Differential Geometry And Difference Geometry?

[adityatatu]

HELLO ALL,
I AM StUDYING DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS IN IMAGE PROCESSING. WHAT EVERYBODY DOES IS THAT TREATS THE IMAGE OR OBJECT AS A CURVE OR A SURFACE AND USES VARIOUS DIFFERENTIAL GEOMETRY OPERATORS TO GET THE DESIRED RESULT, FIRST IN THE CONTINUOS DOMAIN AND THEN FOR IMPLEMENTATION PURPOSES, DISCRETIZES THE RESULTS AND EQUATION.
I WANTED TO KNOW WHETHER SOMETHING LIKE DIFFERENCE GEOMETRY EXISTS, WHICH BEGINS ITSELF ON THE DISCRETE DOMAIN AND OPERATES IN THE DISCRETE MODE, SO AS NOT TO TREAT THE IMPLEMENTATION ISSUE SEPARATELY.
YOUR HELP WILL B HIGHLY APPRECIATED.
aditya_tatu@da-iict.org

 

[robphy]

These may be of interest to you:

http://www.lgep.supelec.fr/mse/perso/ab/bossavit.html
http://citeseer.ist.psu.edu/560328.html
http://www.cap-lore.com/MathPhys/Regge.html

 

[M98Ranger]

Hello Aditya,

Thank you for sharing your interest in differential geometry and its applications in image processing. Differential geometry is a branch of mathematics that deals with the study of curves, surfaces, and their properties using the tools of calculus. It has numerous applications in various fields, including computer vision and image processing.

As you mentioned, differential geometry is typically used in the continuous domain, where images or objects are treated as curves or surfaces. However, there is also a concept called "discrete differential geometry" which deals with the study of discrete objects, such as meshes or graphs, using similar differential geometric concepts. This approach is often used in computer graphics and computer-aided design.

In terms of your question about "difference geometry," there is no specific branch of mathematics with this name. However, there are various approaches and techniques that use discrete methods to solve problems in geometry and image processing. For example, discrete differential geometry is one such approach, as mentioned above.

Additionally, there are other methods such as discrete calculus and discrete differential operators that operate on discrete domains and can be used in image processing applications. These techniques aim to bridge the gap between continuous and discrete approaches, making it easier to implement solutions in the discrete domain.

I hope this helps to answer your question. Keep studying and exploring the fascinating world of differential geometry and its applications! Best of luck in your studies.