1Ch. 8 Math Preliminaries for Lossy Coding8.3 Distortion Criteria or Measure2Structure of Lossy Coding Recall Slight variation on Fig 8.1 in textbook: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 discretizationIn 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 continuumy[n] as taking discrete values3• How do we check how close y[n] is to the original x[n]?• We must define a distortion measure d(x,y)Comparing Original & Compressed Signals Most Common:()2(, )dxy x y=−Square Error (SE) Measure• Now, usually we have N samples to compare, so we use:()1211(,) ([],[])1[] []NnNnddxnynNxnynN====−∑∑xyVectors of SamplesIf SE measure is used• In practice we’ll want to adapt comp. algo. to give the smallest value of d(x,y) for the particularx you are processing…“Operational Distortion” Viewpoint An “operational”Mean Square Error (MSE) Distortion4•In Theorywe 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’saccording to some probability model.{}{}{}1(,)1([],[])(, )NnDEdEdxn ynNEdxy====∑xyif stationary processIf SE is used (& stationary): (){}22errDEyxσ=−=Mean Square Error (MSE)5• 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 is easy (relative to that needed for non-MSE)– It gives decentresults– It is usually part of the non-MSE measuresNon-MSE Distortion Measures6SNR = Signal-to-Noise (Power) RatioSince…Relating Distortion To SNR()2([],[]) [] []dxn yn yn xn=−e[n][] [] []ynxnen=+Reconstructed SignalOriginal Signal“Noise” due to CompressionThe reconstructed signal has an SNR of222{[]}{[ ]}xxepower x nSNRpower e nDσσσ===2xSNRDσ=SNR has the advantage that it measures distortion relativeto the signal power7SNR is typically stated in dB form:210210logxdBeSNRσσ=In image compression it is common to use peak SNR:210210logpeakdBexPSNRσ=max [ ]peaknxxn=8Goals of Lossy CompressionDistortion, DRate, RThink of x[n] as DT and Continuum-Valued (in practice, x[n] is “finely” discrete-valued)Think of y[n] as DT and Discrete-ValuedIn Practice, rate is a measure of how many bits are used to represent the compressed signal:• bits/sample• bits/second• bits/fixed duration•Etc.Goals of Compression1. Reduce distortion for a fixed rate2. Reduce rate for a fixed distortion9Goals of Lossy CompressionMethod #1 (Worse Method)Method #2 (Better Method)RRdesiredDdesiredDGoals of Compression1. For a given Rdesired, find a method that minimizes D2. For a given Ddesired, find a method that minimizes RDesigning Lossy Comp. Algorithms involves solving constrained optimization