3D Computer Vision and Video Computing 3D Vision Topic 8 of Part 2 Visual Motion (II) CSC I6716 Spring 2004 Zhigang Zhu, NAC 8/203A Cover Image/video credits: Rick Szeliski, MSR

3D Computer Vision and Video Computing Outline of Motion n Problems and Applications (Topic 8 Motion I) l The importance of visual motion l Problem Statement n The Motion Field of Rigid Motion (Topic 8 Motion I) l Basics – Notations and Equations l Three Important Special Cases: Translation, Rotation and Moving Plane l Motion Parallax n Optical Flow (Topic 8 Motion II) l Optical flow equation and the aperture problem l Estimating optical flow l 3D motion & structure from optical flow n Feature-based Approach (Topic 8 Motion II) l Two-frame algorithm l Multi-frame algorithm l Structure from motion – Factorization method (*) n Advanced Topics (Topic 8 Motion II; Part 3) l Spatio-Temporal Image and Epipolar Plane Image l Video Mosaicing and Panorama Generation l Motion-based Segmentation and Layered Representation

3D Computer Vision and Video Computing The Motion Field of Rigid Objects n Motion: l 3D Motion ( , T): n camera motion (static scene) n or single object motion n Only one rigid, relative motion between the camera and the scene l Image motion field (velocity field): n 2D vector field of velocities of the image points induced by the relative motion. n Data: Image sequence l Many frames n captured at time t=0, 1, 2, … l Basics: only two consecutive frames n We consider a reference frame and its consecutive frame n Image motion field can be viewed disparity map of the two frames captured at two consecutive camera locations ( assuming we have a moving camera) Motion Field of a Video Sequence (Translation) p O X PV f Z Y v

3D Computer Vision and Video Computing The Motion Field of Rigid Objects n Notations l P = (X,Y,Z) T : 3-D point in the camera reference frame l p = (x,y,f) T : the projection of the scene point in the pinhole camera n Relative motion between P and the camera l T= (T x,T y,T z ) T : translation component of the motion l  x  y  z   : the angular velocity – rotation angles around three axes n Note: l How to connect this with stereo geometry (with R, T)?

3D Computer Vision and Video Computing Basic Equations of Motion Field n Notes: l Take the time derivative of both sides of the projection equation l The motion field is the sum of two components n Translational part n Rotational part l Assume known intrinsic parameters Rotation part: no depth information Translation part: depth Z

3D Computer Vision and Video Computing Special Case 1: Pure Translation n Pure Translation (  =0) n Radial Motion Field (Tz <> 0) l Vanishing point p0 =(x 0, y 0 ) T : n motion direction l FOE (focus of expansion) n Vectors away from p0 if Tz < 0 l FOC (focus of contraction) n Vectors towards p0 if Tz > 0 l Depth estimation n depth inversely proportional to magnitude of motion vector v, and also proportional to distance from p to p 0 n Parallel Motion Field (Tz= 0) l Depth estimation: n depth inversely proportional to magnitude of motion vector v Tz =0

3D Computer Vision and Video Computing Special Case 2: Pure Rotation n Pure Rotation (T =0) l Does not carry 3D information n Motion Field (approximation) l Small motion l A quadratic polynomial in image coordinates (x,y,f) T n Image Transformation between two frames (accurate) l Motion can be large l Homography (3x3 matrix) for all points n Image mosaicing from a rotating camera l 360 degree panorama

3D Computer Vision and Video Computing Special Case 3: Moving Plane n Planes are common in the man-made world n Motion Field (approximation) l Given small motion l a quadratic polynomial in image n Image Transformation between two frames (accurate) l Any amount of motion (arbitrary) l Homography (3x3 matrix) for all points l See Lecture 10 n Image Mosaicing for a planar scene l Aerial image sequence l Video of blackboard Only has 8 independent parameters (write it out!)

3D Computer Vision and Video Computing Special Cases: A Summary n Pure Translation l Vanishing point and FOE (focus of expansion) l Only translation contributes to depth estimation n Pure Rotation l Does not carry 3D information l Motion field: a quadratic polynomial in image, or l Transform: Homography (3x3 matrix R) for all points l Image mosaicing from a rotating camera n Moving Plane l Motion field is a quadratic polynomial in image, or l Transform: Homography (3x3 matrix A) for all points l Image mosaicing for a planar scene

3D Computer Vision and Video Computing Motion Parallax n [Observation 1] The relative motion field of two instantaneously coincident points l Does not depend on the rotational component of motion l Points towards (away from) the vanishing point of the translation direction (the instantaneous epipole) Epipole (x 0, y 0 ) At instant t, three pairs of points happen to be coincident The difference of the motion vectors of each pair cancels the rotational components. … and the relative motion field point in ( towards or away from) the VP of the translational direction (Fig 8.5 ???) Motion Field of a Video Sequence (Translation)

3D Computer Vision and Video Computing Motion Parallax n [Observation 2] The motion field of two frames after rotation compensation l only includes the translation component l points towards (away from) the vanishing point p0 ( the instantaneous epipole) l the length of each motion vector is inversely proportional to the depth, l and also proportional to the distance from point p to the vanishing point p0 of the translation direction (if Tz <> 0) Question: how to remove rotation? n Active vision : rotation known approximately? n Rotation compensation can be done by image warping FOE p0p0 p v

3D Computer Vision and Video Computing Notion of Optical Flow n The Notion of Optical Flow l Brightness constancy equation n Under most circumstance, the apparent brightness of moving objects remain constant l Optical Flow Equation n Relation of the apparent motion with the spatial and temporal derivatives of the image brightness n Aperture problem l Only the component of the motion field in the direction of the spatial image gradient can be determined l The component in the direction perpendicular to the spatial gradient is not constrained by the optical flow equation ?

