CS 223-B Part A
Lect. : Advanced Features
Sebastian Thrun
Gary Bradski
http://robots.stanford.edu/cs223b/index.html
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Readings
This lecture is in 2 separate parts: “A” - Fourier, Gabor,
SIFT and “B” - Texture and other operators”. B is
optional due to time limitations. Good to look through
nevertheless.
Read:
• Computer Vision, Forsyth & Ponce
– Chapters 7 and (optional for texture) 9 … but do it lightly just for
the gist.
• David G. Lowe, “Distinctive Image Features from Scale-Invariant
Keypoints”, IJCV’04.
– Just read/take notes on basic flow of the algorithm.
• W. Freeman and E. Adelson, “The Design and Use of Steerable
Filters”, IEEE Trans. Patt. Anal. and Machine Intell., Vol. 13, No. 9.
– Read pages 1-15.
2
Left over questions…
• Calibration question – the optimization is based on gradient descent
iterations which depend on finding a good initial starting guess.
• How do we scale image derivatives?? Great question…
Brightness
– Images exist as brightness values over pixels. What are the units then
of a simple derivative operator like [-1 0 1]?
1-D image:
Pixels
Ix: [-1
0
1],
the spatial derivative,
has units 2*brightness/pixels
Sobel
1/8
operator
2/8
needs to
be normalized
1/8
0
-1/8
0
-2/8
0
-1/8
In the features lecture, we only wanted
to find edges (identification), but what if we had
instead wanted to make measurements?
In optical flow, we end up wanting to calculate
the velocity v which is found (in the optical flow
lecture) to be equal to It, the temporal derivative
(image difference) I(t+1) – I(t) which is in pixels
divided by the spatial derivative Ix in brightness/pixel
vx [pixels] = It / Ix [brightness/(brightness/pixel)]
Oops! Our derivative is a factor of 2 too great =>
NEED TO NORMALIZE: Ix: [-1/2 0 1/2].
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Good Features
beat
Good Algorithms
For tasks such as recognition, tracking,
and segmentation, experience shows:
• With the “right” features, all algorithms will
work well.
• With the “wrong” features, “good”
algorithms will work marginally better than
“bad/simple” algorithms, but it won’t work
well.
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Fourier Transform 1
• Foundational trick: represent signal/data in terms of an orthogonal
basis. For example, a vector v in 3 space can be represented as a
projection onto 3 orthonormal vectors:
• In the same way, a function can be represented as a point projected into
a space of (infinitely many) orthogonal functions. For Fourier
transforms, we project a function into a space of cos and sin
• Intuitively, how do we know this sin, cos basis is orthogonal?
– Sin or Cos periodically spend as much time above as below the axis. If the
frequency is mismatched, the functions will cancel each other out over
minus to plus infinity.
Formally, one could use
To prove
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* Eqns from Computer Vision IT412
Fourier Transform 2
Fourier transform is defined as continuous
Inverse transform gets rid of freq. components
In general, Fourier transform is complex
The Fourier Spectrum is then
The Phase is then
We often view the Power Spectrum
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Fourier Properties
Fourier Transform:
Is linear
Its spatial scale is inverse to frequency
Shift goes to phase change
Fourier Transform Symmetries are:
* Is the complex conjugate
Convolution Property
Note that scale property implies delta function goes to uniform
7
Fourier Discrete (DFT)
Animals and Machines live in a discrete world. To move the continuous
Fourier world to its discrete version, we sample
• => Multiply by infinite series of delta functions spaced  apart
• => Convolve with a uniform function inversely spaced 1/ 
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Fourier Discrete (DFT) 2
All real world signals are “band limited” That is, they don’t have infinite frequencies
nor infinite spatial extend. This is good, otherwise our discrete Fourier copies would
collide and alias together. But, what if we still sample too seldom? Even band limited
will eventually collide.
How do we keep the copies
apart? Sample at at least
twice the signal’s band limit
frequency => Niquist Criterion
1
2
where  is our sample interval.
c 
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2D DFT
Discrete Fourier Transform (DFT)
Inverse DFT
Optimally implemented on serial machines via the
“Fast Fourier Transform” (FFT), DFT is faster on
parallel machines.
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Fourier Examples
Raw Image
Fourier Amplitude
Sinusoid,
higher frequency
DC term + side lobes
wide spacing
Sinusoid,
lower frequency
DC term+ side lobes
close spacing
Sinusoid,
tilted
Titled spectrum
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Images from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory
More Fourier Examples
Fourier basis element
e i2  uxvy
example, real part
Fu,v(x,y)
Fu,v(x,y)=const. for
(ux+vy)=const.
