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

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Presentation transcript:

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

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

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

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

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

For Example A = 4 x 4 =

U = ∑ = V T =

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

The Pictures The original, k=126 k=4

The Pictures k=8 k=20

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

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