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## 3D Vision

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**3D Vision**Interest Points**correspondence and alignment**• Correspondence: matching points, patches, edges, or regions across images ≈**correspondence and alignment**• Alignment: solving the transformation that makes two things match better T**Example: estimating “fundamental matrix” that**corresponds two views**Example: tracking points**x x x frame 22 frame 0 frame 49 Your problem 1 for HW 2!**Interest points**• Suppose you have to click on some point, go away and come back after I deform the image, and click on the same points again. • Which points would you choose? original deformed**Overview of Keypoint Matching**1. Find a set of distinctive key- points 2. Define a region around each keypoint A1 B3 3. Extract and normalize the region content A2 A3 B2 B1 4. Compute a local descriptor from the normalized region 5. Match local descriptors**Goals for Keypoints**Detect points that are repeatable and distinctive**Choosing interest points**Where would you tell your friend to meet you?**Moravec corner detector (1980)**• We should easily recognize the point by looking through a small window • Shifting a window in anydirection should give a large change in intensity**Moravec corner detector**flat edge**Moravec corner detector**corner isolated point flat edge**Window function**Shifted intensity Intensity Moravec corner detector Change of intensity for the shift [u,v]: Four shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1) Look for local maxima in min{E}**Problems of Moravec detector**• Noisy response due to a binary window function • Only a set of shifts at every 45 degree is considered • Responds too strong for edges because only minimum of E is taken into account • Harris corner detector (1988) solves these problems.**Harris corner detector**Noisy response due to a binary window function • Use a Gaussian function**Harris corner detector**Only a set of shifts at every 45 degree is considered • Consider all small shifts by Taylor’s expansion**Harris corner detector**Equivalently, for small shifts [u,v] we have a bilinear approximation: , where M is a 22 matrix computed from image derivatives:**Harris corner detector**Responds too strong for edges because only minimum of E is taken into account • A new corner measurement**Harris corner detector**Intensity change in shifting window: eigenvalue analysis 1, 2 – eigenvalues of M direction of the fastest change Ellipse E(u,v) = const direction of the slowest change (max)-1/2 (min)-1/2**Harris corner detector**2 edge 2 >> 1 Corner 1 and 2 are large,1 ~ 2;E increases in all directions Classification of image points using eigenvalues of M: 1 and 2 are small;E is almost constant in all directions edge 1 >> 2 flat 1**Harris corner detector**Measure of corner response: (k – empirical constant, k = 0.04-0.06)**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**threshold**R R x(image coordinate) x(image coordinate) Harris Detector: Some Properties • Partial invariance to affine intensity change • Only derivatives are used => invariance to intensity shift I I+b • Intensity scale: I aI**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**Harris Detector is rotation invariant**Repeatability rate: # correspondences# possible correspondences**Harris Detector: Some Properties**• But: non-invariant to image scale! All points will be classified as edges Corner !**Harris Detector: Some Properties**• Quality of Harris detector for different scale changes Repeatability rate: # correspondences# possible correspondences