Vector (u,v)
• Magnitude gives frequency
• Direction gives orientation.
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Slides from Marc Pollefeys, Comp 256 lecture 7
More Fourier Examples
Here u and v
are larger than
in the previous
slide.
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Slides from Marc Pollefeys, Comp 256 lecture 7
More Fourier Examples
And larger still...
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Slides from Marc Pollefeys, Comp 256 lecture 7
Fourier Filtering
Multiply by a filter in the
frequency domain =>
convolve with the fiter in
spatial domain.
Fourier
Amplitude
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Images from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory
Fourier Lens
Remember that Fourier transform takes delta functions to uniform, and uniform to delta?
Well, when focused at infinity (parallel rays to a point), so do lenses!
A lens approximates a Fourier transform processed at the speed of light
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Figures from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory
Phase Caries More Information
Raw
Images:
Magnitude
and
Phase:
Reconstruct
(inverse FFT)
mixing the
magnitude and
phase images
Phase “Wins”
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Phase Coherence for Feature Detection?
Note that the Fourier components for a square wave cohere (are in phase) at the
step junction Here, they must all pass through zero right at the step edge, and
achieve local maximums at the “corners”.
Phase coherence is maximal at “corner points” of triangle and trapezoid waves too
Triangle Wave
Trapezoid Wave
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Images: Peter Kovesi, Proc. VIIth Digital Image Computing: Techniques and Applications, Sun C., Talbot H., Ourselin S. and Adriaansen T. (Eds.), 10-12 Dec. 2003, Sydney
Phase Coherence for Feature Detection
Gist of the idea: Fourier transform yields a series of real and imaginary sinusoidal terms.
At any point x, the local Fourier components will each have an amplitude An(x) and a
phase angle φn(x). Vector addition of these terms yields an vector E(x) at the average
phase angle.
Morrone defined a measure that at absolute phase coherence will be 1 – everything
points in the same direction -- and for no phase coherence will be zero. Local maximums
indicate edges and corners, insensitive to contrast in the image.
In practice, these local components are calculated with Gabor filters at several
orientations that can yield oriented edges and corners.
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Images: Peter Kovesi, Proc. VIIth Digital Image Computing: Techniques and Applications, Sun C., Talbot H., Ourselin S. and Adriaansen T. (Eds.), 10-12 Dec. 2003, Sydney
Phase Coherence for Feature Detection
Comparison of phase vs. Harris Corner detector. Harris response varies by 2 or more
orders of magnitude…threshold? Phase can only vary between 0 and 1 and is
not sensitive to contrast or lighting.
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Images: Peter Kovesi, Proc. VIIth Digital Image Computing: Techniques and Applications, Sun C., Talbot H., Ourselin S. and Adriaansen T. (Eds.), 10-12 Dec. 2003, Sydney
Gabor filters and Jets
Global information is used for physical systems
identification.
– Impulse response of a centrifuge to identify resonance
points which indicate which spin frequencies to avoid.
Local information is used for physical signal analysis.
– In images, it is the relationship of details that matter, not
(usually) things like average brightness.
In 1946, Gabor suggested representing signals over
space and time called Information diagrams. He
showed that a Gaussian occupies minimal area in
such diagrams. Time and Frequency analysis are
the two extremes of such an analysis.
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Gabor filters and Jets
Gabor filters are formed by modulating a
complex sinusoid by a Gaussian function.
Gabor filters
became popular in
vision partly
because J.G
Daugman (1980, ‘88, ‘90) showed that the
receptive fields of most orientation receptive
neurons in the (cat’s) brain looked very much
like Gabor functions.
As with Gabor filters, the brain often makes use
of over complete, non-orthogonal functions.
J.G.Daugman, “Two dimensional spectral analysis of cortical receptive field profiles,” Vision Res., vol.20.pp.847-856.1980
J. Daugman, “Complete discrete 2-d gabor transforms by neural network for image analysis and compression,” IEEE Transactions on Acoustics, Speech, and
Signal Processing, vol. 36, no. 7, pp. 1169–1179, 1988.
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Daugman, J.G. (1990) An information–theoretic view of analogue representation in striate cortex, Computational Neuroscience, Ed. Schwartz, E. L.,
Cambridge, MA: MIT Press, 403–424.
