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3D Computer Vision and Video Computing 3D Vision Lecture 3 - Part 1 Camera Models CS80000 Section 2 Spring 2005 Professor Zhigang Zhu, Rm 4439

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3D Computer Vision and Video Computing 3D Vision n Closely Related Disciplines l Image Processing – images to mages l Computer Graphics – models to images l Computer Vision – images to models l Photogrammetry – obtaining accurate measurements from images n What is 3-D ( three dimensional) Vision? l Motivation: making computers see (the 3D world as humans do) l Computer Vision: 2D images to 3D structure l Applications : robotics / VR /Image-based rendering/ 3D video n Lectures on 3-D Vision Fundamentals l Camera Geometric Models (this lecture and omni cameras) l Camera Calibration (1 lecture) l Stereo (1 lecture) l Motion (1 lecture)

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3D Computer Vision and Video Computing Lecture Outline n Geometric Projection of a Camera l Pinhole camera model l Perspective projection l Weak-Perspective Projection n Camera Parameters l Intrinsic Parameters: define mapping from 3D to 2D l Extrinsic parameters: define viewpoint and viewing direction n Basic Vector and Matrix Operations, Rotation n Camera Models Revisited l Linear Version of the Projection Transformation Equation n Perspective Camera Model n Weak-Perspective Camera Model n Affine Camera Model n Camera Model for Planes n Summary

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3D Computer Vision and Video Computing Lecture Assumptions n Camera Geometric Models l Knowledge about 2D and 3D geometric transformations l Linear algebra (vector, matrix) l This lecture is only about geometry n Goal Build up relation between 2D images and 3D scenes -3D Graphics (rendering): from 3D to 2D -3D Vision (stereo and motion): from 2D to 3D -Calibration: Determning the parameters for mapping

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3D Computer Vision and Video Computing Image Formation

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3D Computer Vision and Video Computing Image Formation Light (Energy) Source Surface Pinhole Lens Imaging Plane WorldOpticsSensorSignal Camera: Spec & Pose 3D Scene 2D Image

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3D Computer Vision and Video Computing Pinhole Camera Model Pin-hole is the basis for most graphics and vision Derived from physical construction of early cameras Mathematics is very straightforward n 3D World projected to 2D Image l Image inverted, size reduced l Image is a 2D plane: No direct depth information n Perspective projection l f called the focal length of the lens l given image size, change f will change FOV and figure sizes

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3D Computer Vision and Video Computing Focal Length, FOV n Consider case with object on the optical axis: f z n Optical axis: the direction of imaging n Image plane: a plane perpendicular to the optical axis n Center of Projection (pinhole), focal point, viewpoint, nodal point n Focal length: distance from focal point to the image plane n FOV : Field of View – viewing angles in horizontal and vertical directions viewpoint Image plane

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3D Computer Vision and Video Computing Focal Length, FOV n Consider case with object on the optical axis: f z Out of view Image plane n Optical axis: the direction of imaging n Image plane: a plane perpendicular to the optical axis n Center of Projection (pinhole), focal point, viewpoint,, nodal point n Focal length: distance from focal point to the image plane n FOV : Field of View – viewing angles in horizontal and vertical directions n Increasing f will enlarge figures, but decrease FOV

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3D Computer Vision and Video Computing Equivalent Geometry n Consider case with object on the optical axis: f z n More convenient with upright image: n Equivalent mathematically

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3D Computer Vision and Video Computing Perspective Projection n Compute the image coordinates of p in terms of the world (camera) coordinates of P. n Origin of camera at center of projection n Z axis along optical axis n Image Plane at Z = f; x // X and y//Y

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3D Computer Vision and Video Computing Reverse Projection n Given a center of projection and image coordinates of a point, it is not possible to recover the 3D depth of the point from a single image. In general, at least two images of the same point taken from two different locations are required to recover depth.

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam : what do you see in this picture? l straight line l size l parallelism/angle l shape l shape of planes l depth

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam straight line l size l parallelism/angle l shape l shape of planes l depth

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam straight line size l parallelism/angle l shape l shape of planes l depth

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam straight line size parallelism/angle l shape l shape of planes l depth

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam straight line size parallelism/angle shape l shape of planes l depth

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3D Computer Vision and Video Computing Pinhole camera image Photo by Robert Kosara, Amsterdam straight line size parallelism/angle shape l shape of planes parallel to image l depth

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3D Computer Vision and Video Computing Pinhole camera image - We see spatial shapes rather than individual pixels - Knowledge: top-down vision belongs to human - Stereo &Motion most successful in 3D CV & application - You can see it but you don't know how… Amsterdam: what do you see? straight line size parallelism/angle shape l shape of planes parallel to image l Depth ? l stereo l motion l size l structure …

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3D Computer Vision and Video Computing Yet other pinhole camera images Markus Raetz, Metamorphose II, , cast iron, 15 1/4 x 12 x 12 inches Fine Art Center University Gallery, Sep 15 – Oct 26 Rabbit or Man?

