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UCSB ECE 181B - FINAL REVIEW

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1Final Review6-6-06(Starts on Slide 17)CS/ECE 181bMidterm ReviewMay 11, 20062• Image Formation• Projective Geometry• Camera Models•Stereo• Edge DetectionImage Formation• Pinhole camera geometry– Perspective projection• Vanishing point– What is it?– How do you geometrically construct thevanishing point of a line in 3-D• Orthographic and Parallel Projections32-D Projective Geometry• Homogeneous coordinates• Points from lines and lines from points– Intersection of parallel lines– Line at infinity• Duality principle• Conics and Conic Sections– Five points define a conic– Tangent lines and dual conicsHomography• Projective Transformations– Isometries– Similarities– Affine Transformation– General projective transformation4Central Projection• Pinhole camera geometry revisited– Central projection– Camera calibration matrix– Camera and World coordinate frames• Interior and Exterior Orientation– Finite Projective cameraCamera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal rayCamera anatomy5[][ ] =0010ppppp43212Column Vectors: image of the world origin.The columns of the projectivecamera are 3-vectors that have ageometric meaning as particularimage points.P1: vanishing point of the world coordinate x-axisP2: vanishing point of y-axisP3: vanishing point of z axisPrincipal Planes• Row vectors of the camera projectionmatrix• Principal axis• Estimating the camera matrix P6Topics• Image Formation• Projective Geometry• Camera Models•Stereo• Edge DetectionStereo• Epipolar Geometry• Essential Matrix• Fundamental Matrix7Epipolar geometry• Epipolar Plane• Epipoles• Epipolar Lines• BaselineC1C2geometric derivationxHx'=x'e'l' =[]FxxHe'==mapping from 2-D to 1-D family (rank 2)Fundamental Matrix F8The eight-point algorithm0=pFpTInvert and solve for FThe Correspondence Problem• Establishing correspondence between thetwo views• Disparity map and depth computations– Random dot stereograms9June 6, 2006• Linear Filtering & Edge detection• Eigenfaces and face recognition• Motion and optic flow• Shape from shadingEdge detectors• Gradient-based edge detectors– Approximate a spatial derivative– X and Y directions, or at various orientations– Fundamentally high-pass (accentuates noise)• Roberts, Sobel, Prewitt, Canny….• Laplacian and other band-pass edgedetectors10Digital Implementations• 1st order operator - 1x2 or 2x1 mask– simple– unbalanced (forward differencing)– sensitive to noiseEij,Eij+1,ExEEij ij +1, ,Eij,Eij, +1EyEEij ij +,,1Another Implementation•1st order operator - 2x2 mask– simple– unbalanced (forward differencing)– more resistive to noiseExEE EEij ij ij ij + ++ + +1211 1 1(( ) ( )),, ,,Eij,Eij+1,Eij++11,Eij, +1Eij,Eij+1,Eij++11,Eij, +1EyEE EEij ij ij ij + ++ + +1211 1 1(( ) ( )),,,,11Observation in 2D• 2D 1st order edge operator– A magnitude– A direction, but ...• Edge direction: iso-brightness direction• Gradient direction: largest brightness changedirectionmagnitudeExEydirectionEyEx=+=()()tan ( )221(, )tan ( )ExEyo>=>=000001(,)tan ( ) ExEyoo>>>>= 00000901(,)tan ( )ExEyo=<<= 0000901Gradient vs. Iso-brightness dirs12(, )tan ( )ExEyo>=>=000001(,)tan ( ) ExEyoo>>>>= 00000901(,)tan ( )ExEyo=<<= 0000901Gradient vs. Iso-brightness dirsMore Edge detectors• Sobel detector• Prewitt detector10-120-210-1121000-1-2-1GxGy|G| = Gx2 + Gy2G = atan Gy/Gx10-110-110-1111000-1-1-1GxGy13LenaVertical edgesHorizontal edgesOriginalEdge magnitudeEdge detectors: second order operators• Laplacian detectorsEdge detectors are not limited to 3x3 kernels22111112211222211 11224ExExExEE EEEEEEyEEELaplacianExEyEEEE Eij i ji j ij ij i jij ij ijij ij iji j i j ij ij ij  =  + ++ ++++ + + +  + ,,,, , ,,,,,,,,,,, ,()():Eij +11,Eij1,Eij11,Eij,Eij+1,Eij++11,Eij, +1Eij, 1Eij+ 11,-4111114Edge detectors: second order operators• Laplacian detectorsEdge detectors are not limited to 3x3 kernels22111112211222211 11224ExExExEE EEEEEEyEEELaplacianExEyEEEE Eij i ji j ij ij i jij ij ijij ij iji j i j ij ij ij  =  + ++ ++++ + + +  + ,,,, , ,,,,,,,,,,, ,()():Eij +11,Eij1,Eij11,Eij,Eij+1,Eij++11,Eij, +1Eij, 1Eij+ 11,-41111Smoothing Filter• Differentiation enhances noise (as well asedges).• Smooth the image before edge detection– Helps in minimizing false positives.– Edges at different scales.• Gaussian smoothing: Why?– G * G is also a Gaussian• Efficient multi-scale convolutions– Central limit theorem—smooth many times ==Gaussian smoothing with an appropriate sigma.– Gaussians are separable==good for implementation.15Laplacian of the Gaussian(LOG)• Smooth with the Gaussian + Laplacian +zero-crossing detector==gives edges• Equivalent to convolving the image withthe Laplacian of the Gaussian kernel.– Note: differentiation is linear and shiftinvariant.– Convolution is associative.• Can be well approximated by theDifference of the Gaussians (DoG)• Marr-Hildreth operator.Linear Filtering & Convolution16Convolution• Think of 2D convolution as the following procedure• For every pixel (i,j):– Line up the image at (i,j) with the filter kernel– Flip the kernel in both directions (vertical and horizontal)– Multiply and sum (dot product) to get output value R(i,j)(i,j)Convolution: example1 1 4 10 2 5 3 0 1 2x(m,n)1 1 10 1 -1 0 1h(m,n)mnmn-1 1 1 1h(-m, -n)-1 1 1 1h(1-m, n)y(1,0) =  k,l x(k,l)h(1-k, -l) = 0 0 0 00 -2 5 00 0 0 0= 31 5 5 13 10 5 22 3 -2 -3mny(m,n)=verify!17Convolution and correlation• Back to convolution/correlation• Convolution (or FT/IFT pair) is equivalent to linear filtering– Think of the filter kernel as a pattern, and convolution checks theresponse of the pattern at every point in the image– At each point, it is a dot product of the local image area with the filterkernelnjmiMmNnnmijFHR===,1010• Conceptually, the image responds best to the pattern of the filterkernel (similarity)– An edge kernel will produce high responses at edges, a facekernel will produce high


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