3D Computer Vision and Video Computing Estimating Optical Flow n Constant Flow Method l Assumption: the motion field is well approximated by a constant vector within any small region of the image plane l Solution: Least square of two variables (u,v) from NxN Equations – NxN (=5x5) planar patch l Condition: A T A is NOT singular (null or parallel gradients) n Weighted Least Square Method l Assumption: the motion field is approximated by a constant vector within any small region, and the error made by the approximation increases with the distance from the center where l Solution: Weighted least square of two variables (u,v) from NxN Equations – NxN patch n Affine Flow Method l Assumption: the motion field is well approximated by a affine parametric model u T = Ap T +b (a plane patch with arbitrary orientation) l Solution: Least square of 6 variables (A,b) from NxN Equations – NxN planar patch

3D Computer Vision and Video Computing Using Optical Flow n 3D motion and structure from optical flow (p ) l Input: n Intrinsic camera parameters n dense motion field (optical flow) of single rigid motion l Algorithm n ( good comprise between ease of implementation and quality of results ) n Stage 1: Translation direction n Epipole (x0, y0) through approximate motion parallax n Key: Instantaneously coincident image points n Approximation: estimating differences for ALMOST coincident image points n Stage 2: Rotation flow and Depth n Knowns: flow vector, and direction of translational component n One point, one equation (without depth)– w Least square approximation of the rotational component of flow n From motion field to depth l Output n Direction of translation (f Tx/Tz, f Ty/Tz, f) = (x0, y0, f) n Angular velocity n 3-D coordinates of scene points (up to a common unknown scale)

3D Computer Vision and Video Computing Some Details n Step 1. Get (Tx, Ty, Tz) = s (x0,y0,f) Step 2. For every point (x,y,f) with known v, get one equation about  from the motion equation (by eliminate Z since it’s different from point to point) Step 3. Get Z (up to a scale s) given T/s and  Rotation part: no depth information Translation part: depth Z

3D Computer Vision and Video Computing Feature-Based Approach n Two frame method - Feature matching l An Algorithm Based on the Constant Flow Method n Features – corners detection by observing the coefficient matrix of the spatial gradient evaluation (2x2 matrix A T A) n Iteration approach: estimation – warping – comparison n Multiple frame method - Feature tracking l Kalman Filter Algorithm n Estimating the position and uncertainty of a moving feature in the next frame n Two parts: prediction (from previous trajectory) and measurement from feature matching n Using a sparse motion field l 3D motion and structure by feature tracking over frames l Factorization method n Orthographic projection model n Feature tracking over multiple frames n SVD

3D Computer Vision and Video Computing Motion-Based Segmentation n Change Detection l Stationary camera(s), multiple moving subjects l Background modeling and updating l Background subtraction l Occlusion handling n Layered representation (I)– rotating camera l Rotating camera + Independent moving objects l Sprite - background mosaicing l Synopsis – foreground object sequences n Layered representation (II)– translating (and rotating) camera l Arbitrary camera motion l Scene segmentation into layers

3D Computer Vision and Video Computing An Example: Augmented Classroom n Scenario l Studio of the UMass Video Instruction Program n Pan/Tilt/Zoom (PTZ) camera viewing the instructor and the slide projections n manual operation by technical staff l MANIC (Jim Kurose’s group – online courses) n Multimedia Asynchronous Networked Individualized Courseware n Goal of our current research : Automated camera control & best visual presentation l Instructor tracking and extraction n Background modeling (from slide only frames) n Instructor detection and tracking ( change detection I) n Slide change detection ( change detection II) l High resolution visuals n Slide projections replaced by corresponding digital slides n Slide matching and alignment (Planar perspective mapping) l Visual Effect for better presentation n Panoramic representation (Video Registration) n Instructor Avatar ( Virtual Instructor)

3D Computer Vision and Video Computing 2D MANIC Interface Instructor Extraction and High Resolution Slides

3D Computer Vision and Video Computing Integration of Real Image and Digital Slide (1)Figure extraction from video (2)figure-slide alignment How to remove the shadow and fill the holes?

3D Computer Vision and Video Computing How to see the words through the body of the instructor?

3D Computer Vision and Video Computing A silhouette (shadow) or…

3D Computer Vision and Video Computing Or the contour, or an avatar?

3D Computer Vision and Video Computing MANIC 2.0 Interface

3D Computer Vision and Video Computing Turn 2D windows into 3D digital space Panoramic mosaic from video Synthetic projection of digital slide Slide projection in original video

3D Computer Vision and Video Computing Summary n After learning motion, you should be able to l Explain the fundamental problems of motion analysis l Understand the relation of motion and stereo l Estimate optical flow from a image sequence l Extract and track image features over time l Estimate 3D motion and structure from sparse motion field l Extract Depth from 3D ST image formation under translational motion l Know some important application of motion, such as change detection, image mosaicing and motion-based segmentation

3D Computer Vision and Video Computing Next n Advanced Topics on Stereo, Motion and Video Computing Video Mosaicing & Omnidirectional Stereo