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Computer Vision : CISC 4/689
Example: Canny Edge Detection
(Matlab automatically set thresholds)
Computer Vision : CISC 4/689
More: facts and figures
• The convolution of two Gaussians with variances {1}2 and {2}2 is {1}2+{2}2. This is same as consecutive smoothing with the two corresponding SD’s.
• Thus, generic formula is: i{i}2
• Problem: A discrete appx. to a 1D Gaussian can be obtained by sampling g(x). In practice, samples are taken uniformly until the truncated values at the tails of the distribution are less than 1/1000 of the peak value.
a) For =1, show that the filter is 7 pixels wide.
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Answer..
• Lets pick (n+1) pixels from the center of kernel(including center). This way, total kernel size is 2n+1, n pixels on either side of origin.
Exp(-{(n+1)2}/{22}) < 1/1000So, n > 3.7 -1n must be the nearest integer to 3.7 -0.5
For =1, n=3, 2n+1=7.Filter coefficients can be obtained as {-3,-2,-1,0,1,2,3}
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Choice of
• The choice of depends on the scale at which the image is to be displayed.
• Small values bring out edges at a fine scale, vice-versa.
• Noise is another factor to look into the selection, along with computational cost
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Some comparisons
Zero-crossings easy to find than threshold
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Canny
• Many implementations of the Canny edge detector approximate this process by first convolving the image with a Gaussian to smooth the signal, and then looking for maxima in the first partial derivatives of the resulting signal (using masks similar to the Sobel masks).
• Thus we can convolve the image with 4 masks, looking for horizontal, vertical and diagonal edges. The direction producing the largest result at each pixel point is marked.
• Record the convolution result and the direction of the edge at each pixel.
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Marr-Hildreth vs. Canny
• Laplacian is isotropic, computationally efficient: single convolution, look for zero-crossing. (one way to explain zero-crossing is, if first derivative can be looked at as a function, its maximum will be its derivative=0).
• Canny being a directional operator (derivative in 4 or 3 directions), more costly, esp. due to hysterisis.
• Two derivatives -> more sensitive to noise
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Image Pyramids
• Observation: Fine-grained template matching expensive over a full image – Idea: Represent image at smaller
scales, allowing efficient coarse- to-fine search
• Downsampling: Cut width, height in half at each iteration:
from Forsyth & Ponce
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Gaussian Pyramid
• Let the base (the finest resolution) of an n-level Gaussian pyramid be defined
as P0 = I. Then the ith level is reduced from the level below it by:
• Upsampling S"(I): Double size of image, interpolate missing pixels
courtesy of Wolfram
Gaussian pyramid
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Laplacian Pyramids
• The tip (the coarsest resolution) of an n-level Laplacian pyramid is the same as the Gaussian pyramid at that level: Ln(I) = Pn(I)
• The ith level is expanded from the level above according to Li(I) = Pi(I) ¡ S"(Pi+1(I))
• Synthesizing the original image: Get I back by summing upsampled Laplacian pyramid levels
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Laplacian Pyramid
• The differences of images at successive levels of the Gaussian pyramid define the Laplacian pyramid. To calculate a difference, the image at a higher level in the pyramid must be increased in size by a factor of four prior to subtraction. This computes the pyramid.
• The original image may be reconstructed from the Laplacian pyramid by reversing the previous steps. This interpolates and adds the images at successive levels of the pyramid beginning with the lowest level.
• Laplacian is largely uncorrelated, and so may be represented pixel by pixel with many fewer bits than Gaussian.
courtesy of Wolfram
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Reconstruction
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Splining
• Build Laplacian pyramids LA and LB for A & B images
• Build a Gaussian pyramid GR from selected region R
• Form a combined pyramid LS from LA and LB using nodes of GR as weights:
LS(I,j) = GR(I,j)*LA(I,j)+(1-GR(I,j))*LB(I,j)
Collapse the LS pyramid to get the final blended image
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Splining (Blending)
• Splining two images simply requires: 1) generating a Laplacian pyramid for each image, 2) generating a Gaussian pyramid for the bitmask indicating how the two images should be merged, 3) merging each Laplacian level of the two images using the bitmask from the corresponding Gaussian level, and 4) collapsing the resulting Laplacian pyramid.
