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。这个 例如，傅立叶变换很好地适用于分析 信号的频率分量。香农的抽样理论是好的 适用于模拟/数字转换，因此非常适合 数字计算机对信号和图像的处理和存储。 然而，香农的抽样理论仅限于带限 功能。我们已经制定了一个通用的抽样程序，用于 非带宽限制信号。特别是，通过将采样视为一种 平移不变函数空间中的逼近问题，我们 已经证明了信号的最小二乘近似包含 一种推广经典抽样程序的抽样程序。它由以下内容组成 最佳预滤波、纯抖动稳定采样和 用于重建的后处理滤波。我们还开发了一种 使用新概念的表示法的一般理论 适用于多尺度的多分辨率和小波 信号处理、边缘检测任务、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 analog/digital conversions, and is therefore well suited to the processing and storage of signals and images by digital computers. However, Shannon's sampling theory is restricted to bandlimited functions. We have developed a general sampling procedure for non-bandlimited 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 classic 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.