2D Real-valued Discrete Gabor Transform Based On 2D DCT

In 1946, Dennis Gabor presented the complex Gabor transform based on Fourier transform by multiplying the kernel function of Fourier transform with time-shiftable window functions. Although people thought it was useful, its applications were restrainted for a long time because of the difficulties in calculating its coefficients.In recent years, professor Liang Tao introduced a series of time-shiftable and frequency-shiftable basis functions based on DHT and presented 1D and 2D real-valued discrete Gabor transform(RDGT) by bi-orthogonal analysis method and discussed its serial and parallel fast algorithms; By the way, the 1D real-valued discrete Gabor transform based on DCT and its fast algorithms were also proposed. All the research above indicates that the practical application of Gabor transform will be greatly enhanced and there will be a good prospect of application in analyzing and processing nonstationary signals.In this dissertation, the theory of time-frequency analysis and the development of Gabor transform are firstly reviewed. Based on the kernel function of the traditional 2D DCT, we present a new transform by bi-orthogonal analysis method: 2D real-valued discrete Gabor transform(2D RDGT) based on 2D DCT in the case of critical sampling condition. First, the completeness property of the theory is proved; second, by matrix transform, the serial fast algorithm is introduced, so the practical application of the theory is enhanced; third, the complexity of algorithm between the 2D real-valued discrete Gabor transform(2D RDGT) and the 2D complex-valued discrete Gabor transform(2D CDGT) is compared, in allusion to real-valued signals, the computational complexity is greatly reduced because only real-valued operations are involved in our theory, as a result, hardware implement can be much more easier; fourth, we further verify our theory by experimentation; finally, as an application, we compare the differences between 2D RDGT and 2D DCT in image coding, which demonstrates that the proposed transform is more attractive in recovery of image details, so our algorithm is surely a good pre-processing tool in image recognition and texture segmentation.