Class Note for EECS 841 with Professor Potetz at KU 11
Class Note for EECS 841 with Professor Potetz at KU 11
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Date Created: 02/06/15
EECS 841 Computer Vision Brian Potetz Fall 2008 Lecture 20 Photometric Stereo amp ShapefromShading Suggested Reading Photometric Stereo Forsyth amp Ponce quotComputer Vision A Modern Approach Section 54 Shape from Shading Zhang Tsai Cryer and Shah quotShape from shading A surveyquot IEEE Trans Pattern Analysis and Machine Intelligence 218690 706 1999 Computing Surface Normals Suppose that points ma 2 in the 3D scene were described by z rm SI fth t tl39 39 Z i Z 139 opeo e angen Ine ax 7 AZ A 32 i q PAM 3 Tangent vector Ax 0 A2 X 1 010 Surface normal is perpendicu ar to the tangent plane 07171 X 170717qu7 1 Note that his is not a unit vector 3 Re ectance map For a xed lighting arrangement viewing direction and surface BRDF the Re ectance Map Rp q describes the pixel intensity given a particular surface normal For a surface of constant albedo and BRDF if the illumination conditions don t vary across the surface then L06 y P411067 mm y where L is pixel intensity lfalbedo varies across the surface but the BRDF is otherwise static we can Write LOW 9zyyRqx7typw7 31 Lambertian Reflectance Map Lambertian Equation L 1 cos 9 Lam bertian p 1 psp M Reflectance Map RC 1 7 1 7 7T Z1P2q21Pgqg 1 pm M c 1 p f1 pf 11 quot e V For Lambem39an surface contours of constant brightness Rq c are nested conic surface sections in the p q plane Maximum ofRCpq is at pq Sunace nuvmal mav l2 arrWhale on the sumo mthls Dune Lambertian Re ectance Map lsurhvluhtness cuntuuv e 90 f aa10 Note Rpq l5 maximum when pq p34 NonLambertian Re ectance Map Glossy surfaces TorranceSparrow re ectance model z mosei 9quot psGRpq 7 cost diffuse term specular term Diffuse peak Photometric Stereo Photometrrc Stereo What if I have different images of o The same scene From the same viewing angle Lit by different lighting conditions The BRDF is known for every point in the scene Generally the BRDF will be constant throughout the scene and o The illumination conditions are known throughout the scene Generally this means no shadows lighting directions are known and all incoming light rays are parallel a Photometric Stereo Photometric Stereo I hotometric Stereo Solving the Equations More then I hree Light Sources e Get better results by using more I gnting angles L1 81 E 5 7 L2 ee g p77 1 1 e L3 egf on T s e LN S N 3 3e 3151 L Se Nel mxax1gt SelL STL STS Discrete Approximations of Differentiation 77 ew e solve for p n as before n 0 Color Images Depth from Normals The Case of R68 Images L Assuming Orthographic Project on 7 get three eete of equetiene one per Color ehennei V V V K e ltxy y ltpltxy y qltxy y 1 7 const LR e pR 87 7 39 3e pe y 3x LG pG 977 gt LB pB 977 r az e lt G e Denvetwe of eeueeen am e 5839 o empe eeuten rst eeive for n ueng one ehenne z e Then substitute Known 1 into above equations to get ihits Di srs csi a sltx 47 1 2 s zltxy 2 Computing Surface Normals Suppose that points ac y z in the 3D scene were described by z fx y dz Slope of the tangent line p z T 87 AZ z I q l Ax l 9y Tangent vector A33 0 Az I X 1 019 Surface normal is perpendicu ar to the tangent plane 07 17 X 17 p7 q7 Note that his is not a unit vector 19 Depth from Normals Assuming Orthographic Projection Wm P7yaQ7y 1const 82 Discrete Approximations of Differentiation 82 8 a Ea Derivative of Gaussian gtllt z Finite Differences g Z z Ly zy ac Each normal gives us two linear constraints on 2 One for dzdx one for dzdy Together all constraints form a matrix equation A2 pq that is both overconstrained and underconstrained does not specify absolute depth MoorePenrose matrix pseudoinverse A A391A can also solve underconstrained linear systems For all x that satisfy Ax b A A391A x has the least magnitude Limitations Big problems Doesn t work for shiny things semitranslucent things Shadows interreflections Smaller problems Camera and lights have to be distant Calibration requirements measure light source direc ions intensities camera response function Original Images Results Shape V 22 a Fde Edlt Vlew Insert tools Desktop Window Help Dene l sewage ma sen File Edit View Insert mots Deskmu WindDW Help BET gilt 795 F U l 4 Ii E E Shallow reconstruction effect of interreflections 41x y Accurate reconstruction after removing interreflections Figure Z a File Edit View Insert Tools Desktop Window Help D h QWGD E DIE E Results Albedo No Shading Information Original Images Results Shape Results Albedo v w Iran Tunis Deskl p Wndnw Help as is DEE hiQ Wi Dl EiUE TEI Results Estimate light source directions Compute surface normals Compute albedo values Estimate depth from surface normals Relight the object with original texture and uniform albedo 1 2 3 4 5
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