1) f (x 0 ) = x 0. 2) For any x X, f n (x) x 0 as n. (Iterates converge to single point x 0.) 3) h(x, f (x)) ɛ h(x, x 0 ) ɛ 1 s. (x f (x) x x 0 )

Iterated Functions Systems and Fractal Coding

Qing Jun He 90121047 Math 308 Essay Iterated Functions Systems and Fractal Coding 1. Introduction Fractal coding techniques are based on the theory of Iterated Function Systems (IFS) founded by Hutchinson