COMPUTER VISION Robust estimation Emanuel Aldea < emanuel.aldea@u-psud.fr > http://hebergement.u-psud.fr/emi/ Computer Science and Multimedia Master - University of Pavia
Back to our simple motivator Objective of the procedure E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (2/23)
Panoramic reconstruction Problem ◮ Corner detection and association ◮ Observation ( x , y , x ′ , y ′ ) : the corner ( x , y ) in the first image is associated to the corner ( x ′ , y ′ ) in the second image x ′ = H ˜ ◮ if pure camera rotation pure between the two images ˜ x where : wx ′ h 00 h 01 h 02 x = wy ′ h 10 h 11 h 12 y 1 w h 20 h 21 h 22 ◮ by developping, we get : � h 00 x + h 01 y + h 02 x ′ = h 20 x + h 21 y + h 22 h 10 x + h 11 y + h 12 y ′ = h 20 x + h 21 y + h 22 E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (3/23)
Panoramic reconstruction Problem ◮ the unknowns are the different h ij � x ′ ( h 20 x + h 21 y + h 22 ) = h 00 x + h 01 y + h 02 y ′ ( h 20 x + h 21 y + h 22 ) = h 10 x + h 11 y + h 12 h 00 h 01 h 02 � x h 10 − x ′ x − x ′ y − x ′ � � � y 1 0 0 0 0 h 11 = − y ′ x − y ′ y − y ′ 0 0 0 x y 1 0 h 12 h 20 h 21 h 22 E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (4/23)
Panoramic reconstruction h 00 h 01 − x ′ − x ′ x ′ x 1 y 1 1 0 0 0 1 x 1 1 y 1 1 h 02 − y ′ − y ′ y ′ 0 0 0 x 1 y 1 1 1 x 1 1 y 1 1 h 10 . . . . . . . . . . . . . . . . . . = . . . . . . . . . h 11 − x ′ − x ′ x ′ x n y n 1 0 0 0 n x n n y n h 12 n − y ′ − y ′ y ′ 0 0 0 x n y n 1 n x n n y n h 20 n h 21 H is determined modulo a multiplicative factor, thus we can set h 22 to 1. We note that in order to estimate the homography we need n = 4 observations. We must solve Ah = b - easy ! E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (5/23)
Panoramic reconstruction h 00 h 01 − x ′ − x ′ x ′ x 1 y 1 1 0 0 0 1 x 1 1 y 1 1 h 02 − y ′ − y ′ y ′ 0 0 0 x 1 y 1 1 1 x 1 1 y 1 1 h 10 . . . . . . . . . . . . . . . . . . = . . . . . . . . . h 11 − x ′ − x ′ x ′ x n y n 1 0 0 0 n x n n y n h 12 n − y ′ − y ′ y ′ 0 0 0 x n y n 1 n x n n y n h 20 n h 21 If n > 4, then the system is overdetermined. In order to find the least square solution for Ah = b , one has to : 1. compute the Singular Value Decomposition (the SVD) of A : A = UDV T 2. compute b ′ = U T b 3. find y defined as y i = b ′ i / d i 4. the solution is h = Vy E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (6/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution ◮ but even for an over determined system, the outlier(s) will have a significant impact (even one outlier may be very detrimental) E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution ◮ but even for an over determined system, the outlier(s) will have a significant impact (even one outlier may be very detrimental) ◮ all least-square based optimizations are sensitive to outliers E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution ◮ but even for an over determined system, the outlier(s) will have a significant impact (even one outlier may be very detrimental) ◮ all least-square based optimizations are sensitive to outliers Objective ◮ solve a Computer Vision problem which requires observations E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution ◮ but even for an over determined system, the outlier(s) will have a significant impact (even one outlier may be very detrimental) ◮ all least-square based optimizations are sensitive to outliers Objective ◮ solve a Computer Vision problem which requires observations ◮ ... while at the same time, pruning the bad observations E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation What if some of the n observations are wrong ? ◮ this will create major problems ◮ obviously for n = 4 we will get a different solution ◮ but even for an over determined system, the outlier(s) will have a significant impact (even one outlier may be very detrimental) ◮ all least-square based optimizations are sensitive to outliers Objective ◮ solve a Computer Vision problem which requires observations ◮ ... while at the same time, pruning the bad observations ◮ underlying idea : outliers participate to “strange” solutions E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (7/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. ◮ a significant part of the observations is generated by a mathematical model characterized by a set of parameters θ E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. ◮ a significant part of the observations is generated by a mathematical model characterized by a set of parameters θ Objective ◮ d´ etermine the parameters θ E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. ◮ a significant part of the observations is generated by a mathematical model characterized by a set of parameters θ Objective ◮ d´ etermine the parameters θ ◮ in robotics : often a movement estimation/information E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. ◮ a significant part of the observations is generated by a mathematical model characterized by a set of parameters θ Objective ◮ d´ etermine the parameters θ ◮ in robotics : often a movement estimation/information ◮ tracking some targets E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
Robust estimation Problem framework : ◮ observations provided by images ◮ interest points (but sometimes contours, regions etc.) ◮ associations : matches, optical flow fields, etc. ◮ a significant part of the observations is generated by a mathematical model characterized by a set of parameters θ Objective ◮ d´ etermine the parameters θ ◮ in robotics : often a movement estimation/information ◮ tracking some targets ◮ the state of a physical system etc. E. Aldea (CS&MM- U Pavia) COMPUTER VISION Chap II : Robust estimation (8/23)
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