Miniature faking In close-up photo, the depth of field is limited. http://en.wikipedia.org/wiki/File:Jodhpur_tilt_shift.jpg CS 376 Lecture 16: Stereo 1

Miniature faking

Miniature faking http://en.wikipedia.org/wiki/File:Oregon_State_Beavers_Tilt-Shift_Miniature_Greg_Keene.jpg CS 376 Lecture 16: Stereo 1

Review Previous section: Model fitting and outlier rejection

Review: Hough transform y m b x x y m 3 5 2 7 11 10 4 1 b 􀁑 Basic ideas • A line is represented as . Every line in the image corresponds to a point in the parameter space • Every point in the image domain corresponds to a line in the parameter space (why ? Fix (x,y), (m,n) can change on the line . In other words, for all the lines passing through (x,y) in the image space, their parameters form a line in the parameter space) • Points along a line in the space correspond to lines passing through the same point in the parameter space (why ?) Slide from S. Savarese

Review: RANSAC Algorithm: Sample (randomly) the number of points required to fit the model (#=2) Solve for model parameters using samples Score by the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence Let me give you an intuition of what is going on. Suppose we have the standard line fitting problem in presence of outliers. We can formulate this problem as follows: want to find the best partition of points in inlier set and outlier set such that… The objective consists of adjusting the parameters of a model function so as to best fit a data set. "best" is defined by a function f that needs to be minimized. Such that the best parameter of fitting the line give rise to a residual error lower that delta as when the sum, S, of squared residuals

Review: 2D image transformations Szeliski 2.1

What if you want to align but have no prior matched pairs? Hough transform and RANSAC not applicable Important applications Medical imaging: match brain scans or contours Robotics: match point clouds

Iterative Closest Points (ICP) Algorithm Goal: estimate transform between two dense sets of points Initialize transformation (e.g., compute difference in means and scale) Assign each point in {Set 1} to its nearest neighbor in {Set 2} Estimate transformation parameters e.g., least squares or robust least squares Transform the points in {Set 1} using estimated parameters Repeat steps 2-4 until change is very small

Example: aligning boundaries Extract edge pixels 𝑝1..𝑝𝑛 and 𝑞1..𝑞𝑚 Compute initial transformation (e.g., compute translation and scaling by center of mass, variance within each image) Get nearest neighbors: for each point 𝑝𝑖 find corresponding match(i)= argmin 𝑗 𝑑𝑖𝑠𝑡(𝑝𝑖,𝑞𝑗) Compute transformation T based on matches Warp points p according to T Repeat 3-5 until convergence q p

Example: solving for translation (tx, ty) Problem: no initial guesses for correspondence ICP solution Find nearest neighbors for each point Compute transform using matches Move points using transform Repeat steps 1-3 until convergence

Algorithm Summaries Least Squares Fit Robust Least Squares closed form solution robust to noise not robust to outliers Robust Least Squares improves robustness to outliers requires iterative optimization Hough transform robust to noise and outliers can fit multiple models only works for a few parameters (1-4 typically) RANSAC works with a moderate number of parameters (e.g, 1-8) Iterative Closest Point (ICP) For local alignment only: does not require initial correspondences

This section – multiple views Today – Intro to multiple views and Stereo Monday– Camera calibration Wednesday– Fundamental Matrix Friday– Optical Flow CS 376 Lecture 16: Stereo 1

Oriented and Translated Camera jw t kw Ow iw

Degrees of freedom 5 6 How many known points are needed to estimate this?

How to calibrate the camera?

Computer Vision James Hays Stereo: Intro Computer Vision James Hays Slides by Kristen Grauman CS 376 Lecture 16: Stereo 1

Multiple views stereo vision structure from motion optical flow Lowe Hartley and Zisserman CS 376 Lecture 16: Stereo 1

Why multiple views? Structure and depth are inherently ambiguous from single views. Images from Lana Lazebnik CS 376 Lecture 16: Stereo 1

Phil Tippett and Jon Berg stop-motion animate a shot with all three walkers (the background walkers are cutouts). Originally the plan had been to photograph the walkers against four-by-five Ektachrome transparencies shot in Norway; when these didn’t work as planned, matte artist Mike Pangrazio copied select Ektachromes onto large scenic backings.

Why multiple views? Structure and depth are inherently ambiguous from single views. P1 P2 P1’=P2’ Optical center CS 376 Lecture 16: Stereo 1

What cues help us to perceive 3d shape and depth?

Shading [Figure from Prados & Faugeras 2006] CS 376 Lecture 16: Stereo 1 24

Focus/defocus Images from same point of view, different camera parameters 3d shape / depth estimates [figs from H. Jin and P. Favaro, 2002] CS 376 Lecture 16: Stereo 1

Texture CS 376 Lecture 16: Stereo 1 26 [From A.M. Loh. The recovery of 3-D structure using visual texture patterns. PhD thesis] CS 376 Lecture 16: Stereo 1 26

Perspective effects Image credit: S. Seitz

Motion CS 376 Lecture 16: Stereo 1 Figures from L. Zhang http://www.brainconnection.com/teasers/?main=illusion/motion-shape CS 376 Lecture 16: Stereo 1

Occlusion CS 376 Lecture 16: Stereo 1 Rene Magritt'e famous painting Le Blanc-Seing (literal translation: "The Blank Signature") roughly translates as "free hand“. 1965 http://rene-magritte-paintings.blogspot.com/2009/11/le-blanc-seing.html CS 376 Lecture 16: Stereo 1

If stereo were critical for depth perception, navigation, recognition, etc., then this would be a problem

