FUNCTIONAL ANALYSIS AND APPLICATIONS TO BIOMEDICAL IMAGE PROCESSING

生物医学数据和图像处理的一个重要方面是找到 适用于应用程序的信号表示形式。《傅立叶》 例如，变换很好地适应于分析频率 信号的组成部分。香农的抽样理论很好地适用于 模拟数字转换。因此，它非常适合于加工。 以及通过数字计算机存储信号和图像。然而， 香农的抽样理论仅限于带限函数。我们 开发了非带限信号的通用采样程序。 特别是，通过将采样视为一个近似问题， 平移不变函数空间，我们已经证明了最小 信号的平方逼近由采样过程组成 概括一下古典的。它包括最优的预滤波， 纯抖动稳定采样和后滤波 重建。我们还发展了一种普遍的理论 使用多分辨率和多分辨率新概念的 适合多尺度信号处理的小波，EDGE 检测任务、信号分析、编码和压缩。

An important aspect of biomedical data and image processing is to find signal representations that are adapted to the application. The Fourier transform, for example, is well adapted to analyzing the frequency components of signals. Shannon's sampling theory is well adapted to analogue digital conversions. Thus, it is well suited to the processing and storage of signals and images by digital computers. However, Shannon's sampling theory is restricted to band limited functions. We have developed a general sampling procedure for non-band limited signals. In particular, by considering sampling as a problem of approximation in translation-invariant function spaces, we have shown that the least squares approximation of a signal consists of a sampling procedure generalizing the classical one. It consists of an optimal prefiltering, a pure jitter-stable sampling, and a postfiltering for the reconstruction. We have also developed a general theory of representation that uses the new concepts of multiresolutions and wavelets, which are well adapted to multiscale signal processing, edge detection tasks, signal analyses, coding, and compression.