# Theoretical foundations of transform coding

@article{Goyal2001TheoreticalFO, title={Theoretical foundations of transform coding}, author={Vivek K Goyal}, journal={IEEE Signal Process. Mag.}, year={2001}, volume={18}, pages={9-21} }

Discusses various aspects of transform coding, including: source coding, constrained source coding, the standard theoretical model for transform coding, entropy codes, Huffman codes, quantizers, uniform quantization, bit allocation, optimal transforms, transforms visualization, partition cell shapes, autoregressive sources, transform optimization, synthesis transform optimization, orthogonality and independence, and departures form the standard model.

## 346 Citations

### Theorems and Counterexamples in Transform Coding

- Computer Science

Results presented here to some extent reveal the importance of orthogonality and of independent transform coefficients when the source is not necessarily Gaussian and the rates are not necessarily high.

### Source Coding: Part I of Fundamentals of Source and Video Coding

- Computer ScienceFound. Trends Signal Process.
- 2010

Based on the fundamentals of information and rate distortion theory, the most relevant techniques used in source coding algorithms are described: entropy coding, quantization as well as predictive and transform coding.

### Towards Optimality in Multiterminal Transform Coding

- Computer Science2012 Data Compression Conference
- 2012

This paper is concerned with transform coding of correlated sources in conjunction with variable rate quantization at high resolution, and derived the necessary and sufficient condition for optimality of a unitary transform in the side information setting is derived.

### Quantization and Entropy Coding in the Versatile Video Coding (VVC) Standard

- Computer ScienceIEEE Transactions on Circuits and Systems for Video Technology
- 2021

The paper provides an overview of the quantization and entropy coding methods in the Versatile Video Coding (VVC) standard and discusses motivations and implementation aspects.

### High-rate quantization and transform coding with side information at the decoder

- Computer ScienceSignal Process.
- 2006

### Multi-rate adaptive transform coding for video compression

- Computer Science
- 2022

This study proposes learned transforms and entropy coding that may either serve as (non)linear drop-in replacements, or enhancements for linear transforms in existing codecs, and augmented the DCT with learned quantization matrices and adaptive entropy coding to compress intra-frame AV1 block prediction residuals.

### Hybrid Video Coding with Trellis-Coded Quantization

- Computer Science2019 Data Compression Conference (DCC)
- 2019

It is shown that the coding efficiency of transform coding can be improved by replacing scalar quantization with trellis-coded quantization (TCQ) and using advanced entropy coding techniques for coding the quantization indexes.

### Optimization of Product Code

- Computer Science
- 2004

This work proposes an optimization method for a product coding, which avoids this computational explosion problem by applying a reasonable restriction to the model architecture.

### Rate-Distortion Analysis of Sparse Overcomplete Codes

- Computer Science
- 2007

A rate-distortion theory for source codes based on sparse approximations is developed and can be used to compare different overcomplete transforms by estimating these parameters for a class of signals to be encoded.

### Identification of Transform Coding Chains

- Computer ScienceIEEE Transactions on Image Processing
- 2016

This paper considers the problem of identifying the transform matrix as well as the quantization step sizes, and proposes an algorithm that is able to find the optimal solution and formally study its convergence properties.

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