February 13, 2014 Sam Siewert

Computer and Machine Vision

Lecture Week 5 Part-1

Outline of Week 5 Background on 2D and 3D Geometric Transformations – Chapter 2 of CV Fundamentals of 2D Image Transformations and Convolutions (CV Chapters 3 & 4) Introduction to More Advanced 2D Transformations – Bottom Up C without OpenCV Simple 2D Center-of-Mass Tracking Introduction to 2D Pattern Matching – Next Time

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Backgrounder

2D and 3D Geometric Math Overview (Chapter 2 – CV)

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Understand Geometric Transform Types

Figure 2.4 – 2D Planar Transformations

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Primitive 2D Transformation Properties

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Primitive 3D Transformation Properties

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Translation and Rotation Studied in Physics (and Linear Systems) Vectors, Rotation Matrices – 2D & 3D Simple C++ Starter – vectmat.zip Ground-up OpenCV Tutorial - http://www.cs.iit.edu/~agam/cs512/lect-notes/opencv-intro/ https://www.khanacademy.org/math/linear-algebra Sam Siewert 7

OpenCV Linear Algebra

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http://www.cs.iit.edu/~agam/cs512/lect-notes/opencv-intro/

OpenCV Linear Algebra

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http://www.cs.iit.edu/~agam/cs512/lect-notes/opencv-intro/

Based on Studies of Perspective by Artists and Physics of Light

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Silk Screen

Illumination

Observer Viewpoint

Rays

Painted/Traced Edge

Orthographic Parallel Projection Cartography – Orthographic – Mercator

(Cylindrical)

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http://en.wikipedia.org/wiki/File:Axonometric_projection.svg

http://en.wikipedia.org/wiki/File:Cylindrical_Projection_basics2.svg

Vanishing Point Linear Perspective (1 to 3 Vanishing Points) Curvilinear (4 or 5 Vanishing Points)

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Single Vanishing Point Double Vanishing Point

http://en.wikipedia.org/wiki/File:Zentralperspektive.png http://en.wikipedia.org/wiki/File:TwoPointPerspective.png

http://en.wikipedia.org/wiki/File:Frankfurt_Airport_tunnel.JPG

Perspective Concept

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viewpoint

Object

Render Plane Intersections

http://en.wikipedia.org/wiki/File:Perspectiva-1.svg

Perspective Projection Normal Projection Used in Ray Tracing – Define Extents of Render Plane – l=left, r=right, t=top, b=bottom – Viewpoint “e” is shown on Render Plane – Move e out of Plane to Location of Camera/Viewer

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Fundamentals of Computer Graphics, 3rd Edition - page 75

2D Image Transformations

Basic PSF Convolutions

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Pixel Neighborhood Convention {P} – Originally Sampled Frame or Sub-frame of Pixels {Q} – First Transformation of {P} – Negative Image – {Q}=Saturation – {P} – Where Saturation for 8-bit Gray-level is 255

– Difference Image – {R} = {Q} – {P} is the time of sample of {Q} is greater than {P}

– A threshold (Part 2) – From Histogram

P0 is the Pixel of Interest in a Neighborhood – So, for all pixels in {P}, if Q0=P1, then {Q} is the same image shifted

one pixel to the left – If {Q}={P}+beta, modifies brightness – If {Q}={P} x gamma, modifies contrast – If {Q}={P} x gamma + beta, modifies both contrast and brightness – This Can Define a PSF for an Image Convolution – e.g. Sharpen

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P4 P3 P2P5 P0 P1P6 P7 P8

Common PSF Convolution See www.dspguide.com/CH24.PDF

– For all pixels, do {Q0 = P0*h0+P1*h1+P2*h2+P3*h3+P4*h4+P5*h5+P6*h6+P7*h7+P8*h8; }

Convolution of causes no change to image

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0 0 00 1 00 0 0 http://www.dspguide.com/ - Chapter 24

Median Filter Concept

For all Pixels in Image do – minPixel = min(P1, P2, P3, P4, P5, P6, P7, P8) – maxPixel = max(P1, P2, P3, P4, P5, P6, P7, P8)

– If(P0 < minPixel) Q0 = minPixel – Else If (P0 > maxPixel) Q0 = maxPixel – Else Q0 = P0

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Standard Median Filter Algorithm

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// zero histogram for 0…255 for (i=0; i<=255; i++) hist[i]=0; for all pixels in Image P do { // All pixels in the kernel note value for (m=0; m<=8; m++) hist[P[m]]++; i=0; sum=0; while (sum < 5) { sum = sum + hist[i]; i = i + 1; } Q0 = i – 1; for(m=0; m<=8; m++) hist[P[m]]=0; }

Q4 Q3 Q2Q5 Q0 Q1Q6 Q7 Q8

Median Filter Algorithm – Nearest Neighbor (p. 108-109)

The Median Filter Uses Binning in range of 0 to Saturation of PGM (0 to 255) – Obviates need to Sort the Value of Neighbors in Kernel

3x3 – Q0 is the 5th largest neighbor in the Kernel – Must Zero Histogram of size 256 for Each Pixel – Loop through all neighbors and bin them – Each P[m] neighbor’s value is the index into the histogram

of counts for pixels at that value

This filter provides noise suppression, but softens edges Truncated Median Filter Removes Noise and Sharpens

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