Scene Modeling for a Single View
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1 on to 3D Scene Modeling for a Single View We want real 3D scene walk-throughs: rotation translation Can we do it from a single photograph? Reading: A. Criminisi, I. Reid and A. Zisserman, Single View Metrology (ICCV 99) B. Zisser And Mundy, appendix rotations with homographies Original image translation Does it work? synthetic PP PP1 Virtual camera rotations St.Petersburg photo by A. Tikhonov PP2 Yes, with planar scene (or far away) So, what can we do here? PP1 PP3 Model the scene as a set of planes! PP2 Now, just need to find the orientations of these planes. PP3 is a projection plane of both centers of projection, so we are OK!
2 Silly Euclid: Trix are for kids! Vanishing points image plane vanishing point camera center ground plane Vanishing point projection of a point at infinity Caused by ideal line Parallel lines??? Vanishing points (2D) Vanishing lines image plane vanishing point v1 v2 camera center line on ground plane Multiple Vanishing Points Any set of parallel lines on the plane define a vanishing point The union of all of these vanishing points is the horizon line also called vanishing line Note that different planes define different vanishing lines Vanishing lines Multiple Vanishing Points Any set of parallel lines on the plane define a vanishing point The union of all of these vanishing points is the horizon line also called vanishing line Note that different planes define different vanishing lines Fitting the box volume Q: What s the significance of the vanishing point location? A: It s at eye level: ray from COP to VP is perpendicular to image plane. 2
3 Comparing heights Vanishing Point Measuring height height Example of user input: vanishing point and back face of view volume are defined Example of user input: vanishing point and back face of view volume are defined High High Example of user input: vanishing point and back face of view volume are defined Example of user input: vanishing point and back face of view volume are defined Low Low
4 Comparison of how image is subdivided based on two different camera positions. You should see how moving the vanishing point corresponds to moving the eyepoint in the 3D world. Another example of user input: vanishing point and back face of view volume are defined Left High Low Another example of user input: vanishing point and back face of view volume are defined Another example of user input: vanishing point and back face of view volume are defined Left Right Another example of user input: vanishing point and back face of view volume are defined Right Comparison of two camera placements left and right. Corresponding subdivisions match view you would see if you looked down a hallway. Left Right
5 Comparing heights Perspective cues Perspective cues Fun with vanishing points Tour into the Picture (SIGGRAPH 97) Create a 3D theatre stage of five billboards Specify foreground objects through bounding polygons Use camera transformations to navigate through the scene The idea Many scenes (especially paintings), can be represented as an axis-aligned box volume (i.e. a stage) Key assumptions: All walls of volume are orthogonal view plane is parallel to back of volume up is normal to volume bottom How many vanishing points does the box have? Three, but two at infinity Single-point perspective Can use the vanishing point to fit the box to the particular Scene!
6 2D to 3D conversion 2D to 3D conversion First, we can get ratios left right top vanishing point back plane bottom Depth of the box Size of user-defined back plane must equal size of camera plane (orthogonal sides) Use top versus side ratio to determine relative left right height and width top dimensions of box Left/right and top/bot camera ratios determine part of bottom pos 3D camera placement DEMO Now, we know the 3D geometry of the box We can texture-map the box walls with texture from the image Can compute by similar triangles (CVA vs. CV A ) Need to know focal length f (or FOV) Note: can compute position on any object on the ground Simple unprojection What about things off the ground? Foreground Objects Use separate billboard for each For this to work, three separate images used: Original image. Mask to isolate desired foreground images. Background with objects removed Foreground Objects Add vertical rectangles for each foreground object Can compute 3D coordinates P0, P1 since they are on known plane. P2, P3 can be computed as before (similar triangles) 6
7 Quiz: which is 1,2,3-point perspective Image B Automatic Photo Pop-up Original Image Geometric Labels Image A Image C Fit Segments Cut and Fold Novel View Foreground DEMO Results Input Image Automatic Photo Pop-up How can we model more complex scene? Finding world coordinates (X,Y,Z) Find world coordinates (X,Y,Z) for a few points Connect the points with planes to model geometry Texture map the planes Define the ground plane (Z=0) Compute points (X,Y,0) on that plane Compute the heights Z of all other points
8 Measurements on planes Unwarp ground plane x p p Approach: unwarp, then measure What kind of warp is this? Our old friend the homography Need 4 reference points with world coordinates p = (x,y) p = (X,Y,0) Finding world coordinates (X,Y,Z) Preliminaries: projective geometry Define the ground plane (Z=0) Compute points (X,Y,0) on that plane Compute the heights Z of all other points Ames Room The projective plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection, multi-view relationships What is the geometric intuition? a point in the image is a ray in projective space Projective lines What does a line in the image correspond to in projective space? (0,0,0) z y x (x,y,1) (sx,sy,s) image plane Each point (x,y) on the plane is represented by a ray (sx,sy,s) all points on the ray are equivalent: (x, y, 1) (sx, sy, s) A line is a plane of rays through origin all rays (x,y,z) satisfying: ax + by + cz = 0 x in vector notation : 0 = [ a b c] y z l p A line is also represented as a homogeneous 3-vector l
9 Point and line duality A line l is a homogeneous 3-vector It is to every point (ray) p on the line: lp=0 Computing vanishing points V l p 1 p 2 What is the line l spanned by rays p 1 and p 2? l is to p 1 and p 2 l = p 1 p 2 l is the plane normal What is the intersection of two lines l 1 and l 2? p is to l 1 and l 2 p = l 1 l 2 Points and lines are dual in projective space given any formula, can switch the meanings of points and lines to get another formula l 1 l 2 p P t P 0 PX + tdx PX / t + DX PY + tdy PY / t + DY P + / + Z tdz PZ t DZ 1 1/ t t = Properties v = ΠP P is a point at infinity, v is its projection They depend only on line direction Parallel lines P 0 + td, P 1 + td intersect at P P t D P = 0 P + td DX DY P D Z 0 Computing vanishing points Vanishing Points and Projection Matrix V p 2 p 1 What is the line l spanned by rays p 1 and p 2? l is to p 1 and p 2 l = p 1 p 2 l is the plane normal What is the intersection of two lines l 1 and l 2? v is to l 1 and l 2 v = l 1 l 2 What is the intersection of a set of lines l 1, l i l n? M = l i l i Eigenvector of M with smalest eigenvalues is v T i Measuring height without a ruler Measuring Heights C Z ground plane Compute Z from image measurements Need more than vanishing points to do this
10 The Cross Ratio Measuring height C b t r H T (top of object) R (reference point) R T B R R B T scene cross ratio t b v r r b Z v t image cross ratio Z H = R H = R v Z B (bottom of object) ground plane scene points represented as X Y P = Z 1 image points as x p = y 1 Measuring height v z r Measuring Height v x vanishing line (horizon) v t 0 H t R H v y b 0 b Measurements Within Reference Plane
11 Assignment 4 Implement Technique in Criminisi et al. Load in an image Click on parallel lines defining X, Y, and Z directions Compute vanishing points Specify points on reference plane, ref. height Compute 3D positions of several points Create a 3D model from these points Extract texture maps using Assignment 2 warping code Output a VRML model Due: Never
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