Image Formation. Image Formation occurs when a sensor registers radiation. Mathematical models of image formation:
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1 Image Formation Image Formation occurs when a sensor registers radiation. Mathematical models of image formation: 1. Image function model 2. Geometrical model 3. Radiometrical model 4. Color model 5. Spatial Frequency model 6. Digitizing model E.G.M. Petrakis Image Formation 1
2 E.G.M. Petrakis Image Formation 2
3 1. Image Function Mathematical representation of a (digital) image. Relates to digitization: conversion from continuous signal to discrete function Black & White image r f x = f x, y = r r r f x = f x, f x, f Color image ( ) ( ) d ( ) { ( ) ( ) ( x) } Multispectral image f = (f 1, f 2,, f n ) R G B r E.G.M. Petrakis Image Formation 3
4 2. Geometrical Model Determines where in the image plane the projection of a point will be located. the projected image is inverted (x,y,z) is projected on (x,y ) f: focal length E.G.M. Petrakis Image Formation 4
5 Avoid inversion by assuming that the image plane is in front of the center of projection done automatically by cameras or by the human brain Apply Euclidean geometry x = x f /z and y = y f/z depth z is lost! E.G.M. Petrakis Image Formation 5
6 Depth Computation Acquire a pair of images of the same scene using two cameras (or two images by a moving camera) Two identical cameras separated in the x direction by a baseline distance b The image planes are coplanar E.G.M. Petrakis Image Formation 6
7 A point is projected at two different positions on the two camera planes their displacement is called disparity z = ' bf x x l r ' E.G.M. Petrakis Image Formation 7
8 In certain systems (human eyes) the optical axes of the cameras intersect in space for any angle there is a surface in space corresponding to d = 0. the disparities may be d = 0, d < 0 or d > 0. E.G.M. Petrakis Image Formation 8
9 Epipolar constraint: even if the cameras are in arbitrary positions and orientation the projections lie on the intersection of camera - epipolar planes E.G.M. Petrakis Image Formation 9
10 Correspondence problem: detection of conjugate pairs in stereo images: for each point in the left image find the corresponding point in the right image measure the similarity between points the points to be matched should be distinctly different from their surrounding points both region and edge features can be used in stereo matching the epipolar constraint limits the search space for finding conjugate pairs. E.G.M. Petrakis Image Formation 10
11 3. Radiometrical Model Measures the intensity of the reflected light at a point (x,y ) of the image plane it is determined by the physics of imaging The proper term of image intensity is image irradiance but intensity, brightness, gray value are also used Image irradiance is the power per unit area of radiant energy falling into the image plane Irradiance is incoming energy Radiance is outgoing energy (from reflecting surface) E.G.M. Petrakis Image Formation 11
12 The irradiance at point (x,y ) of the image plane depends on the amount of energy radiated by points (x,y,z) in the scene Two factors determine the radiance emitted by a patch of scene surface: 1. The illumination falling on a surface (depends in its position relative to the distribution of light) 2. The fraction of incident illumination reflected by the surface (depends on surface properties e.g., dull, flat, mirror-like etc.) The reflectance of a surface is given by the Bi-directional Reflectance Distribution Function (BRDF) E.G.M. Petrakis Image Formation 12
13 Scene Radiance Φ: light energy flux Α: area of source L 2 d Φ da cosθ dω m = 2 θ: angle (surface normal & direction of emission) dω: incremental solid angle Wallts steradiar E.G.M. Petrakis Image Formation 13
14 Image irradiance E 1 D = cos 4 α 4 f p 2 L Ideally, an imaging device should be calibrated so that the variation in sensitivity as a function of a is removed. E.G.M. Petrakis Image Formation 14
15 4. Color Model Visible light is an electromagnetic wave in the 400nm 700nm range The light we see is combination of many wavelengths spectra: the profile below E.G.M. Petrakis Image Formation 15
16 Each neuron on the retina is either a rod or a cone (rods are not sensitive to color). Cons come in 3 types: red, green, blue each responds differently to various frequencies of light. Spectral response functions of cones: E.G.M. Petrakis Image Formation 16
17 The color signal to the brain is obtained by adding the responses of the 3 cones the color signal consists of 3 numbers. R,G, B sensors filter the scene radiance E(λ). each sensor has a different spectral response S(λ). E.G.M. Petrakis Image Formation 17
18 CIE primaries: this figure shows the amounts of the 3 primaries needed to match all the wavelengths of the visible spectrum the negative value indicates that some colors cannot be exactly produced by adding up the 3 primaries. E.G.M. Petrakis Image Formation 18
