Stanford CS223B Computer Vision, Winter 2006 Lecture 2 Lenses, Filters, Features

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Stanford CS223B Computer Vision, Winter
2006Lecture 2 Lenses, Filters, Features

• Professor Sebastian Thrun
• CAs Dan Maynes-Aminzade and Mitul Saha
• with slides by D Forsyth, D. Lowe, M.
  Polleyfeys, C. Rasmussen, G. Loy, D. Jacobs, J.
  Rehg, A, Hanson, G. Bradski,

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Todays Goals

• Thin Lens
• Aberrations
• Features 101
• Linear Filters and Edge Detection

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Pinhole Camera (last Wednesday)
-- Brunelleschi, XVth Century
Marc Pollefeys comp256, Lect 2
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Snells Law
Snells law n1 sin a1 n2 sin a2
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Thin Lens Definition
Spherical lense surface Parallel rays are
refracted to single point
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Thin Lens Projection
optical axis
Image plane
z
f
Spherical lense surface Parallel rays are
refracted to single point
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Thin Lens Projection
optical axis
Image plane
z
f
f
Spherical lense surface Parallel rays are
refracted to single point
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Thin Lens Properties

1. Any ray entering a thin lens parallel to the
   optical axis must go through the focus on other
   side
2. Any ray entering through the focus on one side
   will be parallel to the optical axis on the other
   side

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Thin Lens Model
Q
P
O
Fr
Fl
p
R
Z
f
f
z
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Transformation
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A Transformation
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The Thin Lens Law
Q
P
O
Fr
Fl
p
R
Z
f
f
z
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The Thin Lens Law
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Limits of the Thin Lens Model
3 assumptions

• all rays from a point are focused onto 1 image
  point
• Remember thin lens small angle assumption

2. all image points in a single plane
3. magnification is constant
Deviations from this ideal are aberrations
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Todays Goals

• Thin Lens
• Aberrations
• Features 101
• Linear Filters and Edge Detection

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Aberrations
2 types
geometrical geometry of the lense, small for
paraxial rays
chromatic refractive index function of
wavelength
Marc Pollefeys
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Geometrical Aberrations

• spherical aberration
• astigmatism
• distortion
• coma

aberrations are reduced by combining lenses
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Astigmatism

• Different focal length for inclined rays

Marc Pollefeys
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Astigmatism

• Different focal length for inclined rays

Marc Pollefeys
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Spherical Aberration
rays parallel to the axis do not converge outer
portions of the lens yield smaller focal lenghts
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Distortion

• magnification/focal length different
• for different angles of inclination

pincushion (tele-photo)
barrel (wide-angle)
Can be corrected! (if parameters are know)
Marc Pollefeys
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Coma

• point off the axis depicted as comet shaped blob

Marc Pollefeys
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Chromatic Aberration
rays of different wavelengths focused in
different planes cannot be removed
completely
Marc Pollefeys
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Vignetting
Effect Darkens pixels near the image boundary
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CCD vs. CMOS

• Mature technology
• Specific technology
• High production cost
• High power consumption
• Higher fill rate
• Blooming
• Sequential readout

• Recent technology
• Standard IC technology
• Cheap
• Low power
• Less sensitive
• Per pixel amplification
• Random pixel access
• Smart pixels
• On chip integration with other components

Marc Pollefeys
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Todays Goals

• Thin Lens
• Aberrations
• Features 101
• Linear Filters and Edge Detection

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Todays Question

• What is a feature?
• What is an image filter?
• How can we find corners?
• How can we find edges?
• (How can we find cars in images?)

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What is a Feature?

• Local, meaningful, detectable parts of the image

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Features in Computer Vision

• What is a feature?
• Location of sudden change
• Why use features?
• Information content high
• Invariant to change of view point, illumination
• Reduces computational burden

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(One Type of) Computer Vision
Feature 1 Feature 2 Feature N
Computer Vision Algorithm
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Where Features Are Used

• Calibration
• Image Segmentation
• Correspondence in multiple images (stereo,
  structure from motion)
• Object detection, classification

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What Makes For Good Features?

• Invariance
• View point (scale, orientation, translation)
• Lighting condition
• Object deformations
• Partial occlusion
• Other Characteristics
• Uniqueness
• Sufficiently many
• Tuned to the task

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Todays Goals

• Features 101
• Linear Filters and Edge Detection
• Canny Edge Detector

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What Causes an Edge?