• i.e. GS = Gaussian pyramid of bitmask LA = Laplacian pyramid of image "A" LB = Laplacian pyramid of image "B" therefore, "Lout = (GS)LA + (1-GS)LB"
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Example images from GTech
Image-1 bit-mask image-2
Direct addition splining bad bit-mask choice
Computer Vision : CISC 4/689
Outline
• Corner detection
• RANSAC
Computer Vision : CISC 4/689
Matching with Invariant Features
Darya Frolova, Denis Simakov
The Weizmann Institute of Science
March 2004
Computer Vision : CISC 4/689
Example: Build a Panorama
M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003
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How do we build panorama?
• We need to match (align) images
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Matching with Features
•Detect feature points in both images
Computer Vision : CISC 4/689
Matching with Features
•Detect feature points in both images
•Find corresponding pairs
Computer Vision : CISC 4/689
Matching with Features
•Detect feature points in both images
•Find corresponding pairs
•Use these pairs to align images
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Matching with Features
• Problem 1:– Detect the same point independently in both images
no chance to match!
We need a repeatable detector
Computer Vision : CISC 4/689
Matching with Features
• Problem 2:– For each point correctly recognize the corresponding one
?
We need a reliable and distinctive descriptor
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More motivation…
• Feature points are used also for:– Image alignment (homography, fundamental matrix)
– 3D reconstruction
– Motion tracking
– Object recognition
– Indexing and database retrieval
– Robot navigation
– … other
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Corner Detection
• Basic idea: Find points where two edges meet—i.e., high gradient in two directions
• “Cornerness” is undefined at a single pixel, because there’s only one gradient per point– Look at the gradient behavior over a small window
• Categories image windows based on gradient statistics– Constant: Little or no brightness change– Edge: Strong brightness change in single direction– Flow: Parallel stripes– Corner/spot: Strong brightness changes in orthogonal directions
Computer Vision : CISC 4/689
Corner Detection: Analyzing Gradient Covariance
• Intuitively, in corner windows both Ix and Iy should be high– Can’t just set a threshold on them directly, because we want rotational invariance
• Analyze distribution of gradient components over a window to differentiate between types from previous slide:
• The two eigenvectors and eigenvalues ¸1, ¸2 of C (Matlab: eig(C)) encode the predominant directions and magnitudes of the gradient, respectively, within the window
• Corners are thus where min(¸1, ¸2) is over a threshold courtesy of Wolfram
Covariance is 1/n sum(1 to n) (xi-x’)(xi-x’)^t
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Contents
• Harris Corner Detector
– Description
– Analysis
• Detectors
– Rotation invariant
– Scale invariant
– Affine invariant
• Descriptors
– Rotation invariant
– Scale invariant
– Affine invariant
Computer Vision : CISC 4/689
Harris Detector: Mathematics
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
Change of intensity for the shift [u,v]:
IntensityShifted intensity
Window function
orWindow function w(x,y) =
Gaussian1 in window, 0 outside
Taylor series:I(x+u,y+v) = I(x,y)+Ix(x,y)u+Iy(x,y)v+…http://mathworld.wolfram.com.TaylorSeries.html
Computer Vision : CISC 4/689
Harris Detector: Mathematics
( , ) ,u
E u v u v Mv
For small shifts [u,v] we have a bilinear approximation:
2
2,
( , ) x x y
x y x y y
I I IM w x y
I I I
where M is a 22 matrix computed from image derivatives:
Computer Vision : CISC 4/689
Harris Detector: Mathematics
( , ) ,u
E u v u v Mv
Intensity change in shifting window: eigenvalue analysis
1, 2 – eigenvalues of M
1
2
Ellipse E(u,v) = const
If we try every possible orientation n,the max. change in intensity is 2
Computer Vision : CISC 4/689
Harris Detector: Mathematics
1
2
“Corner”1 and 2 are large,
1 ~ 2;
E increases in all directions
1 and 2 are small;
