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compression5 - Compression Outline 15-853:Algorithms in the...

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1 15-853 Page 1 15-853:Algorithms in the Real World Data Compression 4 15-853 Page 2 Compression Outline Introduction : Lossy vs. Lossless, Benchmarks, … Information Theory : Entropy, etc. Probability Coding : Huffman + Arithmetic Coding Applications of Probability Coding : PPM + others Lempel-Ziv Algorithms : LZ77, gzip, compress, … Other Lossless Algorithms: Burrows-Wheeler Lossy algorithms for images: JPEG, MPEG, ... Scalar and vector quantization JPEG and MPEG Compressing graphs and meshes: BBK 15-853 Page 3 Scalar Quantization Quantize regions of values into a single value: input output uniform input output non uniform Can be used to reduce # of bits for a pixel 15-853 Page 4 Generate Output Vector Quantization Generate Vector Find closest code vector Codebook Index Index Codebook Out In Encode Decode
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2 15-853 Page 5 Vector Quantization What do we use as vectors? Color (Red, Green, Blue) Can be used, for example to reduce 24bits/pixel to 8bits/pixel Used in some terminals to reduce data rate from the CPU (colormaps) K consecutive samples in audio Block of K pixels in an image How do we decide on a codebook Typically done with clustering 15-853 Page 6 Vector Quantization: Example 15-853 Page 7 Linear Transform Coding Want to encode values over a region of time or space Typically used for images or audio Select a set of linear basis functions f i that span the space sin, cos, spherical harmonics, wavelets, … Defined at discrete points 15-853 Page 8 Linear Transform Coding Coefficients: = = Θ j ij j j i j i a x j x ) ( φ ) ( t coefficien transform e input valu t coefficien resulting i j ij a j x i th ij th j th i φ = = = = Θ In matrix notation: Where A is an n x n matrix, and each row defines a basis function Θ = = Θ 1 A x Ax
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