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UCSB ECE 181B - Review of Topics

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Review of Topics Midterm Exam on Thursday, Feb 11, 8am.Wednesday, February 10, 2010Image Formation: SummaryProjection Geometry - determines the position of a 3D point in the image.– Perspective projection– approximations using ♦orthographic projection♦parallel projection– terminology♦center of projection♦vanishing point♦optic axis♦focal point, focal lengthWednesday, February 10, 2010Image Features•What features? and where?•The correspondence problem•The aperture problem (localization)•Local structure matrix and corner response function•Harris corner detector projectWednesday, February 10, 2010Edges and Edge detection•Linear filtering basics (convolution and correlation)•Digital approximations to gradient edge detectors•The Laplacian of the Gaussian edge detector (zero crossings)Wednesday, February 10, 2010Edges: Summary• Convolution (correlation) defines a shift-invariant linear filter• The Fourier transform is a linear operation that exposes the spatial frequency composition of an image• Sampling and aliasing are directly related to spatial frequency issues– E.g., image pyramids• Correlation can be viewed as template matching (or pattern matching)• If the template is a gradient/derivative operator, correlation implements edge detection• Edge detection by itself doesn’t work very well, although it can be useful if its limitations are understood– Noise, missing edges, complex scenes, …Wednesday, February 10, 2010SIFT Descriptor1. Create the Gaussian pyramid and the DoG pyramid at multiple scales.2. find the key point locations through a scale space analysis. Prune the keypoints based on contrast and corner strength.3. compute the dominant edge directions in the neighborhood of the keypoints.4. so, we have the location, scale and orientation of the keypoints at this point in computation.We now need a representation of the image information at the keypoint location, i.e., a description.Wednesday, February 10, 20106. An invariant descriptor.•A weighted orientation histogram is now computed, relative to the keypoint orientation, on 4x4 pixel neighborhoods.Image gradientsKeypoint descriptorFigu re 7: A keypoin t descriptor is create d by first computing the gradient magnitude and orientationat e ach image sample point in a region around the keypoint location, as shown on the le ft. These areweighted by a Gaussia n win dow, indicated by the overlaid circle. These samples are then accumulatedinto orie ntation histog rams summarizing the co ntents over 4x 4 subregions, as shown on the right, withthe leng th of each arrow corresponding to the sum of the grad ient magnitudes near that d irection withinthe region. This figure shows a 2x2 descriptor array computed from an 8x8 set of samples, whereasthe experiments in this paper use 4x4 descriptors computed from a 16x16 sample ar ray.6.1 Descriptor representationFigure 7 illustrates the computation of the keypoint descriptor. First the image gradient mag-nitudes and orientations are sampled around the keypoint location, using the scale of thekeypoint to select the level of Gaussian blur for the image. In order to achieve orientationinvariance, the coordinates of the descriptor and the gradient orientations are rotated relativeto the keypoint orientation. For efficiency, the gradients are precomputed for all levels of thepyram id as described in Section 5. These are illustrated with small arrow s at each samplelocation on the left side of Figure 7.A Gaussian weighting function with σ equal to one half the width of the descriptor win-dow is used to assign a weight to the magnitude of each sample point. This is illustratedwith a circular window on the left side of Figure 7, although, of course, the w eight falls offsmoothly. The purpose of this Gaussian window is to avoid sudden changes in the descriptorwith small changes in the position of the window, and to give less emphasis to gradients thatare far from the center of the descriptor, as these are most affected by misregistration errors.The keypoint descriptor is shown on the right side of Figure 7. It allow s for significantshift in gradient positions by creating orientation histograms over 4x4 sample regions. Thefigure shows eight directions for each orientation histogram, with the length of each arrowcorresponding to the magnitude of that histogram entry. A gradient sample on the left canshift up to 4 sample positions while still contributing to the same histogram on the right,thereby achieving the objective of allowing for larger local positional shifts.It is important to avoid all boundary affects in which the descriptor abruptly changes as asample shifts smoothly from being within one histogram to another or from one orientationto another. Therefore, trilinear interpolation is used to distribute the value of each gradientsample into adjacent histogram bins. In other words, each entry into a bin is multiplied by aweight of 1 − d for each dimension, where d is the distance of the sample from the centralvalue of the bin as measured in units of the histogram bin spacing.154 x 4 windows x 8 orientation bins = 128 dimensional descriptorsWednesday, February 10, 2010Project #2•Implement a SIFT descriptor.•Evaluation of the descriptor: Take one image and transform it with slight rotation, scaling and by adding noise.•compute the descriptors on both original and modified images.•what percentage of the descriptors match in terms of location and nearest neighbors?Wednesday, February 10, 2010back to Projective Geometry•Homogeneous coordinates•points and lines, and duality •ideal points and line at infinity•Conics/conic sections/degenerate conics•Projective transformation: Homography•Isometries, similarities, affine and full projective•invariants under different transformationsWednesday, February 10, 2010Projective Geometry 2DMidterm • everything as of projective geometry lecture will be included.• closed book/notes exam. • no calculators/electronic devices are needed.• review continues tomorrow during the discussion session.10Wednesday, February 10,


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