Gabor filters and Jets
Rotated
Gaussian
Oriented Complex
Sinusoid
2D Gabor filter:
where  x2 and  x2 control the spatial extent of the filter,  is the
orientatio n of the filter and W is the radial frequency of the sinusoid.
Depending on one’s task (object ID, texture analysis, tracking,…) one must then
decide what size filters, in what orientations and what frequencies to use.
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Gabor filters and Jets
In practice, once the scales, orientation and radial frequencies are chosen
one usually sets up filters in quadrature (90o phase shift) pairs and just
empirically normalizes them such that the response is zero to a uniform
background.
Quadrature pairs, in practice the center point (p,q) is set to (0,0).
The magnitude response is then calculated as:
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Gabor filters and Jets
Von Der Malsburg organized Gabor filters at multiple scales and orientations
in a vector, or “Jet”
A graph of such Jets (“Elastic Graph Matching”) has proven to be a good “primitive”
for object recognition.
L. Wiskott, J-M. Fellous, N. Kuiger, C. Malsburg, “Face Recognition by Elastic Bunch Graph Matching”, IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol.19(7), July 1997, pp. 775-779.
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Image from Laurenz Wiskott, http://itb.biologie.hu-berlin.de/~wiskott/
Gabor filters and Jets Example
Gabor Filters used
Training and Recognition Flow Chart
BayesNet Facial Model Instead of an
Malsburg Elastic Graph Model (EGM).
Pose variable added
Results: BN Pose Face Rec. vs. EGM
Pose
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Gang Song, Tao Wang, Yimin Zhang, Wei Hu, Guangyou Xu, Gary Bradski, “Face Modeling and Recognition Using Bayesian Networks”, Submitted to
CVPR 2004
Scale
• 3D to 2D Perspective projections give widely
varying scale for the same object. Computer
vision needs to address scale.
• Gabor discussion above addressed image scale
via the sigma of the modulating Gaussians and
the frequency of the complex sinusoid.
• We can directly deal with scale by repeatedly
down-sampling the image to look for courser
and courser patterns. We call this scale space,
or Image Pyramids
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Image Pyramids
Commonly, we
down-sample
by 2 or sqrt(2).
Sqrt(2) obviously
calls for inter-pixel
interpolation
Gaussian
Pyramid
Laplacian
Pyramid
Gaussian
blur
For down-sample by
2, typical Gaussian
sigma is 1.4. For
Sqrt(2) sigma is
typically the
sqrt(1.4).
Full power 2 pyramid
only doubles the number
of pixels to process.
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Laplacian Pyramid
~ “Error Pyramid
Steerability
Bill Freeman, in his 1992 Thesis determined the necessary conditions for “Steerability”
-- the ability to synthesize a filter of any orientation from a linear combination of filters at
fixed orientations.
The simplest example of this is oriented first derivative of Gaussian filters, at 0o and 90o:
Steering Eqn:
0o
90o
Synthesized 30o
Filter Set:
Response:
Raw Image
Taken from:
W. Freeman, T. Adelson, “The Design
and Use of Sterrable Filters”, IEEE
Trans. Patt, Anal. and Machine Intell.,
vol 13, #9, pp 891-900, Sept 1991
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Steerability
Freeman showed that any band limited signal could form a steerable basis with as many
bases as it had non-zero Fourier coefs.
Important example is 2nd derivative of Gaussian
(~Laplacian):
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Taken from: W. Freeman, T. Adelson, “The Design and Use of Steerable Filters”, IEEE Trans. Patt, Anal. and Machine Intell., vol 13, #9, pp 891-900, Sept 1991
Steerable Pyramid
We may combine Steerability with Pyramids to get a Steerable Laplacian Pyramid as
shown below
Decomposition
Reconstruction
High pass, since
band pass in pyramid
low pass at bottom.
Low Pass
2 Level decomposition
of white circle example:
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Images from: http://www.cis.upenn.edu/~eero/steerpyr.html
Scale Invariant Feature Transform
• Idea is to find local features that stay the same
(as much as possible) under:
–
–
–
–
Scale change
2D rotation in the image x,y plane
3D rotation (affine variation)
Illumination
• Collections of such features can be used for
reliable
–
–
–
–
–
3D object recognition
User interface, toy interface
Robot localization, navigation and mapping
Digital image stitching, organization
3D scene understanding
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Scale Invariant Feature Transform
High Level Algorithm
1. Find peak responses (over scale) in
Laplacian pyramid.
