"... Congratulations. It does seem that
you have a novel and powerful method for solving
integral equations. ... . I admire what appears to be a brand new
and promisingly important advancement to
spatial signal processing."
--A Distinguished
Professor of Engineering in his comments on this research. "In
Mathematics,
sometimes, the simplest results are the most useful
results."
-- A Professor of Mathematics in
his
comments on this research.
Self-Published
Book:
"Rao
Transforms: A New Approach to Integral and Differential Equations",
by Dr. Muralidhara SubbaRao
(Rao), Second Edition, June 2007,
self-published
book, 130 pages,
(First Edition U.S. Copyright
Registration No. TX
6-195-821, June 1, 2005).
Buy
online using credit cards MC/VISA/AMEX (through PayPal)
[ OR Make check
payable to: M. Subbarao
for $139 (US)/$149 (non-US), and mail with your shipping address to
M. Subbarao, 95
Manchester Ln, Stony Brook, NY 11790, USA. ]
*Expert and internationally recognized
researchers who would like to
review this research may request a free copy of this book by sending
email to rao@integralresearch.net. The author may provide a copy
of the book to a limited number of experts. Their reviews may be
posted on this website.
“Direct
Vision Sensor for 3D ComputerVision, Digital
Imaging,and Digital
Video", Provisional
patent
application filed in the United States Patent and Trademark
Office, June 18, 2005, Ref. No. 113013 U.S. PTO, 60/691358. Full patent application filed on
6/10/2006. No. 112959 U.S. PTO, 11/450024. Published on the U.S. PTO website. Published Application No.20060285741.
1. Summary
Rao
Transforms (RTs) are new mathematical
transforms with applications in applied sciences, engineering, and
medical instrumentation. RTs provide a unified theoretical and
efficient computational framework for solving general integral
equations. Many fundamental laws of scinece are described by
differential equations which can be
reformulated as integral equations that incorporate boundary
conditions. RTs can be used to solve such differential equations as
well. The solution method
is simple, novel, and powerful, indicating a promisingly important advance. RTs are relevant to linear and non-linear
systems, and signal processing. A book
on this
topic is being sold, and Intellectual
Property (IP) related to this
topic is on offer for licensing. Three related patents
are pending.
The
most common and useful case is that of linear integral equations. For
this case, the solution method based on RTs is shown in Figure 1 . The new method
has significant
computational advantages in many practical applications. Theoretically,
it provides a novel and unifying way of treating a large class of
diverse problems.
Current State of the Art on Solving
Integral Equations
In the current research literature, there
is no unified theory for solving general integral equations. Solution
methods for different cases are
disconnected, lacking a common framework. There are special
methods for Fredholm-type and Volterra-Type, "First Kind" and "Second
Kind", linear and non-linear, symmetric kernels and separable
kernels, etc. Some well known methods are-- Fredholm's method
(determinants), Volterra's method (iterated kernels, Neuman
series), ortho-normal series expansion, undetermined coefficients
or power series expansion, numerical quadrature (e.g. Nystrom) methods,
etc. More importantly, in terms of practical applications, an
expert
reviewer of this research summarized the highly unsatisfactory state of
the current state of the art as follows:
"Numerical analysts hate solving
integral equations because the resulting matrix approximations are
usually
full. This means that it takes O(n^2) operations to evaluate a matrix
vector
multiply and this is too much for large n. To get around this, applied
mathematicians have looked for transformations which increase the
sparsity of
the matrix. Unfortunately, these type of transformations only work for
matrices
with special structure. The classical example is the Fourier
transformation
which reduces a circulant matrix to a diagonal matrix. These ideas go
back at
least to Cauchy. Recent examples are the Fast Multipole algorithm and
some
Wavelet transforms that I haven't paid much attention to. These methods
are
wonderful if your application has the right structure and are useless
if they
don't. [My agency] seems to mostly have problems in the category for
which
these methods are useless."
The
New Approach: Rao Transform (RT) and General
Rao Transform (GRT) RTs
use a strategy of Localize,
Solve, and Synthesize (LSS) to unify the
theory and significantly reduce computations in solving integral
equations. It uses the Rao Localization Equation (RLE) h(x,y)=k(x+y,x)
to convert a "global" form integration kernel to a "local" form
integration kernel and derive an equivalentlocalizedintegral
equation that is simpler and computationally more efficient to solve.
None of the methods in the current literature use RLE or its
equivalent. This simple equation h(x,y)=k(x+y,x) seems to have eluded
all the
past researchers. The localized integral equation is solved
separately at each point, and the resulting solutions are synthesized to
obtain a complete global solution. RLE facilitates a seamless synthesis
of local solutions to obtain the global solution. In comparison with
other methods, the localization of the problem accomplished by RLE is
complete, absolute, and superior. I
discovered
this equation while doing research on inverse optics for 3D computer
vision and image processing. The completely localized nature of
computations in RTs make them ideally suited for implementation on a
highly fine-grained parallel processing hardware (e.g. Neural Nets).
Rao Transform (RT) is useful in
solving linear integral equations such as Fredholm and Volterra
Integral Equations of the First and Second kind. General Rao Transform (GRT) is
useful in solving non-linear integral equations. Together
they provide a unified theoretical and computational framework. The
computational framework is localized, efficient, and provide
computationally efficient solutions to
general integral equations. RT and GRT can be naturally extended from
the case of one-dimensional problems to multi-dimensional cases. The
solution methods can also be extended to linear
combinations of standard form integral equations and simultaneous
integral/integro-differential equations. The methods are thought to
have reasonable numerical stability. Further, theoretical
insights provided by RT and GRT may lead to more significant
practical and theoretical advances. Many
fundamental problems in science, mathematics, and engineering,
including the problems of differential, partial-differential, and
integro-differential equations, can be converted to the problem of
solving integral equations. Therefore, further
research on RT and GRT will expand the areas of their applications.
