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CS 223-B Part ACS 223-B Part A Lect. : Advanced FeaturesLect. : Advanced Features

Sebastian Thrun Sebastian Thrun

Gary BradskiGary 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.

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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…

– 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

Brig

htn

ess

Ix: [-1 0 1], the spatial derivative, has units 2*brightness/pixels

In the features lecture, we only wantedto find edges (identification), but what if we hadinstead wanted to make measurements?

In optical flow, we end up wanting to calculatethe 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 pixelsdivided 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].

1/8

2/8

1/8

-1/8

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Sobel operatorneeds tobe normalized

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Good Features beat

Good AlgorithmsFor 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

* Eqns from Computer Vision IT412

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Fourier Transform 2Fourier 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 PropertiesFourier Transform:

Is linear

Its spatial scale is inverse to frequency

Shift goes to phase change

Fourier Transform Symmetries are:

Convolution Property

Note that scale property implies delta function goes to uniform

* Is the complex conjugate

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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

Fourier Discrete (DFT)

/1

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Fourier Discrete (DFT) 2All real world signals are “band limited” That is, they don’t have infinite frequenciesnor infinite spatial extend. This is good, otherwise our discrete Fourier copies wouldcollide and alias together. But, what if we still sample too seldom? Even band limitedwill eventually collide.

How do we keep the copiesapart? Sample at at least twice the signal’s band limitfrequency => Niquist Criterion

interval. sampleour is where2

1

c

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2D DFTDiscrete 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 ExamplesRaw Image Fourier Amplitude

Sinusoid,higher frequency

Sinusoid,lower frequency

Sinusoid,tilted

DC term + side lobeswide spacing

DC term+ side lobesclose spacing

Titled spectrum

Images from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory

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Fourier basis element

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.

e i2 uxvy

Slides from Marc Pollefeys, Comp 256 lecture 7

More Fourier Examples

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Here u and v are larger than in the previous slide.

Slides from Marc Pollefeys, Comp 256 lecture 7

More Fourier Examples

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And larger still...

Slides from Marc Pollefeys, Comp 256 lecture 7

More Fourier Examples

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Fourier Filtering

Images from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory

FourierAmplitude

Multiply by a filter in thefrequency domain => convolve with the fiter inspatial domain.

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Fourier LensRemember that Fourier transform takes delta functions to uniform, and uniform to delta?

Figures from Steve Lehar http://cns-alumni.bu.edu/~slehar An Intuitive Explanation of Fourier Theory

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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Phase Caries More Information

MagnitudeandPhase:

RawImages:

Reconstruct(inverse FFT)mixing themagnitude andphase images

Phase “Wins”

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Phase Coherence for Feature Detection?

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

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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Morrone defined a measure that at absolute phase coherence will be 1 – everythingpoints in the same direction -- and for no phase coherence will be zero. Local maximumsindicate edges and corners, insensitive to contrast in the image.

In practice, these local components are calculated with Gabor filters at severalorientations that can yield oriented edges and corners.

Phase Coherence for Feature DetectionGist 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.

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

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Phase Coherence for Feature Detection

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

Comparison of phase vs. Harris Corner detector. Harris response varies by 2 or moreorders of magnitude…threshold? Phase can only vary between 0 and 1 and isnot sensitive to contrast or lighting.

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Gabor filters and JetsGlobal 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 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.

Gabor filters and Jets

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.

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.

J.G.Daugman, “Two dimensional spectral analysis of cortical receptive field profiles,” Vision Res., vol.20.pp.847-856.1980

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Gabor filters and Jets

2D Gabor filter:

Rotated Gaussian

Oriented ComplexSinusoid

sinusoid. theoffrequency radial theis andfilter theofn orientatio

theis filter, theofextent spatial thecontrol and where 2x

2x

W

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 JetsVon Der Malsburg organized Gabor filters at multiple scales and orientationsin a vector, or “Jet”

A graph of such Jets (“Elastic Graph Matching”) has proven to be a good “primitive” for object recognition.

