A Generalized Approach to Linear Transform Approximations with Applications to the Discrete Cosine Transform

This paper aims to develop a generalized framework to systematically trade off computational complexity with output distortion in linear transforms such as the DCT, in an optimal manner. The problem is important in real-time systems where the computational resources available are time-dependent. Our approach is generic and applies to any linear transform and we use the DCT as a specific example. There are three key ideas: (a) a joint transform pruning and Haar basis projection-based approximation technique. The idea is to save computations by factoring the DCT transform into signal-independent and signal-dependent parts. The signal-dependent calculation is done in real-time and combined with the stored signal-independent part, saving calculations. (b) We propose the idea of the complexity-distortion framework and present an algorithm to efficiently estimate the complexity distortion function and search for optimal transform approximation using several approximation candidate sets. We also propose a measure to select the optimal approximation candidate set, and (c) an adaptive approximation framework in which the operating points on the C-D curve are embedded in the metadata. We also present a framework to perform adaptive approximation in real time for changing computational resources by using the embedded metadata. Our results validate our theoretical approach by showing that we can reduce transform computational complexity significantly while minimizing distortion.