Photometric Stereo, Shape from Shading SfS Chapter Szelisky
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1 Photometric Stereo, Shape from Shading SfS Chapter Szelisky Guido Gerig CS 6320, Spring 2012 Credits: M. Pollefey UNC CS256, Ohad Ben-Shahar CS BGU, Wolff JUN (
2 Photometric Stereo Depth from Shading? First step: Surface Normals from Shading Second step: Re-integration of surface from Normals
3 Examples Simulated voyage over the surface of Neptune's large moon Triton
4 Credit: Ohad Ben-Shahar CS BGU
5 Shape from Shading
6 Photometric Stereo Assume: a local shading model a set of point sources that are infinitely distant a set of pictures of an object, obtained in exactly the same camera/object configuration but using different sources A Lambertian object (or the specular component has been identified and removed)
7 Setting for Photometric Stereo Multiple images with different lighting (vs binocular/geometric stereo)
8 Goal: 3D from One View and multiple Source positions Input images Usable Data Mask
9 Scene Results Needle Diagram: Surface Normals Albedo Re-lit:
10 Projection model for surface recovery - usually called a Monge patch
11 Lambertian Reflectance Map LAMBERTIAN MODEL X E = r <n,n s > =r COS q Y Z (p,q,-1) q (ps,qs,-1) COSq 1 p 1 2 q pp 2 L qq 1 p L 2 L q 2 L
12 REFLECTANCE MAP IS A VIEWER-CENTERED REPRESENTATION OF REFLECTANCE (f x, f y, -1) = (0,1,f ) x (1,0,f ) y z=f(x,y) Surface Orientation (0,1,f ) x Z (1,0,f y ) (-f x, -f y, 1) Depth X Y dy dx IMAGE PLANE Wolff, November 4,
13 REFLECTANCE MAP IS A VIEWER-CENTERED REPRESENTATION OF REFLECTANCE (-f x, -f y, 1) = (-p, -q, 1) p, q comprise a gradient or gradient space representation for local surface orientation. Reflectance map expresses the reflectance of a material directly in terms of viewer-centered representation of local surface orientation. Wolff, November 4,
14 Reflectance Map (ps=0, qs=0)
15 Reflectance Map Credit: Ohad Ben-Shahar CS BGU
16 Reflectance Map Credit: Ohad Ben-Shahar CS BGU
17 Reflectance Map
18 p q Reflectance Map (General) z x z y p s, q s isophote
19 Reflectance Map p s, q s Isophote I Given Intensity I in image, there are multiple (p,q) combinations (= surface orientations). z p x z q y Use multiple images with different light source directions.
20 Multiple Images = Multiple Maps Can isolate p, q as contour intersection I 2 I 1 (p,q) I 1 I 2
21 Example: Two Views Still not unique for certain intensity pairs.
22 Constant Albedo Solve linear equation system to calculate N.
23 Varying Albedo Solution Forsyth & Ponce: Out of shadow: For each point source, we know the source vector (by assumption). We assume we know the scaling constant of the linear camera (k). Fold the normal (N) and the reflectance (ρ(x,y)) into one vector g, and the scaling constant and source vector into another V j. In shadow: I x, y = 0
24 Multiple Images: Linear Least Squares Approach Combine albedo and normal Separate lighting parameters More than 3 images => overdetermined system : g (x,y)=v 1 i(x,y) How to calculate albedo ρ and N? g x, y = ρ x, y N(x,y) N= g g, ρ x, y = g
25 Example LLS Input Problem: Some regions in some images are in the shadow (no image intensity).
26 Dealing with Shadows (Missing Info) For each point source, we know the source vector (by assumption). We assume we know the scaling constant of the linear camera. Fold the normal and the reflectance into one vector g, and the scaling constant and source vector into another Vj Out of shadow: I j (x, y) kb(x, y) In shadow: I j (x,y) 0 kr(x, y) N(x, y)s j g(x,y) V j No partial shadow
27 Matrix Trick for Complete Shadows Matrix from Image Vector: Multiply LHS and RHS with diag matrix I 2 1 (x,y) I 2 2 (x,y).. I 2 n (x,y) I 1 (x,y) I 2 (x,y) I n (x,y) V 1 T V 2 T.. V n T g(x,y) Known Known Known Unknown Relevant elements of the left vector and the matrix are zero at points that are in shadow.
