Objectives:
Introduction to the theory of stochastic processes.
Presentation of the main methods to process signals involving random components/noises.
Optimal filtering for signal restoration.
Application to biomedical imaging.
Detailed program:
Stochastic processes: random variables and stochastic processes; mean, variance, autocorrelation/covariance function; Gaussian processes;
stationary processes (strict and wide-sense); power spectral density function and Wiener-Khinchin theorem; ensemble and time average statistics, ergodicity, and Slutsky theorem; main examples: Brownian motion, white noise, random walk, fractional brownian motion.
Application to the study of molecular dynamics: mean-squared displacement, stochastic processes vs biophysical constraints; trajectory classification.
Parameter estimation: Bias; unbiased estimators; maximum likelihood estimator, Fisher information, Cramer-Rao inequality.
Filtering random signal: Linear time-invariant filters, properties of filtered stationary signal; Optimal filtering: matched filter, Wiener filter, causal Wiener filter, Kalman filter.
Application to image deconvolution: image acquisition in microscopy and astronomy, spread point function and convolution; deconvolution for image restoration: inverse filter, Wiener filter, Lucy-Richardson algorithm.
- Enseignant: NARDI Giacomo