On empirical cumulant generating functions of code lengths for individual sequences

We consider the problem of lossless compression of individual sequences using finite-state (FS) machines, from the perspective of the best achievable empirical cumulant generating function (CGF) of the code length, i.e., the normalized logarithm of the empirical average of the exponentiated code length. Since the probabilistic CGF is minimized in terms of the R\'enyi entropy of the source, one of the motivations of this study is to derive an individual-sequence analogue of the R\'enyi entropy, in the same way that the FS compressibility is the individual-sequence counterpart of the Shannon entropy. We consider the CGF of the code-length both from the perspective of fixed-to-variable (F-V) length coding and the perspective of variable-to-variable (V-V) length coding, where the latter turns out to yield a better result, that coincides with the FS compressibility. We also extend our results to compression with side information, available at both the encoder and decoder. In this case, the V-V version no longer coincides with the FS compressibility, but results in a different complexity measure.Comment: 15 pages; submitted for publicatio

Lossy compression of discrete sources via Viterbi algorithm

We present a new lossy compressor for discrete-valued sources. For coding a sequence $x^n$, the encoder starts by assigning a certain cost to each possible reconstruction sequence. It then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor. The cost of each sequence is a linear combination of its distance from the sequence $x^n$ and a linear function of its $k^{\rm th}$ order empirical distribution. The structure of the cost function allows the encoder to employ the Viterbi algorithm to recover the minimizer of the cost. We identify a choice of the coefficients comprising the linear function of the empirical distribution used in the cost function which ensures that the algorithm universally achieves the optimum rate-distortion performance of any stationary ergodic source in the limit of large $n$, provided that $k$ diverges as $o(\log n)$. Iterative techniques for approximating the coefficients, which alleviate the computational burden of finding the optimal coefficients, are proposed and studied.Comment: 26 pages, 6 figures, Submitted to IEEE Transactions on Information Theor

A content-aware quantisation mechanism for transform domain distributed video coding

The discrete cosine transform (DCT) is widely applied in modern codecs to remove spatial redundancies, with the resulting DCT coefficients being quantised to achieve compression as well as bit-rate control. In distributed video coding (DVC) architectures like DISCOVER, DCT coefficient quantisation is traditionally performed using predetermined quantisation matrices (QM), which means the compression is heavily dependent on the sequence being coded. This makes bit-rate control challenging, with the situation exacerbated in the coding of high resolution sequences due to QM scarcity and the non-uniform bit-rate gaps between them. This paper introduces a novel content-aware quantisation (CAQ) mechanism to overcome the limitations of existing quantisation methods in transform domain DVC. CAQ creates a frame-specific QM to reduce quantisation errors by analysing the distribution of DCT coefficients. In contrast to the predetermined QM that is applicable to only 4x4 block sizes, CAQ produces QM for larger block sizes to enhance compression at higher resolutions. This provides superior bit-rate control and better output quality by seeking to fully exploit the available bandwidth, which is especially beneficial in bandwidth constrained scenarios. In addition, CAQ generates superior perceptual results by innovatively applying different weightings to the DCT coefficients to reflect the human visual system. Experimental results corroborate that CAQ both quantitatively and qualitatively provides enhanced output quality in bandwidth limited scenarios, by consistently utilising over 90% of available bandwidth