Research On Gabor Transform And The Application Of Gabor Transform In Image Compression

Gabor Transform that can achieve the lower limit of Uncertainty Principle and has good time-frequency characteristic, especially the 2D Gabor elementary function is a suitable model to simulate biological vision system, has attracted a lot of theoretical and applicable researches. Now it has been applied to many areas such as machine vision, texture synthesis and analysis, image compression, signal detection and linear system analysis etc. During the research and development of iris identification system, the 2D Gabor transform are used. So the author made this research on 2D Gabor transform as a tool of image processing. This paper first introduces the most classical mathematical transform, Fourier Transform. But it is not suitable for unstable signal analysis because its deficiency on local time domain. To cover the local time domain information and make up the deficit of Fourier Transform, Windowed Fourier Transform was introduced, which is the so-called Gabor Transform. This paper presents the definition and physical meaning of Gabor Transform in detail, and two underling properties of Gabor Transform in theory. One is that Gabor Transform can achieve the theoretical lower limits for conjoint resolution of information in the 2D spatial and 2D Fourier domains, namely Gabor Transform can attain optimum localization at spatial and frequency domain; the other is that a family of parameterized 2D Gabor function has properties similar to simple cell receptive field profiles (RFP) in visual cortex of mammalian brains, that is, the RFP of simple cells can be best modeled by a family of 2D Gabor functions. However Gabor Transform is non-orthogonal, it is impossible to calculate the Gabor transform coefficients simply through the inner product projection. This makes it very difficult to calculate the Gabor Transform coefficients, which furthermore restricts the application of Gabor Transform. Many solutions have been advanced at home and abroad in recent years. The paper introduces in brief the auxiliary biorthogonal function advanced by Bastiaans, and presents in detail neural network method by Daugman. Complete complex-valued 2D Gabor Transform by Daugman is the conjoint of spatial and spectral representations, which provides a complete image description in terms of locally windowed 2D spectral coordinates embedded within global 2D spatial coordinates. Using complete complex-valued 2D Gabor transform intrinsic redundancies within images are extracted, so the resulted image codes can be very compact. In Daugman's three-layered neural network approach, the first layer and third layer have fixed weights (2D empirical receptive field profiles obtained from orientation-selective neurons in cat visual cortex). The feed-forward signal is the level of activity of the neuron from the first layer, and the feedback signal is the inner product of the weighting function of the corresponding neuron in the third layer with the weighted sum of all the other neighboring neurons in that layer with which it is connected. The second layer contains adjustable weights for multiplying each output, and the adaptive control signal adjusts each of these weights by an amount given by the difference between a feed-forward signal and a feedback signal. When the difference is small enough, the network reaches equilibrium state. The network can find the optimal Gabor coefficients in its stable state. But the convergence speed in the iteration procedure for finding the optimal Gabor coefficients is too slow and causes massive operation because of the randomness of complex-valued 2D Gabor function parameter selection. The author made detail research on the general format of complex-valued 2D Gabor function, especially the parameter selection, and found that parameters selected by Daugman for discrete complex-valued 2D Gabor function are unreasonable. For example sampling interval on frequency domain is too long, which is actually undersampling, and the first reconstruction image has serious distortion. In the following experiments the author improved parameter selection and applied tocomplex-valued 2D Gabor Transform. To spare time we just used the Gabor transform coefficients obtained after one time iteration to reconstruct images, and also made comparison with construction images by Daugman's method. The experiments show that the improved parameters can get much better reconstruction images. Although compared with neutral network method, the parameter -improved 2D complex Gabor Transform can spare a lot of operating time, the operations concerned and data stored are all complex, and the complex operation is very difficult for practical application. So we further advances real-valued 2D Gabor Transform and used the improved parameters. As physiologically experimental studies approved, adjacent simple cells in visual cortex differ by 90o in their phase, which respectively correspond to real and imaginary part of the complex-valued 2D Gabor elementary function. For the real-valued 2D Gabor functions we used that have no imaginary parts, the real and imaginary parts are added together to modulate the Gaussian window function in order to retain this feature of adjacent simple cells. In real-valued 2D Gabor Transform experiments, the complex operations are converted into real ones and spare operating time. The experiment results indicate that the real-valued transform coefficients have smaller information entropy than that of complex-valued transform coefficients, which means more compact information distribution. If these transform coefficients are coded effectively, we may get better compression effect. Based on the fact that Gabor elementary functions have similar characteristics to the receptive field models of biological visual system, we apply real-valued 2D Gabor Transform to image compression. First we subdivide the original image into 16×16 subimages, which are then transformed using real-valued 2D Gabor Transform to generate the transform coefficients for each subiagme, at last quantize the coefficients and code the quantized coefficients. In decompression process, we inverse Gabor transform the coefficients obtained from the compression process to...