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Lecture 8 (Stereo imaging) (Digital Image Processing)
- 1. Stereo Imaging Subject: Image Procesing & Computer Vision Dr. Varun Kumar Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 1 / 11
- 2. Outlines 1 Calibration of camera set-up 2 Mathematical modeling 3 Stereo Imaging 4 References Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 2 / 11
- 3. Calibration of camera set-up Relation for imaging geometry ch = PTRGwh = Awh (1) ⇒ ch → Camera co-ordinates in homogeneous form ⇒ wh → World co-ordinates in homogeneous form ⇒ A → Overall transformation matrix of size 4 × 4. Note : P, T, R, G → 4 × 4 matrix. Aim of calibration set-up Estimate the element value of A for a given imaging set up. A ∼ Aij ∀ 1 ≤ i ≤ 4 and 1 ≤ j ≤ 4 Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 3 / 11
- 4. Continued– ch = Awh ⇒ ch1 ch2 ch3 ch4 = a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 wh1 wh2 wh3 wh4 General Convention ⇒ x = ch1 ch4 , y = ch2 ch4 , z = ch3 ch4 and X = wh1 wh4 , Y = wh2 wh4 , Z = wh3 wh4 . Let wh4 = 1 ⇒ ch1 = xch4 = a11X + a12Y + a13Z + a14 ⇒ ch2 = ych4 = a21X + a22Y + a23Z + a24 ⇒ ch4 = a41X + a42Y + a43Z + a44 ⇒ x(a41X + a42Y + a43Z + a44) = a11X + a12Y + a13Z + a14 a11X + a12Y + a13Z − a41xX − a42xY − a43xZ − a44x + a14 = 0 (2) a21X + a22Y + a23Z − a41yX − a42yY − a43yZ − a44y + a24 = 0 (3) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 4 / 11
- 5. Stereo imaging Note There are two equation (2), (3), and 12 unknown like, a11, a12, a13, a41, a42, a43, a44, a14, a21, a22, a23, a41y, a42, a43, a24. conventionally, it can’t be solved. For solving such type of problem, we use the numerical techniques. Stereo imaging model: Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 5 / 11
- 6. Another approach Note ⇒ Here left image is aligned with the world co-ordinate system. ⇒ Right image is not aligned with the world co-ordinate system. ⇒ Both image have same focal length λ. ⇒ Same perspective and inverse perspective transform. Images are co-planer Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 6 / 11
- 7. Continued– Modified relation Relative modified relation between world and camera co-ordinates : X1 = x1 λ (λ − Z) world co-ordinate with respect to image plane 1. X2 = x2 λ (λ − Z) world co-ordinate with respect to image plane 2. X2 = X1 + B ⇒ x2 λ (λ − Z) = x1 λ (λ − Z) + B Z = λ − λB x2−x1 ⇒ (x2 − x1) → Disparity Stereo correspondence problem Q If a point is located in image 1 then what will be the co-ordinates of mapped point in image 2. Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 7 / 11
- 8. Stereo correspondence problem Let left image a point cL(x1, y1). In right image there are N2 points, where single point cR(x2, y2) is the correspondence point. Searching huge amount of point is computationally complex. Disparity (x2 − x1) helps for finding Z in world co-ordinate system. Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 8 / 11
- 9. Continued– Solution through imaging geometry x1 = λX1 λ − Z1 , y1 = λY1 λ − Z1 ⇒ Left image x2 = λX2 λ − Z2 , y2 = λY2 λ − Z2 ⇒ Right image Note: Z1 = Z2 and Y1 = Y2, but X1 = X2 Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 9 / 11
- 10. Home work question 1 A camera has focal length 0.04, find out the locus of the imaging point, which will be imaged at location (0.2, 0.6) on the image plane. Assume that the camera co-ordinate and world co-ordinates are perfectly aligned. 2 A camera with focal length is 0.04 m is placed at a height of 2m and is looking vertically downwards to take image of XY plane. If the size of image sensor plate is 4mm × 3mm, find the area of the plane that can be imaged. 3 A camera is mounted on a gimble system that enables the camera to pan and tilt at any arbitrary angle. The gimble center is placed at location (0, 0, 5) and the camera center displacement from gimble center is (0.2, 0.1, 0.2) in a world co-ordinate system XYZ. Assuming that the camera has pan 45o and tilt 135o, find out the world co-ordinate of a world point (1, 1, 0.5). Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 10 / 11
- 11. References M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision. Cengage Learning, 2014. D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern approach, vol. 17, pp. 21–48, 2003. L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey, 2001. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB. Pearson Education India, 2004. Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 8 11 / 11