2. Find response with sub-pixel accuracy.
3. Only keep “corner like” responses
4. Assign orientation
5. Create recognition signature
6. Solve affine parameters (~3D rot. changes)
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Scale Invariant Feature Transform
From Gaussian scale pyramid --
create Difference of Gaussian (DOG) images
And find maximum response over space and scale:
34
Images from: David G. Lowe, Object recognition from local scale-invariant features,
International Conference on Computer Vision, Corfu, Greece (September 1999), pp. 1150-1157
Scale Invariant Feature Transform
Use the gradients to only keep “corner like” peaks in manner similar to
Harris corner detector:
At each peak location and scale, use gradients to form slip tolerant
orientation histogram recognition keys:
35
Images from: David G. Lowe, Object recognition from local scale-invariant features,
International Conference on Computer Vision, Corfu, Greece (September 1999), pp. 1150-1157
At the location and scale of peak found, find the gradient orientation:
Scale Invariant Feature Transform
To account for out of image plane (3D) rotation, solve for affine distortion parameters:
For features found, set up system of equations
Which take the form of
. Over determined (least sqrs) solution is then:
Eqns from: David G. Lowe, Object recognition from local scale-invariant features,
International Conference on Computer Vision, Corfu, Greece (September 1999), pp. 1150-1157
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Scale Invariant Feature Transform
Objects may then be found under occlusion and 3D rotation:
37
Images from: David Lowe, Object Recognition from Local Scale-Invariant Features Proc. of
the International Conference on Computer Vision,Corfu (Sept. 1999)
Recognition example. Learned models of SIFT features, and got object outline from
background subtraction:
Scale Invariant Feature Transform
Image stitching example. Attach images together from keypoints:
Finding similar images in a roll and stitching:
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Images from: M. Brown and D. G. Lowe. Recognising Panoramas. In Proceedings of the
9th International Conference on Computer Vision (ICCV2003)
Solving the homography:
Scale Invariant Feature Transform
Localizing Example:
Given key images, find and trigger on them1:
1) David G. Lowe, Distinctive Image Features from Scale-Invariant Keypoints,
Submitted to International Journal of Computer Vision. Version date: June 2003
Find different views of same scene in video2:
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2) Josef Sivic and Andrew Zisserman, Video Google:
A Text Retrieval Approach to Object Matching in Videos,
ICCV 2003
Log-Polar Transform
Go from Euclidian (x,y) to log-polar space log(rei) => (log r, ) space. Log-polar
transform is always done relative to a chosen center point (xc,yc):

y
(xc,yc)
r


x
y
log r
Log-Polar
(xc,yc)
r

x
log r
Log-Polar
Rotation and scale are converted to shifts along the  or log r axis. Shifting back to a canonical
location gives rotation and scale invariance. If used on a Fourier image (translation invariant), we get
rotation, scale and translation invariance (called Fourier-Mellin transform)1.
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1)
Images, further advances in: George Wolberg, Siavash Zokai, ROBUST
IMAGE REGISTRATION USING LOG-POLAR TRANSFORM, ICIP 2000

Bilateral Filtering
We want smoothing that preserves edges.
Typically done via P. Perona and J. Malik anisotropic diffusion. More
clever is the Tomasi and Manduchi* approximation:
• Rather than just convolve with a Gaussian in space
• the convolution weights use a Gaussian in space together with a
Gaussian in gray level values.

=
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* C. Tomasi and R. Manduchi, "Bilateral Filtering for Gray and Color Images", Proceedings of the 1998
IEEE International Conference on Computer Vision, Bombay, India
But Bio-Vision is more dynamic
• Artifacts of competitive edge/diffusion process:
Neon Color Spreading Illusion
Best explanation is Grossberg and Mingolla – edge detectors need to be “shut off”, performed by competitive
inhibition. When weaker edges meet stronger, the weaker edge is suppressed breaking the dikes that hold back
the diffusion process. When the edges are disconnected, the illusion goes away or is diminished below:
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Grossberg, S., & Mingolla, E. (1985). Neural Dynamics of Form Perception: Boundary Completion. Psychol. Rev., 92, 173--211.
Local vs. Global
Still, vision is a stranger thing than simple processing:
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Local vs. Global
Still, vision is a stranger thing than simple processing:
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Computer vision often misses the
fact that vision is an active sense
These lines are straight
Nothing is moving here
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