Multiframe Image Restoration

Outline

• Introduction

• Mathematical Models

• The restoration Problem

• Nuisance Parameters and Blind Restoration

• Applications

Introduction

• Multiframe image restoration is concerned with the improvement of imagery acquired in the presence of varying degradations.

• In most situations digital data are acquired, and the restoration processing is carried out by a generator special-purpose digital computer.

The general idea of restoration processing

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Image Blur and Sampling

• System and environmental blur

• detector sampling

System and Environmental Blur

• f is blurred by the imaging system, and the observable signal is

• the continuous-domain intensity is formed through a convolution relationship with the image intensity:

System and Environmental Blur

• The point-spread function for diffraction is modeled by the space invariant function:

• the inner product operation

System and Environmental Blur

• Imaging systems often suffer from various types of optical aberrations -imperfections in the figure of the system’s focusing element (usually a mirror or lens).

• The point-spread function takes the form:

System and Environmental Blur

• e(u) is the aberration function

• An out-of-focus blur induces a quadratic aberration function:

• where r is the distance to the scene, d is the focal setting, and f is the focal length.

System and Environmental Blur

• Wave propagation through an inhomogeneous medium such as the Earth’s atmosphere can induce additional distortions. These distortions are due to temperature-induced variations in the atmosphere’s refractive index, and they are frequently modeled in a manner similar to that used for system aberrations:

Sampling

• The detection of imagery with discrete detector arrays results in the measurement of the (time-varying) sampled intensity:

Sampling

• A sequence of image frames

is available for detection

•Each frame is recorded at the time t = t k , and the blur parameter takes the value 8k = 8, during the frame so that we write

Nosie Models

• Electromagnetic waves such as light interact with matter in a fundamentally random way

• Quantum electrodynamics (QED) is the most sophisticated theory available for describing the detection of electromagnetic radiation.

• Electromagnetic energy is transported according to the classical theory of wave propagation, and the field energy is quantized only during the detection process

Object Category Recognition

• the expected photocount for the nth detector during the k-th frame is:

• Read-out noise

The Restoration Problem• The intensity function

Restoration as an Optimization Problem

An optimization problem

Methods

• Maximum-Likelihood Estimation

Gaussian Noise

Poisson Noise

Methods

• Sieve-Constrained Maximum-Likelihood Estimation

Methods

• Penalized Maximum-Likelihood Estimation

Methods

• Maximum a Posteriori Estimation

Methods

• Regularized Least-Squares Estimation

Methods

• Minimum I-Divergence Estimation

Linear Methods

• Linear methods for solving multiframe restoration problems are usually derived as solutions to the regularized least-squares problem:

Linear Methods

• Linear methods for solving multiframe restoration problems are usually derived as solutions to the regularized least-squares problem:

Linear Methods

• C is called the regularizing operator

Linear Methods

• In matrix-vector notation, the regularized least-squares optimization problem can be reposed as

with the minimun-norm solution satisfying:

or

Nonlinear (Iterative) Methods

• General optimization problem:

Applications

• Fine-Resolution Imaging from Undersampled Image Sequences

• Ground-Based Imaging through Atmospheric Turbulence

• Ground-Based Solar Imaging I with Phase Diversity

Applications

• Fine-Resolution Imaging from Undersampled Image Sequences

Applications

• Ground-Based Imaging through Atmospheric Turbulence

Applications

• Ground-Based Solar Imaging I with Phase Diversity