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Unformatted text preview: Optimal Transform: The KarhunenLoeve Transform (KLT) Recall: We are interested in unitary transforms because of their nice properties: energy conservation, energy compaction, decorrelation Motivation: (1D Transform; assume separable) x x T W unitary and are random vector fields (i.e., their elements are R.V.s) Unitary transform preserve the energy: T * 1 x x Unitary transforms tend to stack transform energy in the first few , x x x x x x x x I T T T T T T T * * * * * coefficients: code ignore x x EEE 508 1 The KarhunenLoeve Transform (KLT) What do we mean by an optimal transform? The optimal transform packs packs the maximum average energy in a given (specified) number of transform coefficients while completely decorrelating them optimal in the sense of energy packing according to an error criterion How do we find such optimal transform? 9 Optimality measure: meansquare error optimal in the meansquare sense 9 Desirable transform properties (constraints on transform): unitary and separable => 2 1 T T T X X EEE 508 1 The KarhunenLoeve Transform (KLT) Consider the 1D case for the derivation of the optimal transform Forward transform x x x T T x 1 Forward transform: x t x x j j N x x Inverse transform: N x 1 * * 2 * 1 * t t t x x N j j j N x x 1 * t weighted sum of basis vectors EEE 508 1 j 1 The KarhunenLoeve Transform (KLT) Note unitary T * Note: W unitary I otherwise ; ; 1 , * * j i j i j T i i T j G t t t t orthonormal basis vectors Goal: Find unitary transform (i e the vectors ) that will allow for ^ ` N j j 1 * t * Goal: Find unitary transform (i.e. the vectors ) that will allow for the reconstruction of x with as few coefficients as possible for a given mse distortion j t Let x R = reconstructed image after eliminating some transform coefficients EEE 508 1 The KarhunenLoeve Transform (KLT) Assume that we kept M out of the N coefficients and that ^ ` N x 1 we replaced the remaining with some constants which are independent of the input image x j j...
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 Fall '09

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