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Efficient Image Compression with Singular Value Decomposition

Learn how Singular Value Decomposition reduces image size, eliminates redundant data, and saves storage space, all while preserving image quality. Explore the process and results of digital image compression using SVD.

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Efficient Image Compression with Singular Value Decomposition

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  1. Digital Image Compression via Singular Value Decomposition Robert WhiteRay 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*VT • ∑= the normalized singular values (√λ for ATA) • V= columns are eigenvectors of ATA • U= columns are eigenvectors of AAT • [U,S,V]=svd(A) factors A in Matlab

  6. For Example A = 4 x 4 =

  7. U = ∑ = VT =

  8. Taking Care of Business • SVD, singular values = rank(A) • A = σ1u1vT1 + …σkukvTk + 0*uk+1vTk+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

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