June 2014
Intermediate to advanced
464 pages
11h 15m
English
book
Mathematical Foundations of Image Processing and Analysis, Volume 1
by Jean-Charles Jean-Charles Pinoli Pinoli
Content preview from Mathematical Foundations of Image Processing and Analysis, Volume 1Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.
8
Basic Reminders in Topology and Functional Analysis
Topology and Functional Analysis are of key importance in Mathematical Imaging because they provide a lot of notions and tools that will be useful for setting up and studying the mathematical frameworks for both gray-tone and binary images.
8.1. Mathematical disciplines
The basic mathematical discipline is Topology [KEL 75, JÄN 84], which was historically developed from the concepts arising from Geometry and Set Theory. It deals with abstract ‘spatial’ properties, primarily for continuity based on the central concept of neighborhood. Topological spaces are mathematical structures that make it possible to formally define concepts such as continuity, convergence, compactness and connectedness. They appear in virtually every branch of modern Mathematics and are a central unifying notion.
The other main mathematical discipline of reference is Functional Analysis [KOL 99; Original ed., 1954 and 1957] [KAN 82] [KRE 89; 1st ed., 1978] that deals with the study of functions in concrete or abstract forms.
8.2. Topological spaces
8.2.1. Neighborhood systems
A topological space is a set 
in which is defined, for any of its elements 
, a so-called fundamental system of neighborhoods, or for short a neighborhood system, which consists of assigning ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Read now
Unlock full access
More than 5,000 organizations count on O’Reilly
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.