RAO  TRANSFORMS:  A  NEW  APPROACH  TO
INTEGRAL  AND  DIFFERENTIAL  EQUATIONS

        A  BRAND  NEW APPROACH  TO  A CENTURY  OLD  PROBLEM.

UNIFIED, LOCALIZED,  SIMPLIFIED, AND EFFICIENT  SOLUTION.

By  Dr.  Muralidhara SubbaRao (Rao)
rao@integralresearch.net

"We must know. We shall know."  --David Hilbert

Intellectual Property on Sale:  Patent Applications:    linsys.pdf   inteq.pdf  dirvis.pdf 

Book on Sale: "Rao Transforms: A New Approach to Integral and Differential Equations"

WHITE  PAPER 1:  SUMMARY

RAO  TRANSFORMS: A New Approach to Integral and Differential Equations


WHITE PAPER 2: APPLICATION  EXAMPLE

RAO  TRANSFORMS: Application to the Restoration of Shift-Variant Blurred Images


          FLOW-CHART OF  RT  APPROACH

          SEMINAR  PRESENTATION  SLIDES:

           SEMINAR  ABSTRACT

EXPERT  COMMENTS  ON  THIS  RESEARCH  RESULTS:

In summary the proposed research appears to be valid and
has applications. It is guaranteed to produce doctoral dissertations."
    -An expert researcher in the field in his review of this research for a US federal research funding agency.
   CLICK  HERE  FOR  A   COMPLETE  TECHNICAL  REVIEW   OF   THIS  EXPERT   SPECIALIST  ON  THIS  RESEARCH.

"... 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)
For shipping within USA (US$139 including book price, tax, and shipping and handling)

For shipping outside USA (US$149 including book price, tax, and shipping and handling)


[ 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. ]



Selected pages from the book   (click on the link below)
     Front cover,  dedication page, copyright notice page,  Preface, Table of contents,  selected chapters, and Back cover (pdf).

*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.

 
  See http://www.uspto.gov         http://www.freepatentsonline.com
                   
List of Patent Applications:

    1. M. Subbarao (Rao),  linsys.pdf  
    2.  “Methods and apparatus for computing the input and output signals of a linear shift-variant system”, United States Patent and Trademark Office, Provisional patent application filed  on 11/08/2004, Ref. No. 16800 U.S. PTO. 60/626028. Full patent application filed on 9/26/2005.  No. 113010 U.S. PTO, 11/235724. Published on the U.S. PTO website. Published Application No. 20060101106.
    3. (a) M. Subbarao (Rao),   inteq.pdf
    4. “Method and Apparatus for Solving Linear and Non-Linear Integral and Integro-Differential Equations", United States Patent and Trademark Office, Provisional patent application filed ion 11/23/2004, Ref. No. 22151 U.S. PTO.  60/630395.  Combined full patent application  filed on 10/03/2005, No. 112991 U.S. PTO, 11/242191. Published on the U.S. PTO website. Published Application No.20060111882.

      (b) M. Subbarao (Rao),   inteq.pdf
      Unified Method and Apparatus for Solving Linear and Non-Linear Integral, Integro-Differential, and Differential Equations", Provisional patent application filed in the United States Patent and Trademark Office, 11/29/2004, Ref. No. 19249 U.S. PTO, 60/631555.  Combined full patent application filed on 10/03/2005, No. 112991 U.S. PTO, 11/242191. Published on the U.S. PTO website. Published Application No. 20060111882.

    5. M. Subbarao (Rao),    dirvis.pdf     
    6.  Direct Vision Sensor for 3D Computer  Vision, 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 equivalent localized integral 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.

CLICK  HERE  FOR  A   COMPLETE  TECHNICAL  REVIEW   OF   THIS  RESEARCH  BY  AN   EXPERT   SPECIALIST.



To Corporations/Industry:
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 :
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 SVD method in specific applications and is shown to perform as well as SVD in the worst case scenario. The author also compares RT method to Fast Fourier Transform (FFT) to highlight minor improvements in the context of image restoration. Furthermore the proposed method has been shown to have applications in solving more general class of Linear and non-linear differential equations of various important types, such as, Laplace, Poisson, Helmholtz, and Boundary value problems of certain elliptic type PDE's by recasting them as integral equations. However the specific advantages over conventional methods itself has been left for future research. Given, its advantages in solving some types of problems it is likely that the benefits will carry over to solving the similar class of more general problems 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

Photo

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 SUNY Stony Brook University. 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.

LINKS:
Overview of Integral Equations
http://mathworld.wolfram.com/IntegralEquation.html
General Integral Transforms
http://mathworld.wolfram.com/topics/GeneralIntegralTransforms.html
Integral Transforms
http://mathworld.wolfram.com/topics/IntegralTransforms.html
Ramanujan's Master Theorem
http://mathworld.wolfram.com/RamanujansMasterTheorem.html

Mathematicians:  click on these links for brief biographies.
 Fredholm   Volterra   Fourier   Laplace   Hilbert    Ramanujan


©2004-2007  Dr. Muralidhara SubbaRao ( Rao).  All rights reserved. This material is protected by U.S. and other copyrights and may not be copied, sold,  or redistributed in any form without the written permission.

Last Updated: July 7, 2007.