一般化二維小波轉換及影像多重解析分析

Abstract

小波轉換最重要的特性即為可使用少量的小波轉換係數近似一個信號。因為 這個特性，JPEG 2000 納入了小波轉換，做為其演算法的一部分。 然而，此一特性的理論基礎皆是根據一維信號而來。雖然我們可以使用可分 離小波轉換將一維小波轉換擴展至二維小波轉換。但可分離小波轉換乎略了二維 信號相較於一維信號，有較豐富的幾何特性，如邊緣。 既然二維信號有更多的特性，許多學者便開使研究專門針對二維信號的轉 換，使其不但擁用小波轉換所擁有的特性，且提供優於可分離小波轉換的效果。 本論文著眼於分析這些方法的精神、利弊，並加以改善，使其更加符合實用 所需的條件。

The most important feature of the wavelet transform is that we can use few wavelet coefficients to approximate a signal. Because of this property, JPEG 2000 adopted the wavelet transform as a portion of its algorithm. However, the fundamental theory of this feature was derived from one-dimensional signals. For two-dimensional signals, we can use “separable wavelet transform” to extend one-dimensional wavelet transform into two-dimensional wavelet transform. Although this method was used widely, it ignored the geometric properties of the two-dimensional signal such as edges. Since two-dimensional signals have more features, many researchers started to propose a new transform such that the new transform not only has all features of the wavelet transform but also exploit the properties of the two-dimensional signals. Furthermore, the performance is better than that of the separable wavelet transform. This thesis focuses on the ideas, advantages, and disadvantages of these new transforms. After discussing these methods, we propose our method to improve the performance.