The Coding-Optimal Transform

We propose a new transform coding algorithm that integrates all optimiza tion steps into a coherent and consistent framework Each iteration of the algorithm is designed to minimize coding distortion as a function of both the transform and quantizer designs Our algorithm is a constrained version of the LBG algorithm for vector quantizer design The reproduction vectors are constrained to lie at the vertices of a rectangular grid A signi cant result of our approach is a new transform basis speci cally designed to minimize mean squared quantization distortion for both xed rate and entropy constrained coding For Gaussian distributed data this transform reduces to the Karhunen Loeve transform KLT However in general the cod ing optimal transform COT di ers from the KLT enough to provide up to dB improvement in compressed signal to noise ratio SNR on images We de scribe a practical algorithm that nds the COT for a given signal In addition we present image compression results demonstrating the SNR improvement achieved with our algorithm relative to KLT based transform coding Introduction Transform coding is a low complexity alternative to vector quantization and is widely used for image and video compression A transform coder compresses multi dimensional data by rst transforming the data vectors to new coordinates and then coding the transform coe cient values independently with scalar quantizers A key goal of the transform coder is to minimize compression distortion while keeping the compressed signal representation below some target size While quantizers are typically designed to minimize compression distortion this is not the case for the transform The coordinate transform has been xed apriori as in the discrete cosine transform DCT used in the JPEG compression standard The transform has also been adapted to the signal statistics using the Karhunen Loeve transform KLT as in recently pub lished transform coding work These transforms are not designed to minimize compression distortion nor are they designed selected in concert with quantizer development For instance the design goal of the KLT is to concentrate signal energy into a few components In this paper we present a new algorithm for transform coder design that concurrently optimizes both transform and quantizers Our algorithm is a constrained version of the Linde Buzo Gray LBG algorithm for vector quantizer design A signi cant result of our approach is a new transform basis designed to minimize mean squared compression distortion In this paper we derive the conditions this coding optimal transform COT must satisfy to minimize distortion In addition we describe a simple algorithm for determining the transform We conclude by presenting results from image compression experiments that compare the compression performance of COT based transform coders with KLT based transform coders Optimal Transform Coding A transform coder converts a signal to new coordinates and then codes the transform coe cients independently of one another with scalar quantizers One can think of a transform coder as a vector quantizer with the M reproduction vectors constrained to lie at the vertices of a rectangular grid The grid is de ned by orthogonal axes sJ J n and n sets of scalar reproduction values one for each dimension There are MJ possible reproduction values on the sJ axis thus the total number of grid vertices isM Q J MJ Encoding a n dimensional data vector with a vector quantizer requires O Mn add multiply operations for the distance calculations and O M compare operations A transform coder requires O n add multiply operations for the transform and naively O P J MJ compare operations However e cient binary search techniques can be used to encode the scalar transform coe cients reducing the number of compare operations to O log M Figure Orientation of quantizer grid in signal space The quantizer reproduction vectors q M lie at the vertices of a rectangular grid The grid is oriented to the signal vectors x indicated by the gray area with orthogonal transform W The compression restoration process replaces each signal vector with one of a small set of reproduction vectors The encoder assigns the transform coe cients of a data vector to codewords The decoder replaces each codeword with the associated re production value Figure illustrates the structure of a two dimensional transform coder The r values indicate the scalar reproduction values r i J is the i th value along the sJ axis The coordinates of the reproduction vectors q M are combinations of the scalar reproduction values r i r j r k n T i M j M etc A reproduction vector q represents all the data vectors in region R of the data space We will refer to the regions de ned by the assignment of signal values to reproduction values as the partition The n n orthogonal transform W de nes the orientation of the quantizer grid in the data space In the data coordinate basis the reproduction vectors are given by Wq Conversely in the transform basis the data vectors are s W x To develop an optimal transform coder for a particular signal one nds values for the transform coder parameters that minimize compression induced distortion We quan tify distortion with the mean squared di erence between the original and compressed signal vectors D M X Z