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3D Computer Vision and Video Computing Yet other pinhole camera images Markus Raetz, Metamorphose II, , cast iron, 15 1/4 x 12 x 12 inches Fine Art Center University Gallery, Sep 15 – Oct 26 2D projections are not the “same” as the real object as we usually see everyday!

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3D Computer Vision and Video Computing It’s real!

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3D Computer Vision and Video Computing Weak Perspective Projection n Average depth Z is much larger than the relative distance between any two scene points measured along the optical axis n A sequence of two transformations l Orthographic projection : parallel rays l Isotropic scaling : f/Z n Linear Model l Preserve angles and shapes x y Z P(X,Y,Z) p(x, y) Z = f 0 Y X

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3D Computer Vision and Video Computing Camera Parameters n Coordinate Systems l Frame coordinates (x im, y im ) pixels l Image coordinates (x,y) in mm l Camera coordinates (X,Y,Z) l World coordinates (X w,Y w,Z w ) n Camera Parameters l Intrinsic Parameters (of the camera and the frame grabber): link the frame coordinates of an image point with its corresponding camera coordinates l Extrinsic parameters: define the location and orientation of the camera coordinate system with respect to the world coordinate system ZwZw XwXw YwYw Y X Z x y O PwPw P p x im y im (x im,y im ) Pose / Camera Object / World Image frame Frame Grabber

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3D Computer Vision and Video Computing Intrinsic Parameters (I) n From image to frame l Image center l Directions of axes l Pixel size n From 3D to 2D l Perspective projection n Intrinsic Parameters l (o x,o y ) : image center (in pixels) l (s x,s y ) : effective size of the pixel (in mm) l f: focal length x im y im Pixel (x im,y im ) Y Z X x y O p (x,y,f)oxox oyoy (0,0) Size: (s x,s y )

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3D Computer Vision and Video Computing Intrinsic Parameters (II) n Lens Distortions n Modeled as simple radial distortions l r 2 = x d 2 +y d 2 l (x d, y d ) distorted points l k 1, k 2 : distortion coefficients l A model with k 2 =0 is still accurate for a CCD sensor of 500x500 with ~5 pixels distortion on the outer boundary (xd, yd)(x, y) k 1, k 2

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3D Computer Vision and Video Computing Extrinsic Parameters n From World to Camera n Extrinsic Parameters l A 3-D translation vector, T, describing the relative locations of the origins of the two coordinate systems (what’s it?) l A 3x3 rotation matrix, R, an orthogonal matrix that brings the corresponding axes of the two systems onto each other ZwZw XwXw YwYw Y X Z x y O PwPw P p x im y im (x im,y im ) T

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3D Computer Vision and Video Computing Linear Algebra: Vector and Matrix n A point as a 2D/ 3D vector l Image point: 2D vector l Scene point: 3D vector l Translation: 3D vector n Vector Operations l Addition: n Translation of a 3D vector l Dot product ( a scalar): a.b = |a||b|cos l Cross product (a vector) n Generates a new vector that is orthogonal to both of them T: Transpose a x b = (a 2 b 3 - a 3 b 2 )i + (a 3 b 1 - a 1 b 3 )j + (a 1 b 2 - a 2 b 1 )k

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3D Computer Vision and Video Computing Linear Algebra: Vector and Matrix n Rotation: 3x3 matrix l Orthogonal : n 9 elements => 3+3 constraints (orthogonal) => 2+2 constraints (unit vectors) => 3 DOF ? (degrees of freedom) l How to generate R from three angles? (next few slides) n Matrix Operations l R P w +T= ? - Points in the World are projected on three new axes (of the camera system) and translated to a new origin

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix n Rotation around the Axes l Result of three consecutive rotations around the coordinate axes n Notes: l Only three rotations l Every time around one axis l Bring corresponding axes to each other n Xw = X, Yw = Y, Zw = Z l First step (e.g.) Bring Xw to X ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the Z w Axis l Rotate in X w OY w plane l Goal: Bring X w to X l But X is not in XwOYw l Y w X X in X w OZ w ( Y w X w OZ w ) Y w in YOZ ( X YOZ) n Next time rotation around Y w ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the Z w Axis l Rotate in X w OY w plane so that l Y w X X in X w OZ w ( Y w X w OZ w ) Y w in YOZ ( X YOZ) n Z w does not change ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the Y w Axis l Rotate in X w OZ w plane so that l X w = X Z w in YOZ (& Y w in YOZ) n Y w does not change ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the Y w Axis l Rotate in X w OZ w plane so that l X w = X Z w in YOZ (& Y w in YOZ) n Y w does not change ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the X w (X) Axis l Rotate in Y w OZ w plane so that l Y w = Y, Z w = Z (& X w = X) n X w does not change ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix Rotation around the X w (X) Axis l Rotate in Y w OZ w plane so that l Y w = Y, Z w = Z (& X w = X) n X w does not change ZwZw XwXw YwYw Y X Z O

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3D Computer Vision and Video Computing Rotation: from Angles to Matrix n Rotation around the Axes l Result of three consecutive rotations around the coordinate axes n Notes: l Rotation directions The order of multiplications matters: Same R, 6 different sets of R Non-linear function of l R is orthogonal l It’s easy to compute angles from R ZwZw XwXw YwYw Y X Z O Appendix A.9 of the textbook

