Research On Quaternion And Its Applications In Graphics And Image Processing

Quaternion algebra was founded by Irish mathematician Sir William Rowan Hamilton between 1840s and 1860s. Quaternions form a noncommutative extension of complex numbers in the 4-D real number space. It is a finite-dimensional associative division algebra over the real numbers, and is also a subalgebra of Clifford algebra. In the late 1960s, quaternions began to acquire a practical application in the classical mechanics. In 1985, Shoemake introduced quaternions to computer graphics for representing rotations in 3D space. Ever since then quaternions have been widely used in the realms such as computer graphics, computer animation, computer vision and robotics, etc. In 1996, the quaternion model of color images was proposed, and the application research of quaternion in image processing began to develop.In this thesis, we combined the quaternion methods with the specialized knowledge of digital image processing, especially color image processing, taking theories including quaternion matrix singular value decomposition, quaternion Fourier transform, quaternion convolution, quaternion spherical linear interpolation, compact quaternion representation of 3D rotations, etc. as the main mathematical tools, assisted with other signal processing methods, such as principal component analysis, log-polar mapping, phase correlation, Nyquist-Shannon sampling theorem, and so on, to study and discuss some issues in color image processing. The main research work and contributions are as follows:1. On the basis of theories of quaternion and quaternion matrix, we constructed the equivalent real matrix of quaternion matrix, and discussed the relationship between the quaternion matrix singular value decomposition (QSVD) and its equivalent real matrix singular value decomposition. Under the quaternion model of color images, quaternion matrix singular value decomposition is usedfor color image decomposition: (?) . The color image matrix isdecomposed into linear combination of a series of color eigen-images, in which singular value represents the energy of each eigen-image in the original image. With the help of quaternion principal component analysis, we discussed the color image processing, such as compression, denoising, enhancement, edge detection, and so forth.2. A robust color image watermarking scheme based on block QSVD and Arnold transformation with resistance to geometric attacks was proposed. Since singular values possess some excellent properties such as stability, scaling invariance, rotation invariance, translation invariance and transposition invariance, etc., we chose to embed and extract watermark in the QSVD transform domain of color images. In order to improve the QSVD speed and increase the capacity of watermark information embedded, we adopted the block QSVD method. For enhancing the security and robustness against cropping attacks, we employed Arnold scrambling pretreatment before embedding watermark. In order to strengthen the robustness of the proposed watermarking scheme against rotation attacks, we used log-polar mapping (LPM) and phase correlation methods, first calculating transformation parameters of geometric attacks, and then re-synchronizing the watermark embedded in singular values and the host image through inverse transform before extracting watermark. Experimental results show that our watermarking scheme is robust to Gaussian noise, JPEG lossy compression, low-pass filtering, median filtering, cropping, scaling, cyclic translation, rotation and other image attacks.3. Making use of the capability of quaternions to succinctly represent 3D rotations, we constructed quaternion rotation edge detection operator for color images. The edge of color images is defined as the discontinuous jump of colors (including luminance/brightness, hue and saturation). The same or similar color vectors lying inside a locally homogenous region will overlap or approach to overlap after revolving 360 degrees around a fixed axis, the difference between which will be 0 or close to 0 (black or look like black). However different color vectors will not overlap after revolving, which results in the nonzero difference, indicating the current color pixel lies on an edge. Experiments show that the more contour features (including both luminance transition and color transitions) of the original color image can be preserved by using quaternion rotation edge detection operator. The test results illustrate the feasibility and effectivity of the proposed algorithm.4. The relief display of a two dimentional plane image will produce sculptural embossing effect by a certain treatment. It can reproduce art images, highlighting scenery and its level in the plane, dignified and full of infection, giving a strong visual impact. We put forward a new method for image embossed display using quaternion rotation edge detection operator. The experimental results show that the method is easy to implement and executes quickly, the display effect is similar to or even better than that by the generalized fuzzy operator method and morphological edge detection operator method. Through our approach we can quickly and efficiently obtain satisfactory relief images.5. By means of the quaternion spherical linear interpolation (Slerp), we derived a bi-spherical linear interpolation formula (Bi-Slerp). And we experimented on color image magnification in quaternion space using various method: quaternion spherical linear interpolation, bi-spherical linear interpolation, bilinear interpolation, bicubic interpolation, as well as the adaptive osculatory rational interpolation established by Thiele-type continued fractions. The experimental results were compared and analyzed. The enlargement effect of Bi-Slerp method approaches that of mainstream bilinear interpolation method. But due to using 16 neighboring points, the bicubic interpolation method is preferable in the interpolation accuracy to Bi-Slerp method, which uses only the information of four neighboring points. The adaptive osculatory rational interpolation method established by Thiele-type continued fractions is the best, effectively maintaining the high-frequency information of images, namely, edge information and detail information, the enlarged images having better clarity and sharpness.6. Quaternions can represent rotation easily and compactly, but quaternion algebra is mainly used in three-dimensional space. For spaces above 4D, quaternion becomes ineffective. So we introduced the conformal geometric algebra (CGA), which can be extended to n-dimensional space, and more universal, intuitive, simple, efficient in geometric representation and transform calculation of objects. We experimented on the representation of geometric entities and the computation of motions with CGA, and compared the similarities and differences of quaternion and conformal geometric algebra in rotation description.