28 Obtaining Normal and Albedo Given sufficient sources, we can solve the previous equation (most likely need a least squares solution) for g(x, y). Recall that N(x, y) is the unit normal. This means that rx,y) is the magnitude of g(x, y). This yields a check If the magnitude of g(x, y) is greater than 1, there s a problem. And N(x, y) = g(x, y) / rx,y).
29 Example LLS Input
30 Example LLS Result Reflectance / albedo:
31 Recap Obtain normal / orientation, no depth Surface Orientation Depth? IMAGE PLANE
32 Goal Shape as surface with depth and normal
33 Recovering a surface from normals - 1 Recall the surface is written as (x,y, f (x, y)) This means the normal has the form: N(x,y) f 1 x f 2 x f 2 f y y 1 1 If we write the known vector g as g(x,y) g 1 (x, y) g 2 (x, y) g 3 (x, y) Then we obtain values for the partial derivatives of the surface: f x (x,y) g 1 (x, y) g 3 (x, y) f y (x, y) g 2 (x,y) g 3 (x,y)
34 Recovering a surface from normals - 2 Recall that mixed second partials are equal --- this gives us an integrability check. We must have: g 1 (x, y) g 3 (x, y) y g 2 (x, y) g 3 (x, y) x We can now recover the surface height at any point by integration along some path, e.g. f (x, y) y 0 x 0 f x (s, y)ds f y (x,t)dt c
35 Height Map from Integration How to integrate?
36 Possible Solutions Engineering approach: Path integration (Forsyth & Ponce) In general: Calculus of Variation Approaches Horn: Characteristic Strip Method Kimmel, Siddiqi, Kimia, Bruckstein: Level set method Many others.
37 Shape by Integation (Forsyth&Ponce) The partial derivative gives the change in surface height with a small step in either the x or the y direction We can get the surface by summing these changes in height along some path.
38 Simple Algorithm Forsyth & Ponce Problem: Noise and numerical (in)accuracy are added up and result in distorted surface. Solution: Choose several different integration paths, and build average height map.
39 Mathematical Property: Integrability Smooth, C2 continuous surface: Z(x, y) xy =Z(x, y) yx p y = q x check if ( p y q x )2 is small
40 SHAPE FROM SHADING (Calculus of Variations Approach) First Attempt: Minimize error in agreement with Image Irradiance Equation over the region of interest: object 2 ( I( x, y) R( p, q)) dxdy Wolff, November 4, 1998
41 SHAPE FROM SHADING (Calculus of Variations Approach) Better Attempt: Regularize the Minimization of error in agreement with Image Irradiance Equation over the region of interest: px py q x q y ( I( x, y) R( p, q)) dxdy object Wolff, November 4, 1998
42 Horn: Characteristic Strip Method Small step in x,y change in depth: Horn, Chapter11, pp New values of p,q at this new point (x,y): (r, s, t: second partial derivatives of z(x,y) w.r.t. x and y) Hessian: curv. of surface r p x s p y q x t q y
43 Horn: Characteristic Strip Method Irradiance Equation, Reflectance Map: Horn, Chapter11, pp Derivatives (chain rule): Relationship between gradient in the image and gradient in the reflectance map
44 Horn: Characteristic Strip Method 2 Equations for 3 unknowns (r,s,t): We can t continue in artibrary direction. Horn, Chapter11, pp Trick: Specially chosen direction Step in image E(x,y) parallel to gradient in R
45 Horn: Characteristic Strip Method 2 Equations for 3 unknowns (r,s,t): We can t continue in artibrary direction. Horn, Chapter11, pp Trick: Specially chosen direction Solving for new values for p,q: Step in image E(x,y) parallel to gradient in R Change in (p,q) can be computed via gradient of image
46 Horn: Characteristic Strip Method
47 Horn: Characteristic Strip Method Solution of differential equations: Curve on surface
48 Horn: Characteristic Strip Method
49 Horn: Characteristic Strip Method Horn, Chapter11, pp
50 Another Solution to SFS: Kimmel, Siddiqi, Kimia, Bruckstein pdf document x
51 Kimmel, Siddiqi, Kimia, Bruckstein
52 Kimmel, Siddiqi, Kimia, Bruckstein
53 Kimmel, Siddiqi, Kimia, Bruckstein
54 Application Area: Geography
55 Application: Braille Code pdf document
56 Mars Rover Heads to a New Crater NYT Sept 22, 2008
57 Limitations Controlled lighting environment Specular highlights? Partial shadows? Complex interrreflections? Fixed camera Moving camera? Multiple cameras? => Another approach: binocular / geometric stereo
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