Shape from Shading. Computer Vision CS635 Dr. Sukhendu Das, Dept. of Computer Science & Engg.

Transcription

1 Shape from Shading Computer Viion CS635 Dr. Sukhendu Da Dept. of Computer Science & Engg.

2 Introduction An image i eentiall D where a the world i 3D The human viual tem recover hape of object in a 3D cene from a D image b a number of cue Motion paralla Binocular diparit But even a ingle image give a lot of information about hape of an object. Where i the hidden information? Some eample to illutrate the point previoul mentioned

3 An anwer? Ye ou are right.. It hading on the urface that give the depth information and hence a cue to hape of the urface Our viual tem trie to interpret the brightne pattern on the retina a hading due to patial fluctuation of the urface orientation and patial variation in reflecting propertie of the urface.

4 The Reflectance map Reflected Light Camera I v Surface Normal n e i z Surface S ka Incident Light Image intenit can be related to that in the cene b thi equation z φ n v S urface z Point on the urface n urface normal ource direction v viewing direction i incident angle e emergent angle Point on the image plane az Incident brightne at each point on the 3D cene Φnv the reflecting propertie of the urface in cene

5 Reflectance function Ambient light: Ik i where i indee into the object in the cene. Diffued reflection Specular reflection φ n Phong model v 0 φ n v ρ co v n ele φ n v ρ co i ρ Some eample to follow com i d

6 Camera Reflected Light R v d Surface Normal i e n i z Surface S

7 REFLECTANCE MODELS LAMBERTIAN MODEL φ n v ρ co i PHONG MODEL φ n v ρ co i ρ Diffue albedo m co Specular albedo d albedo ρ 0.3 ρ 0.7m ρ 0.7 ρ 0.3 m0.5

8 Reflectance map Contd.. Auming light ource i at a ditance incident light at ever point i aumed to be contant a I kaφ n v Reflectance at each point on urface depend on the urface propertie and hence varie with a function Ønv which i directl proportional to the image intenit I Since and v are contant Ønv i dependent on n alone Surface Normal n can be repreented in gradient pace p-q pace ielding Rpq called the Reflectance map. What i the gradient pace repreentation?. What are the other wa to repreent the urface orientation.

9 Repreentation of urface orientation Surface normal : n n n n 3 Surface gradient: p-q pace Given the equation of a urface in 3D world a : zf The urface gradient i defined a z z p z q z Slant and the Tilt angle σ τ σ i the angle made b the urface normal with z ai 3D world τ i the angle made b the projection of the normal on the image plane with the ai of image plane

10 Slant and the tilt angle

11 Slant and the tilt angle

12 Slant and the tilt angle

13 Gradient pace repreentation A plane parallel to - plane will have the gradient 0 in both and direction From p and q the equation of a plane can be recovered a z z p q c q p 4 3

14 Reflectance map Contd.. The image irradiance can be related to the cene irradiance I R nˆ Since the urface normal can be repreented uing the gradient pace repreentation I Rp q Rpq i called the reflectance map of the image Our aim in the hape from hading problem i to recover the orientation p q of the urface or urface patch given the image I

15 The hape from hading problem Each point in the image ha onl one attribute the intenit and the urface orientation i defined b p q. I it poible to recover thi from a ingle image? Ye Provided We add ome contraint on the object urface Homogeneit aumption Prior knowledge about the hape of the urface If homogeneit aumption i violated the hape perceived i quiet different from the one that actuall i. e.g:- make up

16 The hape from hading problem Contd To formulate the hape from hading problem iue need to be olved Poition of a point in the image with repect to it poition in the 3D Scene Projective Geometr i the anwer What determine the brightne of each point on the urface Reflecting propertie of the urface BRDF Illumination model ued

17 An eample An eample Lambertian diffued urface p q vector normal to the urface p q vector in the direction of ource co q p q p qq pp i co q p q p qq pp i q p R ρ ρ q p q p qq pp c A contour in p-q pace of contant intenit c R i given b: Check two cae: c 0.

18 c0 line c point ck parabola 0 < c < k hperbola k < c < ellipe p 0 q 0. Circle are plotted for different value of c p - q -. p 0.7 q 0. Contour are plotted for different value of c.

19 Hence for each intenit value and for each ource direction we have a contour on which our orientation could lie. But a contour or a curve doe not give a unique value of: p q? what do we do? One olution i to have more than one image. Photometric tereo Add contraint to the Object urface on the cene. Parallel line Teture element on the urface and it variation in the projected image

20 Photometric tereo Ue more than one image. Find the contour or curve for each one. The interection of curve give uch poible point Interection of 3 or more curve will give one unique value for p q R pq R 3 pq R pq

21 Photometric tereo Photometric tereo Mathematical formulation Mathematical formulation So we have 3 et of light ource direction reulting image E E E 3 reulting reflectance map R pq R pq R 3 pq. co n E k ρ n n n E E E ρ S E ρ E S n ρ

22 Adding Geometric Contraint Adding Geometric Contraint to the cene to the cene z 3D world coordinate Image point f Perpective projection If we know m contraint relating n point in the cene then we have the following et of imultaneou equation n m n n h h h &.. n n f f f Solve to get equation relating z to equation of the curve.

23 An eample Projection of parallel line in the cene meet at a vanihing point in the image. An pair of parallel line meet on the vanihing point. Line OP i termed a the vanihing line. P -f p q P l l P z O

24 Generating the contraint Generating the contraint P and P are both vanihing point and OP and OP are perpendicular to the urface normal 0 0 f q p f q p

25 Generating the contraint Generating the contraint f f f f >3 >8 contraint to find unknown Conider thee et of parallel line

26 End of lecture on Shape from Shading Slide courte: Shivani G. Rao