Introduction, numerical images and geometrical representation§

author:	David Coeurjolly

Goals§

Lectures

Introduce fundamental concepts in computer graphics, image processing, digital geometry and computational geometry
Illustrate links between geometrical analysis of shapes and related fields (complexity, arithmetic, word theory, ….)

Practical work (TP)

Implement some image processing/shape analysis tools
Comparative evaluation principles (tests, asymptotic vs. experimental computational costs…)

Infrastructure§

Github project https://github.com/dcoeurjo/lectureDG

Lecture sources
Practical work

git clone https://github.com/dcoeurjo/lectureDG.git

Practical work

Mostly based on DGtal (http://dgtal.org)

Homework, project, final exam

One practical work session will be evaluated
Homework
Final exam

Context: Analysis of geometrical objects§

Geometrical objets

From acquisition devices
- CCD devices
- tomographic images (IRM, scanners X, …)
From modeling processes
- Geometrical modelers, CAD (computer-aided design)
- From mathematical modeling of phenomena

Analysis

… computer-based -> algorithms
… quantitative
- Scalar shape descriptors
- Geometrical paramters
- Topological invariants, …
- -> quality, robustness, certified computations, …

Couple of Acquisition Devices§

CCD Device§

Charged-Coupled Device

Principles photo-active regions/pixels : each unit if surface element accumulate some electrical charges proportional to the intensity of received light

photon -> electron charges


Linear device [1]	2D device [2]	Bayer pattern [3]

Physical notion of pixels

[4]

Note

[1]	http://en.wikipedia.org/wiki/File:CCD_line_sensor.JPG

[2]	http://fr.wikipedia.org/wiki/Fichier:CCD_in_camera.jpg

[3]	http://en.wikipedia.org/wiki/File:Bayer_pattern_on_sensor.*

[4]	http://fr.wikipedia.org/wiki/Fichier:CCD_Sensor_Layout_Evolution.png

Associated Modeling§

By construction

Underlying support geometry is induced by a periodic tiling
Values are quantified on a small number of bits

def.

Image: $\qquad S \subset \mathbb{Z}^n \rightarrow Q\subset \mathbb{Z}^+$

Tomography§

Principles

particles are emitted (ions, protons, photons, …) from a source device to a target (object, body, …)
a detector measures particles intensity after the object traversal. The intensity decay is a function of the time flight and the traversed material)
the image is reconstructed from attenuation measurements using back-projection approaches

[5] [6]

Note

[5]	http://en.wikipedia.org/wiki/File:CT_Scanner_Line_Beam.jpg

[6]	http://en.wikipedia.org/wiki/File:Cone_beam_image_Cam_320x240.jpg

Tomography (bis)§

Basic Idea Radon’s Theorem

$[Rf](t,\theta) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x,y)\delta(t-x \cos(\theta) - y \sin(\theta)) dxdy$

$\delta$ Dirac measures

<demo>

Digital Tomography§

Specificities

Projection directions: rational straight lines $ax-by=c$ ( $a,b,c\in\mathbb{Z}$ )
Projection function: sum of digital point values along the rational straight line
Data to reconstruct: binary values {0,1} or in $\mathbb{Z}$

Problems

Number of required projections
Uniqueness of the reconstruction
Sometimes, further hypotheses are required (convexity, smoothness, …)

Example: Mojette Transform§

Digital Tomography: conclusion§

By construction or for efficiency of the reconstruction process from projections, the result isusually defined in regular lattices in 2D or 3D

def.

Image: $\qquad S \subset \mathbb{Z}^n \rightarrow Q\subset \mathbb{Z}^+$

Acquisition from laser/ultrasound/contact§

Mechanical contacts probe measurements from motorized systems

Approches based on time-of-flight computations (e.g. laser or sonar rangefinder, …)

measures the time taken by the pulse to be reflected off the target and returned to the sender
data = distance of the target from the source en a set of directions. After reconstruction, $\{(x,y,z)\}\in\mathbb{R}^3$

Reconstruction from triangulation principle§

Input: a pulse (or series of) is emitted from the source

and observed if a detector (ex. laser + CCD)
Relative position of the source and the detector is known and thus
From 2D positions in the detector,
data = After calibration and reconstruction, a point cloud $\{(x,y,z)\}\in\mathbb{R}^3$

Reconstruction from images: example with Epipolar Geometry§

[7]

[7]	http://en.wikipedia.org/wiki/File:Epipolar_geometry.svg

Hybrid approach: Kinect§

Color camera (CCD, 640x480)
Pattern projection in infrared + CMOS detector (~640x480)

Raw data = Depth map in the detector plane + color image

Quiz: why infrared ?