Image from Laurenz Wiskott, http://itb.biologie.hu-berlin.de/~wiskott/

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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Gabor filters and Jets Example

Gang Song, Tao Wang, Yimin Zhang, Wei Hu, Guangyou Xu, Gary Bradski, “Face Modeling and Recognition Using Bayesian Networks”, Submitted to CVPR 2004

Gabor Filters used

BayesNet Facial Model Instead of anMalsburg Elastic Graph Model (EGM).

Pose

Pose variable added

Training and Recognition Flow Chart

Results: BN Pose Face Rec. vs. EGM

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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

Gaussianblur

GaussianPyramid

LaplacianPyramid

Commonly, wedown-sampleby 2 or sqrt(2).Sqrt(2) obviouslycalls for inter-pixelinterpolation

Laplacian Pyramid~ “Error Pyramid

For down-sample by2, typical Gaussiansigma is 1.4. For Sqrt(2) sigma istypically the sqrt(1.4).

Full power 2 pyramidonly doubles the numberof pixels to process.

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SteerabilityBill 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:

Filter Set:0o 90o Synthesized 30o

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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SteerabilityFreeman showed that any band limited signal could form a steerable basis with as manybases as it had non-zero Fourier coefs.

Important example is 2nd derivative of Gaussian (~Laplacian):

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

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Steerable PyramidWe may combine Steerability with Pyramids to get a Steerable Laplacian Pyramid as shown below

Images from: http://www.cis.upenn.edu/~eero/steerpyr.html

High pass, sinceband pass in pyramidlow pass at bottom.

Low Pass

Orie

nted

Decomposition Reconstruction

2 Level decompositionof white circle example:

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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 Algorithm1. 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 TransformFrom Gaussian scale pyramid -- create Difference of Gaussian (DOG) images

And find maximum response over space and scale:

Images 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 TransformAt the location and scale of peak found, find the gradient orientation:

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:

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Scale Invariant Feature TransformTo account for out of image plane (3D) rotation, solve for affine distortion parameters:

Eqns from: David G. Lowe, Object recognition from local scale-invariant features, International Conference on Computer Vision, Corfu, Greece (September 1999), pp. 1150-1157

For features found, set up system of equations

Which take the form of . Over determined (least sqrs) solution is then:

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Scale Invariant Feature TransformRecognition example. Learned models of SIFT features, and got object outline frombackground subtraction:

Objects may then be found under occlusion and 3D rotation:

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Scale Invariant Feature TransformImage stitching example. Attach images together from keypoints:

Solving the homography: Finding similar images in a roll and stitching:

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Scale Invariant Feature TransformLocalizing Example:

Given key images, find and trigger on them1: Find different views of same scene in video2:

2) Josef Sivic and Andrew Zisserman, Video Google: A Text Retrieval Approach to Object Matching in Videos, ICCV 2003

1) David G. Lowe, Distinctive Image Features from Scale-Invariant Keypoints, Submitted to International Journal of Computer Vision. Version date: June 2003

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Log-Polar TransformGo from Euclidian (x,y) to log-polar space log(rei) => (log r, ) space. Log-polartransform is always done relative to a chosen center point (xc,yc):

(xc,yc)

r

x

y

log rLog-Polar

r(xc,yc)

x

y

Log-Polarlog r

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 getrotation, scale and translation invariance (called Fourier-Mellin transform)1. 1)

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Bilateral FilteringWe 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.

* C. Tomasi and R. Manduchi, "Bilateral Filtering for Gray and Color Images", Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India

=

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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:

Grossberg, S., & Mingolla, E. (1985). Neural Dynamics of Form Perception: Boundary Completion. Psychol. Rev., 92, 173--211.

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Local vs. GlobalStill, vision is a stranger thing than simple processing:

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Local vs. GlobalStill, 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