computer vision and applications prof. trevor....
TRANSCRIPT
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6.891Computer Vision and Applications
Prof. Trevor. Darrell
Lecture 2:– Linear Filters and Convolution (review)– Fourier Transform (review)– Sampling and Aliasing (review)
Readings: F&P Chapter 7.1-7.6
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Recap: Cameras, lenses, and calibration
Last time:• Camera models• Projection equations• Calibration methods
Images are projections of the 3-D world onto a 2-D plane…
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Recap: pinhole/perspectivePinole camera model -
box with a small hole in it:
Perspective projection:
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Recap: Intrinsic parameters
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Recap: Combining extrinsic and intrinsic calibration parameters
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Other ways to write the same equation
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Recap:Camera calibration
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TodayReview of early visual processing
– Linear Filters and Convolution– Fourier Transform– Sampling and Aliasing
You should have been exposed to this material in previous courses; this lecture is just a (quick) review.
Administrivia: – sign-up sheet– introductions
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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.
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Local image data
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Modified image data
Some function
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Linear functions• Simplest: linear filtering.– Replace each pixel by a linear combination of its neighbors.
• The prescription for the linear combination is called the “convolution kernel”.
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Convolution
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Linear filtering (warm-up slide)
original0Pixel offset
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ient
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Linear filtering (warm-up slide)
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Linear filtering
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shift
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Linear filtering
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original
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Blurring
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Blur examples
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Blur examples
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Smoothing reduces noise• Generally expect pixels to “be like” their neighbours– surfaces turn slowly– relatively few reflectance
changes• Generally expect noise processes to be independent from pixel to pixel
• Implies that smoothing suppresses noise, for appropriate noise models
• Scale– the parameter in the
symmetric Gaussian– as this parameter goes up,
more pixels are involved in the average
– and the image gets more blurred
– and noise is more effectively suppressed
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The effects of smoothingEach row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.
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Linear filtering (warm-up slide)
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Linear filtering (no change)
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Linear filtering
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(remember blurring)
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Linear filtering
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Sharpening
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Sharpening example
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Sharpening
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Gradients and edges• Points of sharp change in an image are interesting:– change in reflectance– change in object– change in illumination– noise
• Sometimes called edge points
• General strategy– linear filters to estimate image gradient
– mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).
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Smoothing and Differentiation• Issue: noise
– smooth before differentiation– two convolutions to smooth, then differentiate?– actually, no - we can use a derivative of Gaussian filter
• because differentiation is convolution, and convolution is associative
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The scale of the smoothing filter affects derivative estimates, and alsothe semantics of the edges recovered.
1 pixel 3 pixels 7 pixels
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Oriented filtersGabor filters (Gaussian modulated harmonics) at differentscales and spatial frequencies
Top row shows anti-symmetric (or odd) filters, bottom row thesymmetric (or even) filters.
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Linear image transformations• In analyzing images, it’s often useful to
make a change of basis.
Fourier transform, orWavelet transform, orSteerable pyramid transform
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Self-inverting transforms
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An example of such a transform: the Fourier transform
discrete domain
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To get some sense of what basis elements look like, we plot a basis element --- or rather, its real part ---as a function of x,y for some fixed u, v. We get a function that is constant when (ux+vy) is constant. The magnitude of the vector (u, v) gives a frequency, and its direction gives an orientation. The function is a sinusoid with this frequency along the direction, and constant perpendicular to the direction.
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Here u and v are larger than in the previous slide.
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And larger still...
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Phase and Magnitude• Fourier transform of a real function is complex– difficult to plot, visualize– instead, we can think of the
phase and magnitude of the transform
• Phase is the phase of the complex transform
• Magnitude is the magnitude of the complex transform
• Curious fact– all natural images have
about the same magnitude transform
– hence, phase seems to matter, but magnitude largely doesn’t
• Demonstration– Take two pictures, swap the
phase transforms, compute the inverse - what does the result look like?
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This is the magnitude transform of the cheetah pic
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This is the phase transform of the cheetah pic
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This is the magnitude transform of the zebrapic
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This is the phase transform of the zebrapic
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Reconstruction with zebra phase, cheetah magnitude
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Reconstruction with cheetah phase, zebra magnitude
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Example image synthesis with fourier basis.