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3D Computer Vision and Video Computing Rotation- Axis and Angle According to Euler’s Theorem, any 3D rotation can be described by a rotating angle, , around an axis defined by an unit vector n = [n 1, n 2, n 3 ] T. n Three degrees of freedom – why? Appendix A.9 of the textbook

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3D Computer Vision and Video Computing Linear Version of Perspective Projection n World to Camera l Camera: P = (X,Y,Z) T l World: P w = (X w,Y w,Z w ) T l Transform: R, T n Camera to Image l Camera: P = (X,Y,Z) T l Image: p = (x,y) T l Not linear equations n Image to Frame l Neglecting distortion l Frame (x im, y im ) T n World to Frame l (X w,Y w,Z w ) T -> (x im, y im ) T l Effective focal lengths n f x = f/s x, f y =f/s y n Three are not independent

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3D Computer Vision and Video Computing Linear Matrix Equation of perspective projection n Projective Space l Add fourth coordinate n P w = (X w,Y w,Z w, 1 ) T l Define (x 1,x 2,x 3 ) T such that n x 1 / x 3 =x im, x 2 / x 3 =y im n 3x4 Matrix M ext l Only extrinsic parameters l World to camera n 3x3 Matrix M int l Only intrinsic parameters l Camera to frame n Simple Matrix Product! Projective Matrix M= M int M ext l (X w,Y w,Z w ) T -> (x im, y im ) T l Linear Transform from projective space to projective plane l M defined up to a scale factor – 11 independent entries

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3D Computer Vision and Video Computing Three Camera Models n Perspective Camera Model l Making some assumptions n Known center: Ox = Oy = 0 n Square pixel: Sx = Sy = 1 l 11 independent entries 7 parameters n Weak-Perspective Camera Model Average Distance Z >> Range Z l Define centroid vector P w l 8 independent entries n Affine Camera Model l Mathematical Generalization of Weak-Pers l Doesn’t correspond to physical camera l But simple equation and appealing geometry n Doesn’t preserve angle BUT parallelism l 8 independent entries

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3D Computer Vision and Video Computing Camera Models for a Plane n Planes are very common in the Man-Made World l One more constraint for all points: Zw is a function of Xw and Yw n Special case: Ground Plane l Z w =0 l P w =(X w, Y w,0, 1) T l 3D point -> 2D point n Projective Model of a Plane l 8 independent entries n General Form ? l 8 independent entries

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3D Computer Vision and Video Computing Camera Models for a Plane n A Plane in the World l One more constraint for all points: Zw is a function of Xw and Yw n Special case: Ground Plane l Z w =0 l P w =(X w, Y w,0, 1) T l 3D point -> 2D point n Projective Model of Z w =0 l 8 independent entries n General Form ? l 8 independent entries

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3D Computer Vision and Video Computing Camera Models for a Plane n A Plane in the World l One more constraint for all points: Zw is a function of Xw and Yw n Special case: Ground Plane l Z w =0 l P w =(X w, Y w,0, 1) T l 3D point -> 2D point n Projective Model of Z w =0 l 8 independent entries n General Form ? l n z = 1 l 8 independent entries n 2D (x im,y im ) -> 3D (X w, Y w, Z w ) ?

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3D Computer Vision and Video Computing Applications and Issues n Graphics /Rendering l From 3D world to 2D image n Changing viewpoints and directions n Changing focal length l Fast rendering algorithms n Vision / Reconstruction l From 2D image to 3D model n Inverse problem n Much harder / unsolved l Robust algorithms for matching and parameter estimation l Need to estimate camera parameters first n Calibration l Find intrinsic & extrinsic parameters l Given image-world point pairs l Probably a partially solved problem ? l 11 independent entries 10 parameters: f x, f y, o x, o y, , Tx,Ty,Tz

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3D Computer Vision and Video Computing 3D Reconstruction from Images (1) Panoramic texture map (2)panoramic depth map Flower Garden Sequence Vision: Camera Calibration l Motion Estimation l 3D reconstruction

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3D Computer Vision and Video Computing Image-based 3Dmodel of the FG sequence

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3D Computer Vision and Video Computing Rendering from 3D Model Graphics: Virtual Camera l Synthetic motion l From 3D to 2D image

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3D Computer Vision and Video Computing Camera Model Summary n Geometric Projection of a Camera l Pinhole camera model l Perspective projection l Weak-Perspective Projection n Camera Parameters (10 or 11) l Intrinsic Parameters: f, o x,o y, s x,s y,k 1: 4 or 5 independent parameters l Extrinsic parameters: R, T – 6 DOF (degrees of freedom) n Linear Equations of Camera Models (without distortion) l General Projection Transformation Equation : 11 parameters l Perspective Camera Model: 11 parameters l Weak-Perspective Camera Model: 8 parameters l Affine Camera Model: generalization of weak-perspective: 8 l Projective transformation of planes: 8 parameters

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3D Computer Vision and Video Computing Next n Determining the value of the extrinsic and intrinsic parameters of a camera Calibration (Ch. 6)

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