IMAGE FORMATION PROCESS

Imaging process maps three-dimensionalscene points onto a two-dimensionalimage plane.

Imaging (Perspective) Transformation

3D-Scene ==============> 2D-Image

Perspective Transformation is key to theImaging Process. It projects 3D-pointsonto a plane.

An analytical model of PerspectiveTransformation can be found byestablishing:

● A world co-ordinate systemrepresenting 3D scene.

● Camera co-ordinate system (x, y, z)which has the image plane.

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IMAGING PROCESS MODEL

For the camera coordinate system (x, y, z)assume:

● Its xy-plane is coincident with theimage plane.

● Its z-axis is along the optical axis ofcamera or line of sight of the observer.

● Centre of camera lens is at (0, 0, λ)where λ is the focal length of lens.

If (X, Y, Z) is the world co-ordinates ofany point in a 3-D scene and (x, y) is itsprojection onto the image plane.

Perspective transformation is the relationbetween (X, Y, Z) and (x, y) which can beobtained by using similar triangles.

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PERSPECTIVE TRANSFORMATION

x/λ = - X/(Z - λ) = X/(λ - Z)y/λ = - Y/(Z - λ) = Y/(λ - Z)

The image plane co-ordinates of theprojected 3-D point are:

x = λX/(λ - Z)y = λY/(λ - Z)

These are non-linear equations. It is oftenconvenient to express them in linearmatrix form.

A point in cartesian world co-ordinatesystem can be represented as a vector incartesian and homogeneous co-ordinates.

| X | | kX |w = | Y | or wh = | kY |

| Z | | kZ || k |

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PERSPECTIVE TRANSFORMATION(cont. 1)

Perspective matrix

| 1 0 0 0 || 0 1 0 0 |

P = | 0 0 1 0 || 0 0 -1/λ 1 |

ch = P wh

| 1 0 0 0 | | kX |= | 0 1 0 0 | | kY |

| 0 0 1 0 | | kZ || 0 0 -1/λ 1 | | k |

| kX |= | kY |

| kZ || -kZ/λ + k |

ch elements are the camera coordinates inhomogeneous form.

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PERSPECTIVE TRANSFORMATION(cont. 2)

The cartesian coordinates of any point inthe camera coordinates system are:

| x | | λX/λ − Z |c = | y | = | λY/λ − Z |

| z | | λZ/λ − Z |

z component is of no interest in termsof image formation model.

It acts as a free variable in the inverseperspective transformation.

The Inverse Perspective Transformationmaps an image point back into 3-D

wh = P−1 ch

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PERSPECTIVE TRANSFORMATION(cont. 3)

| 1 0 0 0 |P−1 = | 0 1 0 0 |

| 0 0 1 0 || 0 0 1/λ 1 |

An image point has co-ordinates (x0, y0, 0)where image plane is located at z = 0.

This point in homogeneous vector ch

| kx0 |ch = | ky0 |

| 0 || k |

wh = P−1 ch

| kx0 |= | ky0 |

| 0 || k |

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PERSPECTIVE TRANSFORMATION(cont. 4)

In cartesian coordinates

| X | | x0 |w = | Y | = | y0 |

| Z | | 0 |

It gives Z = 0 for any 3-D point

☛ Problem

● The problem here is caused bymapping a 3-D scene onto the imageplane.

● It is a many-to-one transformation.

● The image point (x0, y0) correspondsto all the colli near 3-D points that lieon the line passing through (x0, y0, 0)and (0, 0, λ)

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PERSPECTIVE TRANSFORMATION(cont. 5)

The equations of the line joining (x0, y0, 0)and (0, 0, λ) are:

X = (x0/λ)(λ - Z)

Y = (y0/λ)(λ - Z)

They show unless some-thing is knownabout the 3-D scene point, it is notpossible to completely recover the 3-Dpoint from its image.

☛ This is the basic computer visionproblem which every one in this areais researching on.

☛ There are some limited solutions.

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COMPUTER VISION

The purpose of computer vision is toenable a computer to understand itsenvironment from visual information i.e.images using

(grey level, colour, shades, texture etc)

Computer vision is also known as MachineVision and related to:

● Image Processing and Image Analysis

● Image Understanding

● Pattern Recognition

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COMPUTER VISION PROBLEM

Given a 2-D (two-dimensional) image,infer the objects that produce it.

Image I(x, y) is a function of Gray level ateach point in the image.

computer vision processes

I(x, y) ========> Object Information's2-D -------------------> 3-D (three dimensional)

● It is a transformation from 2D to 3D

● A many-to-one transformation. Manysolutions.

● How to identify the correct solution ?

● Third dimension (depth) is missing

● We can not completely recover thethird dimension from a 2-D image.

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COMPUTER VISION PROCESSES

● Sensing

How to get a digital image ?Image Sampling and Quantization

● Video or a TV camera(Monochrome and Colour Images)● IR camera (Thermal Image)● Range finder/Laser etc. (Range Images)

● Preprocessing

● Noise reduction● Filtering● Smoothing● Enhancement etc.

■ Frequency Domain Methods(also known as Image Processing)

■ Spatial Domain Methods

● Segmentation

Partitioning an Image into Objects ofInterest.

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COMPUTER VISION PROCESSES(cont. 1)

● Shape Extraction

Computation of Depth Information by

● Shape from Contours● Shape from Shading● Stereoscopic Imaging● Shape from Motion● Shape from Texture

● Description

Computation of Features

● Recognition

Identifying the Objects

● Interpretation

Assigning meaning to a group ofrecognised objects

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CLASSES OF VISUAL PROCESSING

Three Levels of Visual Processing

They are based on the sophistication ofimplementation and widely known as

Low, Intermediate and High level Vision

● Low Level Vision

It involves preprocessing and low levelfeature extraction

● Requires no intelli gence

● Involves only the data driven processes

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CLASSES OF VISUAL PROCESSING(cont. 1)

● Intermediate Level Vision

It involves image segmentation andimage analysis type processing. Rawshape feature extraction processes arealso its part.

● Very important processing stage

● It combines the data driven and goal driven processes

● High Level Vision

It involves the goal driven processesconsisting of model and knowledgebased vision.

● Recognition

● Interpretation

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IMAGE SEGMENTATIONThe Image Segmentation Process segment an image.

Segmentation:

* Subdivide an image into its constituents parts/objects.

* The level to which this subdivision is carried out depends on the problem being solved.

Constituent Parts:

* Object Boundaries

* Uniform Intensity or colour regions

* Lines, Points and Curves etc.

Image Segmentation is a very important part ofComputer Vision