1. Digital Image Compression via Singular Value Decomposition Robert White Ray Buhr Math 214 Prof. Buckmire May 3, 2006

2. The Problem High resolution digital images are dense files and take up lots of bandwidth Cost of: time spent online accepting large files capable machinery

3. The Solution Using Singular Value Decomposition, we can reduce the size of the image’s matrix Eliminates the end SVDs Cuts out the boring parts

4. The Matrix/Image Matrix represents a grayscale image (126x128) Each component is represented by a # 0-255

5. The Process A = U* Σ *V T ∑= the normalized singular values (√ λ for A T A) V= columns are eigenvectors of A T A U= columns are eigenvectors of AA T [U,S,V]=svd(A) factors A in Matlab

6. For Example A = 4 x 4 = 3078 5570 2749 5969

7. -0.421990.332350.842790.034276 -0.34099-0.879590.18727-0.27383 -0.523850.33928-0.36806-0.68921 -0.65669-0.02749-0.345210.66995 22.495000 07.082400 006.09010 0000.83996 U = ∑ = -0.3246-0.50153-0.50572-0.62238 -0.40379-0.32057-0.372540.77162 0.16461-0.779460.60210.053009 0.83934-0.1953-0.492880.12012 V T =

8. Taking Care of Business SVD, singular values = rank(A) A = σ 1 u 1 v T 1 + …σ k u k v T k + 0*u k+1 v T k+1 Approximate A by eliminating small singular values

9. The Pictures The original, k=126 k=4

10. The Pictures k=8 k=20

11. The Pictures k=50 Original, again, k=126

12. The Results How much space is this process saving? 4 + 4(126) + 4(128) = 1020 8 + 8(126) + 4(128) = 2040 20 + 20(126) + 20(128) = 5100 (~31.6%) 50 + 50(126) + 50(128) = 12750 (~79.0%) (126)*(128) = 16128! x + x(126) + x(128) = 16128, x ≈ 63.247