• Depth discontinuity
• Surface orientation discontinuity
• Reflectance discontinuity (i.e., change in
  surface material properties)
• Illumination discontinuity (e.g., shadow)

Slide credit Christopher Rasmussen
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Quiz How Can We Find Edges?
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Edge Finding 101

• im imread('bridge.jpg')
• image(im)
• figure(2)
• bw double(rgb2gray(im))
• image(bw)
• gradkernel -1 1
• dx abs(conv2(bw, gradkernel, 'same'))
• image(dx)
• colorbar colormap gray
• dx,dy gradient(bw)
• gradmag sqrt(dx.2 dy.2)
• image(gradmag)

matlab
colorbar colormap(gray(255)) colormap(default)
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Edge Finding 101

• Example of a linear Filter

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Todays Goals

• Thin Lens
• Aberrations
• Features 101
• Linear Filters and Edge Detection

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What is Image Filtering?

• Modify the pixels in an image based on some
  function of a local neighborhood of the pixels

10 5 3
4 5 1
1 1 7

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Some function
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Linear Filtering

• Linear case is simplest and most useful
• Replace each pixel with a linear combination of
  its neighbors.
• The prescription for the linear combination is
  called the convolution kernel.

10 5 3
4 5 1
1 1 7

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0 0 0
0 0.5 0
0 1.0 0.5
kernel
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Linear Filter Convolution
f (i,j)
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Linear Filter Convolution
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Filtering Examples
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Filtering Examples
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Filtering Examples
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Image Smoothing With Gaussian

• figure(3)
• sigma 3
• width 3 sigma
• support -width width
• gauss2D exp( - (support / sigma).2 / 2)
• gauss2D gauss2D / sum(gauss2D)
• smooth conv2(conv2(bw, gauss2D, 'same'),
  gauss2D', 'same')
• image(smooth)
• colormap(gray(255))
• gauss3D gauss2D' gauss2D
• tic smooth conv2(bw,gauss3D, 'same') toc

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Smoothing With Gaussian
Slide credit Marc Pollefeys
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Smoothing Reduces Noise
The effects of smoothing Each row shows
smoothing with gaussians of different width each
column shows different realizations of an image
of gaussian noise.
Slide credit Marc Pollefeys
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Example of Blurring
Image
Blurred Image
-

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Edge Detection With Smoothed Images

• figure(4)
• dx,dy gradient(smooth)
• gradmag sqrt(dx.2 dy.2)
• gmax max(max(gradmag))
• imshow(gradmag)
• colormap(gray(gmax))

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Scale

• Increased smoothing
• Eliminates noise edges.
• Makes edges smoother and thicker.
• Removes fine detail.

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The Edge Normal
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Displaying the Edge Normal

• figure(5)
• hold on
• image(smooth)
• colormap(gray(255))
• m,n size(gradmag)
• edges (gradmag gt 0.3 gmax)
• inds find(edges)
• posx,posy meshgrid(1n,1m)
  posx2posx(inds) posy2posy(inds)
• gm2 gradmag(inds)
• sintheta dx(inds) ./ gm2
• costheta - dy(inds) ./ gm2
• quiver(posx2,posy2, gm2 . sintheta / 10, -gm2 .
  costheta / 10,0)
• hold off

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Separable Kernels
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Combining Kernels / Convolutions
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Effect of Smoothing Radius
1 pixel
3 pixels
7 pixels
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Roberts Cross Operator

S
or
I(x, y) - I(x1, y1) I(x, y1) - I(x1,
y)
S
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Sobel Operator
-1 -2 -1 0 0 0 1 2 1
-1 0 1 -2 0 2 -1 0 1
S1
S2
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The Sobel Kernel, Explained
Sobel kernel is separable!
Averaging done parallel to edge
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Sobel Edge Detector

• figure(6)
• edge(bw, 'sobel')

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Robinson Compass Masks
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Claim Your Own Kernel!
Frei Chen
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Comparison (by Allan Hanson)

• Analysis based on a step edge inclined at an
  angle q (relative to y-axis) through center of
  window.
• Robinson/Sobel true edge contrast less than 1.6
  different from that computed by the operator.
• Error in edge direction
• Robinson/Sobel less than 1.5 degrees error
• Prewitt less than 7.5 degrees error
• Summary
• Typically, 3 x 3 gradient operators perform
  better than 2 x 2.
• Prewitt2 and Sobel perform better than any of the
  other 3x3 gradient estimation operators.
• In low signal to noise ratio situations, gradient
  estimation operators of size larger than 3 x 3
  have improved performance.
• In large masks, weighting by distance from the
  central pixel is beneficial.