RTs
offer significant computational speed-up in many practical applications
involving integral/differential equations. Extensive care has been taken to
fully protect
this intellectual property (IP). Three patent applications
have been filed on the RT technology. This invention is being
offered for licensing to corporations. Both product license and
research license are available. Please contact me by email.
The areas of application of this intellectual property include :
image and signal processing (e.g. image/video
restoration,
filtering),
computer vision (e.g. 3D vision sensor),
optics (e.g. computing the image formed by a lens system),
inverse optics (e.g. inverting the image formation
process
in a lens system to obtain a 3D scene model)
mathematical software (e.g. MatLab, Mathematica)
differential equations (ODE/PDEs),
analysis of linear and non-linear integral systems and
processes,
and
scientific and medical instrumentation.
To the
College Professor:
It
is better to
teach students the best and the simplest methods to solve integral
equations. I wish to bring Rao Transforms to your attention. My
book can be used as a supplement to
a standard text book for teaching. If you teach
methods
for converting differential equations to integral equations and then
solve them, then my book can also be used in a course dealing with
ODEs/PDEs. If you are authoring a text book or reference book, it may
serve the readership to include my new results.
EXPERT REVIEW COMMENTS
ON THIS RESEARCH RESULTS:
A
research proposal was submitted to a U.S. Federal Research Funding
Agency seeking funds for doing further research on RTs. Technical comments of an expert
reviewer of the proposal are included below.
Technical
Review of Proposal 50608-CI: "Rao Transforms: A New Theoretical
and Computational Framework for Linear and Non-linear Integral
Equations."
February, 2007.
Overall: In
summary the proposed research appears to be valid and has applications.
It is guaranteed
to produce doctoral dissertations.
Potential of the proposal to
achieve the stated objective of the research: The
proposal aims at developing
techniques to solve linear and non-linear integral equations. The PI
has
isolated a method called Rao Transforms that he proposes to use to
solve
integral equations. The technique has been proved by the author
to be more
effective than conventional SVD method in handling image restoration
from
blurring. The technique is also more suited for implementing on
parallel and
distributed processing architectures. The author also demonstrates its
effectiveness in solving various more general integral equations viz.
Fredholm,
Volterra, Uryshon, etc., of which image restoration from blurring is a
specific
case of Fredholm.
Likelihood of the proposed
method
to develop new capabilities or enhance existing capabilities: The method of Rao Transforms has
been proven to have computational advantages (polynomial speedup) over
conventional SVDmethod in specific applications and
is shown to perform as well as SVD in the worst case scenario. The
author also
compares RTmethod to Fast Fourier Transform (FFT)
to highlight minor improvements in the context of image restoration.
Furthermore the proposedmethod has been
shown to have applications in solving more general class of Linear and
non-linear differential equations of variousimportant
types, such as, Laplace,
Poisson, Helmholtz, and Boundary value problems of certain elliptic
type PDE's
by recasting them asintegral equations. However the
specific advantages over conventional methods itself has been left for
future
research. Given, itsadvantages in solving some types of
problems it is likely that the benefits will carry over to solving the
similar
class of more generalproblems of both differential and
integral types.
Overall potential of the
proposed
effort: If certain class of differential
equations and boundary value problems could be solved more effectively,
it is
likely to have application in problems of computational and finite
element
fluid mechanics. Specifically could help in the hull design of
underwater
vehicles. Solving integral equation will also likely impact
computational
physics.
Acknowledgements
I have provided my
research monograph to about 20 people including experts in the
field, professors, and students,
seeking comments. I wish to thank those who responded with
valuable
comments. One Professor of Mathematics has gone
through my research monograph and verified some of the key steps in the
derivation of new results. Another Professor of Engineering skimmed
through my monograph with interest and provided useful comments. I
also wish to thank a Program Director and two reviewers of a U.S.
federal research
funding agency for their valuable comments on my research. This
research
is dedicated to my family and my teachers.
Dr.
Muralidhara SubbaRao (Rao)
Contact:
Dr. Muralidhara SubbaRao (Rao)
95 Manchester Ln, Stony Brook, NY 11790,
USA. Email:
rao@integralresearch.net Ph. No.:
1-631-751-2627 www.IntegralResearch.net
Dr.
Rao (Muralidhara SubbaRao / Murali
SubbaRao) graduated with a B. Tech. degree in Electrical
Engineering
from the Indian Institute of Technology, Madras,
and an M.S. and a Ph.D., both in Computer Science, from the University of Maryland
at College Park.
He is a Professor of Electrical and Computer Engineering at SUNYStonyBrookUniversity.
His teaching and research interests are Computer Vision and Digital
Image
Processing. He has been the Principal
Investigator of research grants
from industry and the National Science Foundation. He has authored one
book
(research monograph other than that on RTs), published over 50
papers in
professional journals and conferences, and is the inventor of 4 U.S.
patents that have been licensed to industry. He is a pioneer researcher
in the
field of Computer Vision who invented the Depth-from-Defocus
technique
that uses arbitrarily defocused images (without requirement of any
focused
image) for three-dimensional shape recovery. Ten students have
completed their
Ph.D. thesis research under his supervision in the area of Computer
Vision and Image processing. He was a principal member
and the
Chief Computer Scientist of a high-tech start-up company for
online image
management in 2000-2001.