• 16 images
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2
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Fourier transform magnitude
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67
Masking out the fundamental and harmonics from periodic pillars
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68
Name as many functions as you can that retain that same
functional form in the transform domain
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69Forsyth&Ponce
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70
Oppenheim, Schafer and Buck,Discrete-time signal processing,Prentice Hall, 1999
Discrete-time, continuous frequency Fourier transform
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71
Discrete-time, continuous frequency Fourier transform pairs
Oppenheim, Schafer and Buck,Discrete-time signal processing,Prentice Hall, 1999
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73Bracewell, The Fourier Transform and its Applications, McGraw Hill 1978
Bracewell’s pictorial dictionary of Fourier transform pairs
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74Bracewell, The Fourier Transform and its Applications, McGraw Hill 1978
Bracewell’s pictorial dictionary of Fourier transform pairs
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75Bracewell, The Fourier Transform and its Applications, McGraw Hill 1978
Bracewell’s pictorial dictionary of Fourier transform pairs
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76Bracewell, The Fourier Transform and its Applications, McGraw Hill 1978
Bracewell’s pictorial dictionary of Fourier transform pairs
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77
Why is the Fourier domain particularly useful?
• It tells us the effect of linear convolutions.
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78
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79
hgf ��
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Fourier transform of convolution
Take DFT of both sides
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80
hgf ��� �hgDFTnmF ��],[
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Write the DFT and convolution explicitly
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81
hgf ��� �hgDFTnmF ��],[
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82
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83
hgf ��� �hgDFTnmF ��],[
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84
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85
Analysis of our simple filters
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86
Analysis of our simple filters
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87
Analysis of our simple filters
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88
Analysis of our simple filters
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89
Analysis of our simple filters
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90
Sampling and aliasing
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91
Sampling in 1D takes a continuous function and replaces it with a vector of values, consisting of the function’s values at a set of sample points. We’ll assume that these sample points are on a regular grid, and can place one at each integer for convenience.
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92
Sampling in 2D does the same thing, only in 2D. We’ll assume that these sample points are on a regular grid, and can place one at each integer point for convenience.
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93
A continuous model for a sampled function
• We want to be able to approximate integrals sensibly
• Leads to– the delta function– model on right
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94
The Fourier transform of a sampled signal
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95
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96
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97
Aliasing• Can’t shrink an image by taking every second pixel
• If we do, characteristic errors appear – In the next few slides– Typically, small phenomena look bigger; fast phenomena can look slower
– Common phenomenon• Wagon wheels rolling the wrong way in movies• Checkerboards misrepresented in ray tracing• Striped shirts look funny on colour television
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98
Resample the checkerboard by taking one sample at each circle. In the case of the top left board, new representation is reasonable. Top right also yields a reasonable representation. Bottom left is all black (dubious) and bottom right has checks that are too big.
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99
Constructing a pyramid by taking every second pixel leads to layers that badly misrepresent the top layer
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100
Smoothing as low-pass filtering• The message of the FT is that high frequencies lead to trouble with sampling.
• Solution: suppress high frequencies before sampling– multiply the FT of the signal with something that suppresses high frequencies
– or convolve with a low-pass filter
• A filter whose FT is a box is bad, because the filter kernel has infinite support
• Common solution: use a Gaussian– multiplying FT by Gaussian is equivalent to convolving image with Gaussian.
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101
Sampling without smoothing. Top row shows the images,sampled at every second pixel to get the next; bottom row shows the magnitude spectrum of these images.
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Sampling with smoothing. Top row shows the images. Weget the next image by smoothing the image with a Gaussian with sigma 1 pixel,then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images.
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Sampling with smoothing. Top row shows the images. Weget the next image by smoothing the image with a Gaussian with sigma 1.4 pixels,then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images.
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Thought problemAnalyze crossed
gratings…
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Thought problemAnalyze crossed
gratings…
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Thought problemAnalyze crossed
gratings…
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Thought problemAnalyze crossed
gratings…
Where does perceived near horizontal grating come from?
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What is a good representation for image analysis?
• Fourier transform domain tells you “what” (textural properties), but not “where”.
• Pixel domain representation tells you “where” (pixel location), but not “what”.
• Want an image representation that gives you a local description of image events—what is happening where.