Welcome! The SIAM Activity Group on Imaging Science brings
mathematicians and scientists from diverse disciplines (e.g.,
engineering, radiology and computer science) with an interest in the
mathematical and computational aspects of imaging. Currently, the
officers of the activity group are (1/1/2010 -
The responsibilities and objectives of this committee include:
In recent decades, the analysis of 2D and 3D images and video has
become vital to many areas of science, medicine, engineering,
manufacturing and entertainment. Typical objectives include image
formation, compression, reconstruction, restoration, registration,
motion recovery, tracking, feature extraction and semantic
interpretation. Moreover, recent developments in the technology of
imaging have also contributed to the explosive growth in the highly
interdisciplinary field of "imaging science," generating new
mathematical problems which cut across disciplines.
At the same time, increasingly sophisticated mathematical, statistical, and computational methods are being employed in these areas. The techniques utilized, besides those from traditional areas of mathematics, often originate in fields such as information theory, computer graphics, machine learning and speech technology, and include those based on linear transforms, harmonic analysis, nonlinear optimization, numerical linear algebra, integral equations, partial differential equations, differential geometry, statistical estimation and stochastic modeling.
World of Imaging
Many of the tasks and procedures involved in working with images may be naturally divided into three areas:
The first two areas together constitute traditional "image processing" and the third one is sometimes referred to as "computer vision" or "image understanding." More information about each area is provided below. However, these domains are far from disjoint. On the contrary, acquisition, processing and interpretation are deeply interconnected; for example, effective image restoration depends on a good model for image formation, and efficient image representation is crucial for image interpretation. Moreover, in some lines of research, such as radar automatic target recognition, one proceeds directly from sensor outputs to image descriptions.
Image formation refers to the process of obtaining images from sensor data. With the advent of new devices capable of seeing objects and structures not previously imagined, the reach of science and medicine has been in extended in numerous ways.
In particular, many classical and well-developed imaging techniques use wave propagation to generate data. This is an important modality with examples from radar, ultrasound, and seismic prospecting. Progress in sensor technology has meant that there are new types of data available from which images are to be reconstructed. These advances, coupled with advances in mathematics for integrating the collected data, deliver quantitative information about structures and phenomena long assumed to be inaccessible to imaging. Examples which offer exciting opportunities to mathematicians include nanoscale, quantum state and in vivo cell imaging, and network tomography. An emerging area related to imaging, Integrated Sensing and Processing, is an effort to implement intelligent integration of detection and processing in a systems approach.
The process of image analysis and understanding usually begins with storage and enhancement (e.g., removing blur, noise and artifacts), and often involves the recovery of motion, photometry and surface properties, especially for natural images. Mathematical methods for addressing these problems, including variational, geometric and stochastic, have made a deep and visible impact on the field in recent years, and have brought imaging science to the attention of many practitioners as well as applied mathematicians. For example, the mathematics of film editing and restoration is a singularly exciting area of image processing which naturally brings together applied mathematicians, film editors and post-production artists and engineers. Such areas, together with image formation, have been the main focus of the imaging science activity group since its inception.
Whereas much of the information carried by a digital image is generally directly accessible to a human observer (consider everyday face recognition or a radiologist studying a CT scan) the same cannot be said for automated systems. For instance, no computer program today can reliably detect a lesion in a magnetic resonance image, or convert a digital image representation of a three-dimensional scene into a symbolic description involving semantic categories. The absence of a general solution to the perceptual inverse problem of converting matrices of integers into useful information (the "semantic gap") impedes scientific and technological advances in many areas, e.g., robotics and medical diagnosis. Indeed, understanding how brains interpret sensory data, or how computers might, is arguably one of the great challenges in modern science.
Bayesian modeling and inference is based on the observation that natural images, whereas notoriously ambiguous at a local level, are perceived globally as largely unambiguous due to incorporating pre-observation and post-observation likelihoods. Special cases such as deformable templates and compositional vision lead to the construction of probability measures on complex structures, such as grammars, graphs and spaces of transformations.
The simple observation that the names of objects do not change under
various image transformations has led to another approach to image
interpretation based on "invariant"
geometric and algebraic. Still other methods are based on the
role of "shape," the analysis of which has driven research in
differential geometry, stochastic diffusions and nonlinear partial
differential equations. Finally,
nearly all methods encounter
formidable computational challenges, inspiring new strategies for
simulation, search and optimization.