x 1 so that the components of the resulting vector y 1 are conditionally independent given y 2 Since we are focusing
2.4 ALTERNATIVE TRANSFORMS
Finally, we would also like to point out that, because of the very peculiar structure of the covariance matrix of this example,the local KLT turns out to be always diagonal. However, in most situations the local KLT is not diagonal and is very different from the centralized one. We refer the reader to [12] for examples showing these differences.
2.3.3
The Multiterminal Scenario and the Distributed KLT Algorithm
The general situation in which L terminals have to optimize their local transform
simultaneously is more complicated,and,as pointed out previously,an optimal solution does not appear to exist. However, the local perspective discussed in the previous section can be used to devise an iterative algorithm to find the best distributed KLT. The basic intuition is that if all but one terminal have decided their approximation strategy, the remaining terminal can optimize its local transformation using the local KLT discussed in the previous section. Then in turn, each encoder can optimize its local transform given that the other transforms are fixed. This is a sort of round robin strategy, but it is implemented off-line, and encoders do not need to interact since they
all have access to the global covariance matrix⌺x. The algorithm is not guaranteed to
converge to the global minimum but may get stuck in a local one. The convergence of such algorithm has been studied in [12].
The distributed KLT algorithm for compression operates along the same principles.
Now we have a total rateR available. At each iteration only a fraction of the rate is
allocated. Let us denote withR(i)the rate used in the ith iteration. In turn, each
terminal is allowed to use theR(i)bits and optimizes its local compression scheme,
while all other encoders are kept fixed. The encoder that achieves the best overall
reduction of the distortion is allocated the rateR(i). The process is then iterated on a
new rateR(i⫹1), and the process stops when the total bit budgetRhas been achieved.
Notice that the approach is in spirit very similar to a standard greedy bit allocation strategy used in traditional compression algorithms.
2.4
ALTERNATIVE TRANSFORMS
In many practical situations, the assumption that the source is Gaussian might not be correct, and in fact the problem of properly modeling real-life signals such as images and video remains open. In these cases, the KLT is rarely used and is often replaced by the DCT or WT.
In this section, we discuss how transform coding is used in existing distributed compression schemes of sources such as images and video. We briefly describe how the concept of source coding with side information is combined with the standard model of transform coding to form a distributed transform coding scheme. The main results of high bit-rate analysis of such a scheme will also be presented. Lastly, in
Section2.5, we will look at an alternative approach to distributed coding, which uses
2.4.1
Practical Distributed Transform Coding with Side Information
Existing practical distributed coding schemes are the result of combining transform coding with Wyner–Ziv coding (also known as source coding with side information at the decoder). Wyner–Ziv coding is the counterpart of the Slepian and Wolfs theorem [30], which considers the lossless distributed compression of two correlated discrete
sourcesXandY. The Wyner–Ziv theorem [36] is an extension of the lossy distributed
compression of X with the assumption that the lossless version of Y is available at
the decoder as side information. A Slepian–Wolf coder for distributed source coding with side information employs channel coding techniques. A Wyner–Ziv encoder can be thought of as a quantizer followed by a Slepian–Wolf encoder.
In cases of images and video, existing distributed coding schemes add the Wyner– Ziv encoder into the standard transform coding structure. As with the centralized case, a linear transform is independently applied to each image or video frame. Each transform coefficient is still treated independently, but it is fed into a Wyner–Ziv coder instead of a scalar quantizer and an entropy coder. We refer the reader to [24] for an overview of distributed video coding.
Examples of these coding schemes are the Wyner–Ziv video codec [1, 14], and PRISM [25] where both schemes employ the block-based discrete cosine transform (DCT) in the same way as the centralized case. In theWyner–Ziv video codec,the quan- tized transform coefficients are grouped into bit-planes and fed into a rate-compatible punctured turbo coder (RCPT). Syndrome encoding by a trellis channel coder was used to code the scalar quantized coefficients in [25].
In the work of [8], a distributed coding scheme for multiview video signals was proposed where each camera observes and encodes a dynamic scene from a differ- ent viewpoint. Motion-compensated spatiotemporal wavelet transform was used to explore the dependencies in each video signal.As with other distributed source coding schemes, the proposed scheme employs the coding structure of Wyner–Ziv coding with side information. Here, one video signal was encoded with the conventional source coding principle and was used as side information at the decoder. Syndrome coding was then applied to encode the transform coefficients of the other video sig- nals. Interestingly, it was also shown that, at high rates, for such coding structure the optimal motion-compensated spatiotemporal transform is the Haar wavelet. Another wavelet-based distributed coding scheme was presented in [3], where the wavelet- based Slepian–Wolf coding was used to encode hyperspectral images that are highly correlated within and across neighboring frequency bands. A method to estimate the correlation statistics of the wavelet coefficients was also presented.
2.4.2
High-rate Analysis of Source Coding with Side Information
at Decoder
In this section we briefly describe some of the main results of applying high-rate quantization theory to distributed source coding. We refer the reader to [26, 27] for a detailed explanation and derivation of these results.