Quad-trees are restricted to combined horizontal and vertical decomposition of tiles. If this restriction is dropped, anisotropic rectangular tiles result which can be arranged in a previously developed graph structure called “bush”. Although this graph requires more bits to be encoded, the reduction of the number of tiles by a factor of about two more than compensates this, especially when there is a big payload per tile such as polynomial or DCT approximations of image content. Therefore, together with an existing efficient coding scheme for this graph, a lossy coding scheme based on tile-wise rate-distortion optimized quantization of low-frequency DCT coefficients is developed. It is able to compete with JPEG2000, especially for images with smooth regions.

It was shown that sinusoid synthesis can be implemented efficiently by an inverse Fourier transform on consecutive frames where all but a small number of coefficients per oscillator are dropped. This leads to a compromise between computational complexity and approximation accuracy. The method can be improved by two approaches. First, optimal coefficients can be found by minimizing the average approximation error. Second, the optimal window function can be found through an iterative process. The gain in signal-to-noise ratio (SNR) is between 10 and 40 dB and can be used to reduce computational complexity while satisfying required synthesis quality.

Widespread use of wavelet transforms as in JPEG2000 demands efficient implementations on general purpose computers as well as dedicated hardware. The increasing availability of SIMD technologies is a great challenge since efficient SIMD parallelizations are not trivial. This work presents a parallelized 2D wavelet transform following a single-loop approach, i.e. a loop fusion of the lifting steps of horizontal filtering, and interleaving horizontal and vertical filtering for optimal temporal locality. In this way, each input value is read only once and each output value is written once without subsequent updates. Such an approach turns out to be a necessary basis for an efficient SIMD parallelization. Results are obtained on a general purpose processor with a 4-fold single-precision SIMD extension. Speedups of about 3.7 due to the use of SIMD, 2.55 due to the single-loop approach and up to 6 due to cache effects for pathologic data sizes are obtained, giving total speedups of up to 56.

Video coding algorithms based on 3-D wavelet transforms exist which have a competitive rate-distortion performance and are very fast because motion compensation is not needed. The compression efficiency of these algorithms can be increased by the use of more general wavelet packet transforms. Anisotropic wavelet packets are an even more generalized type of wavelet transform, where multidimensional wavelets do not necessarily have equal dilations in each dimension. This property is advantageous especially for 3-D video coding because video data has different characteristics in the time and spatial dimensions. This paper develops methods to adapt the SMAWZ codec to anisotropic 3-D wavelet packet bases and investigates algorithm complexity and the resulting rate-distortion performance. It can be shown that there exists a fixed anisotropic wavelet packet basis that is able to outperform MPEG-4 by up to 4dB especially for video sequences with low motion.