Multi-view geometry problems Structure: Given projections of the same 3D point in two or more images, compute the 3D coordinates of that point ? Structure from motion solves the following problem: Given a set of images of a static scene with 2D points in correspondence, shown here as color-coded points, find… a set of 3D points P and a rotation R and position t of the cameras that explain the observed correspondences. In other words, when we project a point into any of the cameras, the reprojection error between the projected and observed 2D points is low. This problem can be formulated as an optimization problem where we want to find the rotations R, positions t, and 3D point locations P that minimize sum of squared reprojection errors f. This is a non-linear least squares problem and can be solved with algorithms such as Levenberg-Marquart. However, because the problem is non-linear, it can be susceptible to local minima. Therefore, it’s important to initialize the parameters of the system carefully. In addition, we need to be able to deal with erroneous correspondences. Camera 1 Camera 3 Camera 2 R1,t1 R3,t3 R2,t2 Slide credit: Noah Snavely

Multi-view geometry problems Stereo correspondence: Given a point in one of the images, where could its corresponding points be in the other images? Structure from motion solves the following problem: Given a set of images of a static scene with 2D points in correspondence, shown here as color-coded points, find… a set of 3D points P and a rotation R and position t of the cameras that explain the observed correspondences. In other words, when we project a point into any of the cameras, the reprojection error between the projected and observed 2D points is low. This problem can be formulated as an optimization problem where we want to find the rotations R, positions t, and 3D point locations P that minimize sum of squared reprojection errors f. This is a non-linear least squares problem and can be solved with algorithms such as Levenberg-Marquart. However, because the problem is non-linear, it can be susceptible to local minima. Therefore, it’s important to initialize the parameters of the system carefully. In addition, we need to be able to deal with erroneous correspondences. Camera 1 Camera 3 Camera 2 R1,t1 R3,t3 R2,t2 Slide credit: Noah Snavely

Multi-view geometry problems Motion: Given a set of corresponding points in two or more images, compute the camera parameters Structure from motion solves the following problem: Given a set of images of a static scene with 2D points in correspondence, shown here as color-coded points, find… a set of 3D points P and a rotation R and position t of the cameras that explain the observed correspondences. In other words, when we project a point into any of the cameras, the reprojection error between the projected and observed 2D points is low. This problem can be formulated as an optimization problem where we want to find the rotations R, positions t, and 3D point locations P that minimize sum of squared reprojection errors f. This is a non-linear least squares problem and can be solved with algorithms such as Levenberg-Marquart. However, because the problem is non-linear, it can be susceptible to local minima. Therefore, it’s important to initialize the parameters of the system carefully. In addition, we need to be able to deal with erroneous correspondences. ? Camera 1 ? Camera 3 Camera 2 ? R1,t1 R3,t3 R2,t2 Slide credit: Noah Snavely

Human eye Rough analogy with human visual system: Pupil/Iris – control amount of light passing through lens Retina - contains sensor cells, where image is formed Fovea – highest concentration of cones Fig from Shapiro and Stockman CS 376 Lecture 16: Stereo 1

Human stereopsis: disparity Human eyes fixate on point in space – rotate so that corresponding images form in centers of fovea. CS 376 Lecture 16: Stereo 1 36

Human stereopsis: disparity Disparity occurs when eyes fixate on one object; others appear at different visual angles CS 376 Lecture 16: Stereo 1 37

Stereo photography and stereo viewers Take two pictures of the same subject from two slightly different viewpoints and display so that each eye sees only one of the images. Image from fisher-price.com Invented by Sir Charles Wheatstone, 1838 CS 376 Lecture 16: Stereo 1

http://www.johnsonshawmuseum.org CS 376 Lecture 16: Stereo 1 44

http://www.johnsonshawmuseum.org CS 376 Lecture 16: Stereo 1 45

Public Library, Stereoscopic Looking Room, Chicago, by Phillips, 1923 CS 376 Lecture 16: Stereo 1 46

http://www.well.com/~jimg/stereo/stereo_list.html CS 376 Lecture 16: Stereo 1 47

Autostereograms Exploit disparity as depth cue using single image. (Single image random dot stereogram, Single image stereogram) Images from magiceye.com CS 376 Lecture 16: Stereo 1 48

Autostereograms Images from magiceye.com CS 376 Lecture 16: Stereo 1 49

Estimating depth with stereo Stereo: shape from “motion” between two views We’ll need to consider: Info on camera pose (“calibration”) Image point correspondences scene point image plane optical center CS 376 Lecture 16: Stereo 1 50

Stereo vision Two cameras, simultaneous views Single moving camera and static scene

Camera parameters Camera frame 2 Camera frame 1 Extrinsic parameters: Camera frame 1  Camera frame 2 Intrinsic parameters: Image coordinates relative to camera  Pixel coordinates Extrinsic params: rotation matrix and translation vector Intrinsic params: focal length, pixel sizes (mm), image center point, radial distortion parameters We’ll assume for now that these parameters are given and fixed. CS 376 Lecture 16: Stereo 1 52

Geometry for a simple stereo system First, assuming parallel optical axes, known camera parameters (i.e., calibrated cameras): CS 376 Lecture 16: Stereo 1

optical center (right) optical center (left) World point Depth of p image point (left) image point (right) Focal length optical center (right) optical center (left) baseline CS 376 Lecture 16: Stereo 1 54

Geometry for a simple stereo system Assume parallel optical axes, known camera parameters (i.e., calibrated cameras). What is expression for Z? Similar triangles (pl, P, pr) and (Ol, P, Or): disparity CS 376 Lecture 16: Stereo 1 55

Depth from disparity (x´,y´)=(x+D(x,y), y) image I(x,y) Disparity map D(x,y) image I´(x´,y´) (x´,y´)=(x+D(x,y), y) So if we could find the corresponding points in two images, we could estimate relative depth… CS 376 Lecture 16: Stereo 1

Where do we need to search? To be continued… CS 376 Lecture 16: Stereo 1