One of the hallmarks of the subject is the incredible variety
of mathematical tools and theories that arise. Although perhaps
daunting to the novice, this diversity enhances the range
of possible interactions and cross fertilizations among the
different mathematical disciplines, as each is called into
play in quantifying, analyzing and developing practical algorithms.
A key goal of the multimedia year at the IMA is to foster
the interaction between researchers using similar tools in
different multimedia modalities, and, on the other hand, providing
researchers utilizing different mathematical techniques to
study the same modality to compare results and combine promising
methods. The year will play an important role in exposing
the mathematical community to a new range of challenging and
timely mathematical problems and applications. Particular
attention will be paid to the training of postdoctoral researchers
to be familiar with a wide variety of mathematical tools and
techniques that will hopefully lay the foundations for a genuinely
mathematical discipline that will become known as "multimedia."
The underlying mathematical theories can be broadly divided
into several important, overlapping categories. First, since
vision, language, music, video and other sensory processes
are much less deterministic than ordinary physical phenomena,
a large class of methods rely on probabilistic and stochastic
paradigms. For example, Markov random fields form the basis
of much current research in speech and feature recognition,
in visual tracking, in segmentation, and in signal processing
and storage. Statistical estimation and Bayesian methods feature
in pattern recognition, in language modeling, in optical character
recognition, as well as signal image reconstruction. Monte
Carlo methods have been successfully used in the numerical
analysis of some processes, as well as in computer design
and graphics.
Wavelets and other transform methods appear in a wide variety
of contexts, including image and signal compression and enhancement,
computer graphics, texture analysis, and the recovery of degraded
audio signals. A particularly striking example is the FBI's
adoption of a wavelet-based system for compression of fingerprint
images. (On the other hand, image enhancement techniques based
on nonlinear differential equations have also been dramatically
successfully in forensics and video enhancement.) Wavelet
methods are particularly useful in the as yet poorly understood
theory of image textures, but successfully incorporating them
into the variational and partial differential equation approaches
to segmentation has yet to be completed.
Since digital imagery is a fundamentally discrete data system,
a particular surprise is the relevance of both nonlinear partial
differential equations and variational methods in low level
image and video processing. The mathematical techniques include
the various recent approaches to curve shortening and mean
curvature flows (originally studied by differential geometers)
and the variational approaches of the type used in minimal
surfaces. These methods are used in denoising images and in
edge detecion of features, and have seen success in visual
tracking and recognition of objects, medical imaging, video
enhancement, and collision avoidance. Implementation of the
relevant nonlinear diffusion equations has led to the development
of fast, sophisticated numerical algorithms that have broad
applicability. The passage from discrete pixel-based image
to continuous partial differential equation and back to discrete
finite difference numerical implementation sounds paradoxical,
until one realizes that the fundamental processes in continuum
mechanics and fluid dynamics work in precisely the same paradigm
- the discrete molecular system is modeled by a continuous
system, e.g. the Navier-Stokes equations, which are themselves
integrated numerically via a discrete approximation.
The nonlinear partial differential equation approach to image
processing relies on classical differential geometry. Modern
fractal geometry, which arose in the study of natural phenomena,
also has immediate applications in computer graphics, animation
and the construction of artificial scenes. Fractal-based algorithms
has had dramatic results in practical image compression, while
very recent wavelet-fractal hybrids indicate the potentialities
for cross disciplinary approaches to multimedia processes.
Algebraic theories arise in cryptography and information theoretic
approaches to signal processing. Secure computing, copyright
protection, and language grammars also rely on fundamental
algebraic methods for implementation.
In computer graphics and geometric design, the digital representation
of fully three-dimensional objects has been the subject of
intense research activity in the past decade, leading to powerful
new algorithms. Algebraic geometry, splines, wavelets, optics
and solid geometry are but a few of the mathematical tools
brought to bear on this vital area. Applications include computer
graphics, surface and solid rendering, animation, flight simulation,
virtual reality, and scientific and medical imagery. But real-time
display of fully three-dimensional images remains problematic,
necessitating further research in the fundamental mathematical
issues. Inevitable technological advances in graphics performance
will continuously change the requirements for real-time rendering
and the complexity of models to be visualized.
Group theory is recognized to play an increasingly important
role in media analysis; for instance, both human and camera
visual systems naturally incorporate certain symmetries, including
translations, rotations, scalings, and then progressing on
to the less familiar groups of affine and projective transformations.
Galilean invariance plays an important role in movie restoration
and recognition of moving objects. In general, the reliable
recognition of objects must form an integral part of any functional
multimedia system. These include recognition of visual objects,
target recognition, satellite images, speech and language
recognition, optical character recognition, and language.
Most of these are still in their infancy, with the underlying
mathematical theory still in need of development. Applying
symmetry groups to recognition problems in a natural fashion
requires the adaptation of classical geometric theories of
differential invariants and algebraic invariants, subjects
that formed the core of pure mathematical research in the
last century. Recent work on symmetry-incorporating numerical
approximations hold significant promise for computation of
invariants and object recognition in physical images.
Use of multimedia in manufacturing holds great potential for
industrial applications. Visual control of the robots used
in industrial processes, such as semiconductor manufacturing
and etching, requires efficient, real-time processing of images,
incorporating the denoising, enhancement, segmentation, and
object recognition techniques from a computer vision system
into a broader control-theoretic loop. Some striking experiments,
including an image-based vehicular navigational system, show
significant promise. Fast and efficient numerical implementation
of the analytical algorithms provides the key to real-time
applications and automatic control. Both finite difference
and finite element methods have been successfully employed.
For example, the use of level set methods for front tracking
and interface evolution in phase transitions requires fast
marching algorithms and efficient numerical implementations.
However, to date the marriage of computer vision and control
theory has yet to be properly consummated, requiring a new
synthesis of the underlying mathematical theories.
In summary, while much progress can be seen on individual
aspects, their synthesis into a mathematical theory of multimedia
is as yet unexplored. This year at the I.M.A. will provide
a unique and unprecedented opportunity to bring together researchers
in a wide variety of mathematical disciplines and applications.
Forging these disparate subjects into a new and vital subject
will have long range effects, both within mathematics, and
in the practical applications to multimedia science.
Special note about the worksohps on Mathematical Methods
in Geometric Design and Computer Graphics:
Most multimedia content today consists primarily of text,
2D images, video, and audio. Multimedia of the near future
will additionally rely heavily on 3D graphics. The creation
and use of 3D graphics in turn draws upon the two closely
related disciplines of geometric design (the creation of the
geometric objects), and computer graphics (the animation and
display of the objects). These disciplines have been extraordinarily
vibrant and diverse, with current applications including flight
simulation, medical imaging, scientific visualization, computer-aided
design, and entertainment. Results obtained in research laboratories
over the past twenty years have given rise to a multi-billion
dollar industry. The pending 3D graphics revolution is due
in large part to impressive improvements in processor speeds
and memory capacities. More impressive still are numerous
algorithmic improvements that have opened up new application
areas and have provided remarkable asymptotic reductions in
both running time and memory requirements. Many of the algorithmic
advances are based on mathematical methods such as wavelets,
finite element analysis, Monte Carlo methods, and algebraic
geometry. There is every reason to suspect that this trend
can continue, so it is essential to stimulate collaboration
between the mathematics and graphics communities. The goal
of this pair of workshops is to do just that.