E is almost constant in all directions
“Edge” 1 >> 2
“Edge” 2 >> 1
“Flat” region
Classification of image points using eigenvalues of M:
Computer Vision : CISC 4/689
Harris Detector: Mathematics
Measure of corner response:
2det traceR M k M
1 2
1 2
det
trace
M
M
(k – empirical constant, k = 0.04-0.06)
Computer Vision : CISC 4/689
Harris Detector: Mathematics
1
2 “Corner”
“Edge”
“Edge”
“Flat”
• R depends only on eigenvalues of M
• R is large for a corner
• R is negative with large magnitude for an edge
• |R| is small for a flat region
R > 0
R < 0
R < 0|R| small
Computer Vision : CISC 4/689
Harris Detector
• The Algorithm:– Find points with large corner response function R (R >
threshold)
– Take the points of local maxima of R
Computer Vision : CISC 4/689
Harris Detector: Workflow
Computer Vision : CISC 4/689
Harris Detector: Workflow
Compute corner response R
Computer Vision : CISC 4/689
Harris Detector: Workflow
Find points with large corner response: R>threshold
Computer Vision : CISC 4/689
Harris Detector: Workflow
Take only the points of local maxima of R
Computer Vision : CISC 4/689
Harris Detector: Workflow
Computer Vision : CISC 4/689
Example: Gradient Covariances
Full imageDetail of image with gradient covar-
iance ellipses for 3 x 3 windows
from Forsyth & Ponce
Corners are where both eigenvalues are big
Computer Vision : CISC 4/689
Example: Corner Detection (for camera calibration)
courtesy of B. Wilburn
Computer Vision : CISC 4/689
Example: Corner Detection
courtesy of S. Smith
SUSAN corners
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Harris Detector: Summary
• Average intensity change in direction [u,v] can be expressed as a bilinear form:
• Describe a point in terms of eigenvalues of M:measure of corner response
• A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive
( , ) ,u
E u v u v Mv
2
1 2 1 2R k
Computer Vision : CISC 4/689
Contents
• Harris Corner Detector
– Description
– Analysis
• Detectors
– Rotation invariant
– Scale invariant
– Affine invariant
• Descriptors
– Rotation invariant
– Scale invariant
– Affine invariant
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Tracking: compression of video information
• Harris response (uses criss-cross gradients)• Dinosaur tracking (using features)• Dinosaur Motion tracking (using correlation)• Final Tracking (superimposed)Courtesy: (http://www.toulouse.ca/index.php4?/CamTracker/index.php4?/CamTracker/FeatureTracking.html)
This figure displays results of feature detection over the dinosaur test sequence with the algorithm set to extract the 6 most "interesting" features at every image frame.
It is interesting to note that although no attempt to extract frame-to-frame feature correspondences was made, the algorithm still extracts the same set of features at every frame.
This will be useful very much in feature tracking.
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One More..
• Office sequence
• Office Tracking
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Harris Detector: Some Properties
• Rotation invariance
Ellipse rotates but its shape (i.e. eigenvalues) remains the same
Corner response R is invariant to image rotation
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Harris Detector: Some Properties
• Partial invariance to affine intensity change
Only derivatives are used => invariance to intensity shift I I + b
Intensity scale: I a I
R
x (image coordinate)
threshold
R
x (image coordinate)
Computer Vision : CISC 4/689
Harris Detector: Some Properties
• But: non-invariant to image scale!
All points will be classified as edges
Corner !
Computer Vision : CISC 4/689
Harris Detector: Some Properties• Quality of Harris detector for different scale changes-- Correspondences calculated using distance (and threshold)
-- Improved Harris is proposed by Schmid et al
-- repeatability rate is defined as the number of points
repeated between two images w.r.t the total number of
detected points. Repeatability rate:
# correspondences# possible correspondences
C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000
Imp.Harris uses derivative of Gaussianinstead of standard template used byHarris et al.
