lec4 4pp

lec4 4pp - Lecture 4: Sept 1, 10 Course Enrollment HW1 to...

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USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 1 Lecture 4: Sept 1, 10 • Course Enrollment • HW1 to be posted later today, Due Sept 13 • Make up class today: 5-6:20PM, OHE 120 • Last Class – Intrinsic and extrinsic parameters – Affine cameras • Today’s objective – Camera calibration – Photometry – Color perception USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 2 Homogeneous Coords in 3-D • Point X: normal coords (x,y,z); homogeneous (x, y, z,1) T • Plane Π : ax + by + cz + d =0 is defined by (a, b, c, d) T • Point X is on plane iff Π T X = 0 • 3 points define a plane ( X 1 T , X 2 T , X 3 T ) T Π = 0 – Note: this equation is correct, the outer transpose is simply intended to indicate that the components X 1 T , X 2 T and X 3 T are arranged in a column, not a row. • 3 planes define a point ( Π 1 T , Π 2 T , Π 3 T ) T X = 0 USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 3 Affine Cameras • In many cases, perspective projection can be approximated by simpler camera models – Typically when the objects are far from the camera relative to their size • Projection matrix can be simplified by having only two rows: affine transformation – M becomes a 2 x 4 (3-D point expressed as homogeneous, 2-D as non- homogeneous): examples to come – Smaller number of parameters makes it easier to estimate them. – Images of affine cameras have simpler properties, e.g. parallelism may be preserved under certain models • Some specific affine cameras – Orthographic camera – Weak perspective camera – Para-perspective camera USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 4 Weak Perspective • All points at nearly same distance, but not necessarily very far
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USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 5 Weak Perspective: cont’d • Introduce intrinsic and extrinsic parameters • Since z r is a constant, rewrite as: R 2 is the matrix formed by the first two rows of R and t 2 is the vector formed by first two components of t k,s denote the aspect ration and skew respectively USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 6 Affine Cameras General affine projection matrix M = ( A b ), A is arbitrary rank-2, 2x3 matrix, b is an arbitrary 2 vector Table 2.1 gives a summary of different projection transformations USC CS574: Computer Vision, Fall 2010 Copyright 2010, by R. Nevatia 7 Inverse Problem • In graphics, object point(s) P and camera transformation matrix, M , are known, task is to compute the image p – Matrix multiplication solves the problem • Inverse problem p is given, estimate P M may or may not be known – Even if M is given, P is still not unique, M is not invertible; however, we can put some constraints on P (must lie on a specific line) – Given points in two images (say p1 and p2 ) and M1 and M2 , we can solve for P • Stereo processing, requires finding corresponding points,
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lec4 4pp - Lecture 4: Sept 1, 10 Course Enrollment HW1 to...

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