RAO TRANSFORMS: A New Approach to Integral and Differential Equations

Dr. Muralidhara SubbaRao (Rao), Professor of Electrical and Computer Engineering.

Second Edition 2007 is on sale now.

Third Edition 2026 is under preparation for sale in 2026.

This book presents a novel technique that was invented in 2005 by this author. It has remained unnoticed by others until now (Dec. 2025) due to a lack of demonstration of its applications to important practical problems. Such demonstration has now been found in preliminary studies for some very important practical problems (Dec. 2025). It is found to have great advantages in medical image deblurring and other applications. The contents of this book can be simply attached/uploaded to popular chatbots (e.g. ChatGPT, Gemini, Grok) to get working programs for use in actual applications in many areas including the deblurring of images in MRI, CT, PET, SPECT, Ultrasound, digital photos and videos, etc. Many new theoretical and practical applications are yet to be discovered for the results in this book and therefore it serves as a valuable resource in research and development in industry and academia. The methods in this book are likely to be included in college textbooks on image processing and applied mathematics.

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.

"... 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.

Background

Rao Transforms (RTs) provide a novel approach to the centuries old problem of integral equations. Differential equations are solved by first converting them to integral equations by incorporating boundary conditions, and then solving the resulting integral equations. Since many fundamental laws of physics are stated using differential equations, RTs are expected to have wide applications in scientific, engineering, and medical applications. RTs are based on a breakthrough strategy of “Localize, Solve, and Synthesize” using the simple equation L(u,v)=G(u+v,u) to change a global form integration kernel G to a local form kernel L. This apparently simple idea seems to have eluded researchers until now. RTs were invented by this author while doing research on shift-variant image deblurring which involves solving a Fredholm integral equation of the First Kind, and extending the results of that work to general integral equations. RTs are an extension of the Spatial-Domain Convolution/Deconvolution Transform (S Transform) invented by this author in 1989 related to convolution integral equations. S transform has been successfully used in computer vision and image processing applications such as depth-from-defocus and image restoration, and RTs are expected to be similarly useful. RTs provide both symbolic and numerical solutions. The solution is fully localized and therefore offers significant computational savings and permits extremely fine-grained parallel implementation on a computer. The approach is simple as it is easy to implement and comprehend, and unified as a common framework solves a large class of diverse problems. Therefore RTs have both computational and theoretical advantages in comparison with existing techniques. RTs can be naturally extended from the case of one dimensional problems to multidimensional cases. The basic theory of RTs, and their application to two practical problems are presented.

The Third Edition of this Book is under preparation for publication in 2026. The Rao Transforms will be renamed in the Third Edition in 2026. The naming rights for this Transform is on sale by this author. Contact the author at rao@integralresearch.net to negotiate a mutually agreeable price. The interest in renaming this new transform is expected to be high and therefore the price will be high. The high price is justified by the fact that Rao Transform is expected to be included in college text books on image/signal processing, calculus, and applied mathematics, in the future.