Year

1994

Degree Name

Doctor of Philosophy

Department

Department of Electrical and Computer Engineering

Abstract

Digital images are replacing analogue images such as photographs and x-rays in many different fields. Compression of these digital images is desirable for efficient storage and transmission. Subband coding has proved an effective method of image compression. This thesis investigates subband analysis structures and filters which are optimised for still image compression. Among other results it is shown that a high coding gain, based on a typical image model, and good spatial localisation are desirable filter bank characteristics for subband image coding.

Assuming a high bit rate it is well known that the Karhunen-Loeve Uansform (KLT) is the optimum orthogonal block Uansform in terms of a coding gain metric. It is shown further that the KLT is the optimum invertible block Uansform using the unified coding gain. The coding gain metric is examined under a rate constraint. It is shown that for highly correlated sources increased low frequency subband resolution is required for optimum performance at low rates as compared to high rates: a result that is corroborated using a practical subband coder.

Subband filters (CQF's) that globally maximise the coding gain for all two-band perfect reconstruction orthogonal filter banks are derived. Various characteristics of these filters are predicted using a new theorem on the zeros of an eigenvector of a symmetric Toeplitz matrix corresponding to the minimum (maximum) eigenvalue. These filters are shown to enjoy the three properties of the KLT : namely maximum coding gain, minimum basis restriction error, and subband decorrelation. It is also shown that there is some freedom to select different impulse responses. The design of maximum gain filters is extended to include filters consUained to certain subspaces. For example maximum gain wavelets may be designed.

A modified two-climensional discrete wavelet Uansform (DWT) is proposed based on a typical image model. A generic subband quantisation and encoding method suitable for any subband structure is inuoduced. This method is essentially a generalisation of the JPEG quantisation and encoding method and has good spatial adaptation properties.

Using the generic subband quantisation and encoding method various subband analysis structures and filters are compared for still image compression. The best orthogonal filters, Daubechies wavelets and m a x i m u m gain filters designed using an i m a g e source model, and the discrete cosine transform (DCT) perform in a similar manner. The filters with the minimum spatial (time) width perform better than other impulse responses in a mean square error sense and exhibit significantly less ringing. The performance of cosine modulated filter banks, with poorer spatial resolution, is slightly inferior to the DCT . Preliminary investigations show that biorthogonal filters, with a smaller spatial width and higher coding gain, can outperform the best orthogonal filters, especially at low rates. These biorthogonal filters also exhibit minimal ringing. Finally, the modified D W T is shown to be superior to the D W T for head and shoulders type images.

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.