19 CIE XYZ Based on the CIE primaries negative values are transformed to positive chromaticity values x=x/(x+y+z), y=y/(x+y+z), z=z/(x+y+z) x+y+z=1: two values represent all colors E.G.M. Petrakis Image Formation 19
20 Chromaticity Diagram Visible colors: points in the bell Non-visible colors outside the bell Primaries at edges A white point at centre Saturated colors along the radii from edge E.G.M. Petrakis Image Formation 20
21 Color Representation Several methods Hardware-oriented: defined to properties of devices (TV, printers) that reproduce colors (RGB, CMY etc.) User-Oriented: based on human perception of colors (HIS, L*u*v etc.) Colorimetric (CIE), Physiological (CIE XYZ, RGB), Psychological (HIS, L*u*v etc) E.G.M. Petrakis Image Formation 21
22 RGB Color Space The most popular hardware oriented scheme The colors form a unit cube r = R/(R+G+B) g = G/(R+G+B) b = B/(R+G+B) RGB is good for acquisition and display but not for the perception of colors E.G.M. Petrakis Image Formation 22
23 CMY Color Space Cyan, Magenta, Yellow are complements of Red, Green, Blue Obtained by subtracting light from white For color printing Conversion from RGB to CMY R = 1 C G = 1 M B = 1 Y E.G.M. Petrakis Image Formation 23
24 Munsell Color Space Represented in cylindrical coordinates based on Brightness: vertical axis Hue: angular displacement Saturation: cylindrical radius E.G.M. Petrakis Image Formation 24
25 Color Definitions Brightness: intensity of color, average intensity over all wavelengths Hue:is roughly proportional to the average wavelength of the color percept Saturation: amount of white light in color highly saturated colors have no white deep red has S=1, pinks have S 0 P = SH+(1 S)W: Think of a color P as an additive mixture W and H where S controls the proportions of W and H E.G.M. Petrakis Image Formation 25
26 HIS Color Space Represented as a double cone Intensity: the main axis (white at the top, black at the bottom) Hue: angle around the axis Saturation: distance from axis Saturated colors on maximal circles E.G.M. Petrakis Image Formation 26
27 HSV Color Space Similar to HIS Value Hue H = 1 V = ( R+ G + 3 cos 2 B) 1 2 R G B ( R G ) 2 + ( R B )( G B ) Saturation S 3 = 1 min( R, G, B) R + G+ B H = undefined for S = 0 H = 360 H if B/V > G/V E.G.M. Petrakis Image Formation 27
28 Color Models for Video (YIQ) YIQ is used for color TV broadcasting Y I Q = R G B E.G.M. Petrakis Image Formation 28
29 E.G.M. Petrakis Image Formation Spatial Frequency Model Describe spatial variations in the frequency domain of the Fourier Transform: ( ) ( ) = = + = exp, 1, M m N n N nv M mu j n m f N M v u F π ( ) ( ) = = + = exp,, M u N v N nu M mu j v u F n m f π 1 1, ,0 0 M v M u N n M m
30 f(m,n): linear combination of periodic waveforms exp{j2π(ux + vy)} F(u,v): weight factor of frequency u,v High u,v image detail (edges, points etc.) Low u,v no detail, smooth areas e E.G.M. Petrakis Image Formation 30
31 Fourier Transform Pairs E.G.M. Petrakis Image Formation 31
32 Fourier Transform Pairs (2) E.G.M. Petrakis Image Formation 32
33 Sinc(x,y) E.G.M. Petrakis Image Formation 33
34 (a) Original image (b) Edge Enhanced image (c) Smoothed image Fourier Transform of (a) Fourier Transform of (b) Fourier Transform of (c) E.G.M. Petrakis Image Formation 34
35 Convolution Convolution of f and g: h ( x, y ) = f g = f ( a, b) g ( x a, y b) dadb < a, b< Invert g by 180 0, pass pass g over f and compute h on each point of f Theorem: { } { } { } F f g = F f F g E.G.M. Petrakis Image Formation 35
36 6. Digitizing Model Digitization: Conversion of continuous signals to discrete. f(x,y) f(m,n), 0<= m <= M-1,0<=n<=N-1 f(m,n) = k (intensity value) 0 <= k <= K-1. f(m,n): samples taken at equal intervals. Perfect sampling: It is possible to reconstruct f(x,y) from f(m,n) K,m,n must be large enough E.G.M. Petrakis Image Formation 36
37 Image Sampling Multiply f(x,y) by comb ( x, y) = δ ( x m, y n) m= n= Sampling function E.G.M. Petrakis Image Formation 37
38 One way to reconstruct an image from its samples f(kt) would be to interpolate suitably between the samples Consider one dimensional signals f ( x) = f ( k T ) g( x kt ) k = in the frequency domain ( ) G ω ( ) 2πn F ω = F ω T n= T g(x-kt) interpolation function T: sampling period E.G.M. Petrakis Image Formation 38
39 F(u) for 1D band limited function Non-overlapping copies of F(u) f c : max frequency Overlapping copies of F(u) E.G.M. Petrakis Image Formation 39
40 F(u,v) 2D band-limited function Non-overlapping copies of F(u,v) E.G.M. Petrakis Image Formation 40
41 Select G(w) that isolates F(w) from its samples G ( ω ) = T, ω < 2πf c 0, otherwise Whittaker-Kotelnikov-Shannon theorem: f(x) can be reconstructed if the time distance between the samples is at least 1/2f 2f c : sampling rate If the signal is not band-limited we have aliasing (interference from high frequencies) Smooth before sampling E.G.M. Petrakis Image Formation 41
42 E.G.M. Petrakis Image Formation 42
43 E.G.M. Petrakis Image Formation 43
44 K E.G.M. Petrakis Image Formation 44
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