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Harris Corners16-385 Computer Vision (Kris Kitani) Carnegie Mellon University

How do you find a corner?Easily recognized by looking through a small window Shifting the window should give large change in intensity[Moravec 1980]

"edge": no change along the edge direction"corner": significant change in all directions"flat" region:

no change in all directionsEasily recognized by looking through a small window Shifting the window should give large change in intensity[Moravec 1980]

Design a program to detect corners (hint: use image gradients)

Finding corners (a.k.a. PCA) 1.Compute image gradients over small region2.Subtract mean from each image gradient3.Compute the covariance matrix4.Compute eigenvectors and eigenvalues5.Use threshold on eigenvalues to detect cornersI

x -I -x I y -I -y p-P I x I x p-P I x I y p-P I y I x p-P I y I y u v p-P I x I t p-P I y I t

1. Compute image gradients over a small region (not just a single pixel)

1. Compute image gradients over a small region (not just a single pixel)I

x -I -x I y -I -y array of x gradientsarray of y gradients visualization of gradientsimageX derivativeY derivative I x -I -x I x -I -x I x -I -x I y -I -y I y -I -y I y -I -y What does the distribution tell you about the region? I x -I -x I x -I -x I x -I -x I y -I -y I y -I -y I y -I -y distribution reveals edge orientation and magnitude I x -I -x I x -I -x I x -I -x I y -I -y I y -I -y I y -I -y

How do you quantify orientation and magnitude?

2. Subtract the mean from each image gradient

2. Subtract the mean from each image gradientplot intensitiesconstant intensity gradientintensities along the line

2. Subtract the mean from each image gradientplot intensitiesconstant intensity gradientI

x -I -x I y -I -y intensities along the lineplot of image gradientssubtract mean

2. Subtract the mean from each image gradientplot intensitiesconstant intensity gradientI

x -I -x I y -I -y intensities along the lineplot of image gradientsI x -I -x I y -I -y data is centered ('DC' offset is removed)subtract mean

3. Compute the covariance matrix

3. Compute the covariance matrixWhere does this covariance matrix come from?-

p-P I x I x p-P I x I y p-P I y I x p-P I y I y u v p-P I x I t p-P I y I t I x -I -x I y -I -y array of x gradientsarray of y gradients*.=sum()- p-P I x I x p-P I x I y p-P I y I x p-P I y I y u v p-P I x I t p-P I y I t

Error functionChange of intensity for the shift [u,v]:IntensityShifted intensityWindow functionorWindow function w(x,y) =Gaussian1 in window, 0 outsideError functionSome mathematical background...

Error function approximationChange of intensity for the shift [u,v]:Second-order Taylor expansion of E(u,v) about (0,0) (bilinear approximation for small shifts):first derivativesecond derivative

Bilinear approximationFor small shifts [u,v] we have a 'bilinear approximation':where M is a 2×2 matrix computed from image derivatives:Change in appearance for a shift [u,v]M'second moment' matrix 'structure tensor'

p-P I x I x p-P I x I y p-P I y I x p-P I y I y u v p-P I x I t p-P I y I t

By computing the gradient covariance matrix...we are fitting a quadratic to the gradients over a small image region

Visualization of a quadraticThe surface E(u,v) is locally approximated by a quadratic form

Which error surface indicates a good image feature?What kind of image patch do these surfaces represent?

flatedge 'line'corner 'dot'

4. Compute eigenvalues and eigenvectors

eig(M)

4. Compute eigenvalues and eigenvectorseigenvectoreigenvalueMe=-e(M--I)e=0

1. Compute the determinant of (returns a polynomial)eigenvectoreigenvalueM--IMe=-e(M--I)e=0

4. Compute eigenvalues and eigenvectors

1. Compute the determinant of (returns a polynomial)eigenvectoreigenvalue2. Find the roots of polynomial (returns eigenvalues)det(M--I)=0M--IMe=-e(M--I)e=0

4. Compute eigenvalues and eigenvectors

1. Compute the determinant of (returns a polynomial)eigenvectoreigenvalue2. Find the roots of polynomial (returns eigenvalues)det(M--I)=0M--IMe=-e(M--I)e=0

3. For each eigenvalue, solve (returns eigenvectors)(M--I)e=0

4. Compute eigenvalues and eigenvectors

Visualization as an ellipseSince M is symmetric, we haveWe can visualize M as an ellipse with axis lengths determined by the eigenvalues and orientation determined by Rdirection of the slowest changedirection of the fastest change(λ

max )-1/2 min )-1/2

Ellipse equation:

interpreting eigenvaluesλ 1 2 2 1 1 2 1 -0 2 -0 What kind of image patch does each region represent? interpreting eigenvalues'horizontal' edge'vertical' edgeflatcornerλ 1 2 2 1 1 2 1 2 interpreting eigenvalues'horizontal' edge'vertical' edgeflatcornerλ 1 2 2 1 1 2 1 2

5. Use threshold on eigenvalues to detect corners

flatλ 1 2

5. Use threshold on eigenvalues to detect cornersThink of a function to score 'cornerness'

flatλ 1 2

5. Use threshold on eigenvalues to detect cornersThink of a function to score 'cornerness'strong corner

flatcornerλ 1 2

5. Use threshold on eigenvalues to detect corners

CHWPEVKQPQH^R=min(-

1 2 Use the smallest eigenvalue as the response function flatcornerλ 1 2

5. Use threshold on eigenvalues to detect corners

CHWPEVKQPQH^R=-

1 2 1 2 2 Eigenvalues need to be bigger than one.Can compute this more efficiently... flatcornerλ 1 2

R < 0R > 0R < 0R-0R=det(M)--trace

2 (M)

5. Use threshold on eigenvalues to detect corners

CHWPEVKQPQH^

R=det(M)--trace

2 (M) R= det(M) trace(M)+- Harris & Stephens (1988)Kanade & Tomasi (1994)Nobel (1998)R=min(- 1 2

1.Compute x and y derivatives of image 2.Compute products of derivatives at every pixel 3.Compute the sums of the products of derivatives at each pixelHarris DetectorC.Harris and M.Stephens. "A Combined Corner and Edge Detector."1988. IGIxx∗=σIGIyy∗=σxxxIII⋅=2yyyIII⋅=2yxxyIII⋅=22'xxIGS∗=σ22'yyIGS∗=σxyxyIGS∗='σ

Harris DetectorC.Harris and M.Stephens. "A Combined Corner and Edge Detector."1988. 4.Define the matrix at each pixel 5.Compute the response of the detector at each pixel 6.Threshold on value of R; compute non-max suppression.!!"#$$%&=),(),(),(),(),(22yxSyxSyxSyxSyxMyxyxyx()2tracedetMkMR-=

Corner response

Thresholded corner response

Non-maximal suppression

Harris corner response is rotation invariantEllipse rotates but its shape (eigenvalues) remains the sameCorner response R is invariant to image rotation

intensity changesPartial invariance to affine intensity changeü Only derivatives are used => invariance to intensity shift I → I + bü Intensity scale: I → a IRx (image coordinate)thresholdRx (image coordinate)

The Harris detector not invariant to changes in ...quotesdbs_dbs16.pdfusesText_22

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