eee508_Transforms_KLT

# eee508_Transforms_KLT - Optimal Transform The...

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Unformatted text preview: Optimal Transform: The Karhunen-Loeve Transform (KLT) • Recall: We are interested in unitary transforms because of their nice properties: energy conservation, energy compaction, decorrelation ¾ Motivation: (1D Transform; assume separable) x x Ĳ T ˆ W unitary ¡ and are random vector fields (i.e., their elements are R.V.s) Unitary transform preserve the energy: T * 1 Ĳ Ĳ ¢ x ˆ x Unitary transforms tend to stack transform energy in the first few £ ¤ , x x x x x x x x I Ĳ Ĳ Ĳ Ĳ T T T T T T T * * * * * ˆ ˆ coefficients: ¡ code ignore x x ˆ EEE 508 1 ¡ The Karhunen-Loeve Transform (KLT) • What do we mean by an “optimal” transform? ¾ 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 ¾ How do we find such optimal transform? 9 Optimality measure: mean-square error ¡ optimal in the mean-square sense 9 Desirable transform properties (constraints on transform): unitary and separable => 2 1 ˆ T T T X X EEE 508 1 The Karhunen-Loeve Transform (KLT) Consider the 1D case for the derivation of the optimal transform ¾ Forward transform x x x T T x ¡ » º « ª ˆ ˆ ˆ 1 ¾ Forward transform: x t x x Ĳ j j N x x ¡ » » ¼ « « ¬ ˆ ¡ ¾ Inverse transform: » » º « « ª » » » º « « « ª N x 1 * * 2 * 1 * ˆ ˆ ˆ t t t x x Ĳ ¡ ¢ ¦ » ¼ « ¬ » ¼ « ¬ N j j j N x x 1 * ˆ t ¡ weighted sum of basis vectors EEE 508 1 j 1 The Karhunen-Loeve Transform (KLT) ¾ Note unitary ¡ T * ¾ Note: W unitary ¡ ¡ I Ĳ Ĳ ¢ £ ¯ ® ­ otherwise ; ; 1 , * * j i j i j T i i T j G t t t t ¡ orthonormal basis vectors ¾ Goal: Find unitary transform (i e the vectors ) that will allow for ¯ ^ ` N j j 1 * t * ¾ 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 ¾ Let x R = reconstructed image after eliminating some transform coefficients EEE 508 1 The Karhunen-Loeve Transform (KLT) ¾ Assume that we kept M out of the N coefficients and that ^ ` N x 1 ˆ we replaced the remaining with some constants which are independent of the input image x j j...
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eee508_Transforms_KLT - Optimal Transform The...

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