Computer Vision : CISC 4/689
Contents
• Harris Corner Detector
– Description
– Analysis
• Detectors
– Rotation invariant
– Scale invariant
– Affine invariant
• Descriptors
– Rotation invariant
– Scale invariant
– Affine invariant
Computer Vision : CISC 4/689
We want to:
detect the same interest points regardless of image changes
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Models of Image Change
• Geometry– Rotation
– Similarity (rotation + uniform scale)
– Affine (scale dependent on direction)valid for: orthographic camera, locally planar object
• Photometry– Affine intensity change (I a I + b)
Computer Vision : CISC 4/689
Contents
• Harris Corner Detector
– Description
– Analysis
• Detectors
– Rotation invariant
– Scale invariant
– Affine invariant
• Descriptors
– Rotation invariant
– Scale invariant
– Affine invariant
Computer Vision : CISC 4/689
Rotation Invariant Detection
• Harris Corner Detector
C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000
Computer Vision : CISC 4/689
Contents
• Harris Corner Detector
– Description
– Analysis
• Detectors
– Rotation invariant
– Scale invariant
– Affine invariant
• Descriptors
– Rotation invariant
– Scale invariant
– Affine invariant
Computer Vision : CISC 4/689
Scale Invariant Detection• Consider regions (e.g. circles) of different sizes around a point
• Regions of corresponding sizes will look the same in both images
Computer Vision : CISC 4/689
Scale Invariant Detection• The problem: how do we choose corresponding circles independently
in each image?
Computer Vision : CISC 4/689
Scale Invariant Detection
• Solution:
– Design a function on the region (circle), which is “scale invariant” (the same for corresponding regions, even if they are at different scales)
Example: average intensity. For corresponding regions (even of different sizes) it will be the same.
scale = 1/2
– For a point in one image, we can consider it as a function of region size (circle radius)
f
region size
Image 1 f
region size
Image 2
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Scale Invariant Detection
• Common approach:
scale = 1/2
f
region size
Image 1 f
region size
Image 2
Take a local maximum of this function
Observation: region size, for which the maximum is achieved, should be invariant to image scale.
s1 s2
Important: this scale invariant region size is found in each image independently!
Computer Vision : CISC 4/689
Scale Invariant Detection
• A “good” function for scale detection: has one stable sharp peak
f
region size
bad
f
region size
bad
f
region size
Good !
• For usual images: a good function would be the one with contrast (sharp local intensity change)
Computer Vision : CISC 4/689
Scale Invariant Detection
• Functions for determining scale
2 2
21 22
( , , )x y
G x y e
2 ( , , ) ( , , )xx yyL G x y G x y
( , , ) ( , , )DoG G x y k G x y
Kernel Imagef Kernels:
where Gaussian
Note: both kernels are invariant to scale and rotation
(Laplacian)
(Difference of Gaussians)
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Scale Invariant Detectors• Harris-Laplacian1
Find local maximum of:– Harris corner detector in space
(image coordinates)– Laplacian in scale
1 K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20012 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 2004
scale
x
y
Harris
L
apla
cian
• SIFT (Lowe)2
Find local maximum of:– Difference of Gaussians in
space and scale
scale
x
y
DoG
D
oG
Computer Vision : CISC 4/689
Harris Laplacian
• Characteristic Scale: Given a point in an image, compute the function responses for several factors sn The characteristic scale is the local max. of the function (can be more than one).
• Easy to look for zero-crossings of 2nd derivative than maxima.
• The ratio of the scales, at which the extrema were found for corresponding points in two rescaled images, is equal to the scale factor between the images.
• Then what..?
Computer Vision : CISC 4/689
Harris-Laplacian
• Existing methods search for maxima in the 3D representation of an image (x,y,scale). A feature point represents a local maxima in the surrounding 3D cube and its value is higher than a threshold.
• THIS (Harris-Laplacian) method uses Harris function first, then selects points for which Laplacian attains maximum over scales.
• First, prepare scale-space representation for the Harris function. At each level, detect interest points as local maxima in the image plane (of that scale) – do this by comparing 8-neighborhood. (different from plain Harris corner detection)
• Second, use Laplacian to judge if each of the candidate points found on different levels, if it forms a maximum in the scale direction. (check with n-1 and n+1)
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Scale Invariant Detectors
K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 2001
• Experimental evaluation of detectors w.r.t. scale change
Repeatability rate:
# correspondences# possible correspondences
(points present)
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Scale Invariant Detection: Summary
• Given: two images of the same scene with a large scale difference between them
• Goal: find the same interest points independently in each image• Solution: search for maxima of suitable functions in scale and in space
(over the image)
Methods:
1. Harris-Laplacian [Mikolajczyk, Schmid]: maximize Laplacian over scale, Harris’ measure of corner response over the image
2. SIFT [Lowe]: maximize Difference of Gaussians over scale and space