Bottlenecks/Problems to consider§

Filtering (noise, outliers,…)
Point cloud registration (by device calibration or from data processing) [8]
Local density control
How to deal wit occluded regions ?
…

[8]	Point Cloud Library

Bottlenecks/Problems to consider (bis)§

Geometrical reconstruction define a high quality manifold approximating or interpolating the point cloud. [9]

Manifold cf later
Quality
- distance to samples (which metric ? how to be robust to noise ?…)
- Smoothness of the reconstruction
- Geometrical and topological certificate of the reconstruction
…

[9]

CGAL.org

Some External Devices§

Raster Screens§

Principles

Luminophores with RGB cells (red, green, blue) to render colors by additivity
In our context
- Color image : $[0..n]\times[0..m] -> Q^3$
- $Q$ : range of each color channel (8bits, 24bits, …)

_images/Liquid_Crystal_Display_Macro_Example_zoom_2.jpg

Problems

Rendering problem: geometrical models $\rightarrow$ digital representations
ex: straight lines/circles drawing…

3D printers§

Layer based

Additive approach: material is added layer by layer
Several technologies: heated plastic layers, stereolithography (solidification of the polymer resin from ultra-violet beam), …
Key point : slice based or discrete representation $\,f: \mathbb{Z}^3 \rightarrow \{0,1\}$ of the object to print

Problems

Geometrical model $\rightarrow$ layer based representaiton
How to control the topology/surface orientation during the process ?
Some geometrical analysis can be performed to enforce stability, robustness, …

(Very) Short (Subjective) Glossary§

Image Processing (traitement et analyse d’images)

General image related topic
Image as the “realization” of a bi-directional signal
keywords: image compression, filtering, denoising, color analysis, shape segmentation,…

Computer Vision (Vision par ordinateur, reconnaissance de formes,..)

Focusing on object perception
Keywords: Shape recognition, shape/image indexing and retrieval, 3D reconstruction from images, …

(Very) Short (Subjective) Glossary§

Computational Geometry

Discrete data (point sets, simplicial complexes, …)
Aim to first reconstruct structures and perform geometrical/topological computations
Certified computations
Complexity

Digital Geometry

We focus on discrete data defined on lattices ( $\Rightarrow$ integer coordinates, arithmetical properties of objects…)
Values are usually binary
take geometrical/topological decisions from objects defined by extension (vs. from properties)

(Very) Short (Subjective) Glossary§

Geometric Modeling

Model objects and complex geometrical scenes
Procedural modeling, animation, …

Image Synthesis

Image rendering from geometrical models + material properties + Illumination models
Ray shooting, radiosity, photon path tracing, ….

Overview of the course§

TOC§

Image Processing

Image filtering
Colorimetric (historgram) processing
Mathematical morphology
Segmentation

Digital Geometry

Digital model and Digital topology
Digital object surface analysis
Volumetric processing
Fast arithmetical transforms

Computational Geometry

Geometrical predicates, convex hulls, Delaunay triangulations
Spatial Data structures
…

Introduction, numerical images and geometrical representation§

Preliminaries§

Goals§

Infrastructure§

Context: Analysis of geometrical objects§

Couple of Acquisition Devices§

CCD Device§

Associated Modeling§

Tomography§

Tomography (bis)§

Digital Tomography§

Example: Mojette Transform§

Digital Tomography: conclusion§

Acquisition from laser/ultrasound/contact§

Reconstruction from triangulation principle§

Reconstruction from images: example with Epipolar Geometry§

Hybrid approach: Kinect§

Bottlenecks/Problems to consider§

Bottlenecks/Problems to consider (bis)§

Some External Devices§

Raster Screens§

3D printers§

Topics Overview§

(Very) Short (Subjective) Glossary§

(Very) Short (Subjective) Glossary§

(Very) Short (Subjective) Glossary§

Overview of the course§

TOC§