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# Distortion Criteria

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Ch 8 Math Preliminaries for Lossy Coding 8 3 Distortion Criteria or Measure 1 Structure of Lossy Coding In practice this is how things get implemented The A D not only samples x t in time but it also discretizes the values however the discretization is very fine The Comp Algo often includes further coarser discretization Recall Slight variation on Fig 8 1 in textbook In theory we think of the A D as only sampling the signal in time the Comp Algo handles the discretization Thus we often think of x n as taking values on continuum y n as taking discrete values 2 Comparing Original Compressed Signals How do we check how close y n is to the original x n We must define a distortion measure d x y Square Error Most Common d x y x y SE Measure 2 Now usually we have N samples to compare so we use Vectors of Samples An operational Mean Square Error MSE Distortion 1 d x y N 1 N N d x n y n n 1 N x n y n 2 If SE measure is used n 1 In practice we ll want to adapt comp algo to give the smallest value of d x y for the particular x you are processing Operational Distortion Viewpoint 3 In Theory we don t have a particular x in a theoretical setting we haven t collected a signal yet so we strive to minimize d x y on average a probabilistic average over the ensemble of x s according to some probability model D E d x y 1 N N E d x n y n n 1 E d x y if stationary process If SE is used stationary D E y x 2 2 err Mean Square Error MSE 4 Non MSE Distortion Measures MSE is the most widely used due to its simplicity of application math results are fairly easy to derive But SE doesn t always correspond well with visual audio quality as perceived by humans Compression algorithms intended for video audio often use distortion measures that include ways to capture the psychology of human vision hearing MSE is usually not the best choice when the decompressed signal is going to be used in statistical estimation decision processing See many of my papers posted on my web page So why study MSE Math

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