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Computer Vision Spring 2006 15-385,-685 Instructor: S. Narasimhan Wean 5403 T-R 3:00pm – 4:20pm Lecture #12

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Midterm – March 9 Syllabus – until and including Lightness and Retinex Closed book, closed notes exam in class. Time: 3:00pm – 4:20pm Midterm review class next Tuesday (March 7) (Email me by March 6 specific questions) If you have read the notes and readings, attended all classes, done assignments well, it should be a walk in the park

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Mechanisms of Reflection source surface reflection surface incident direction body reflection Body Reflection: Diffuse Reflection Matte Appearance Non-Homogeneous Medium Clay, paper, etc Surface Reflection: Specular Reflection Glossy Appearance Highlights Dominant for Metals Image Intensity = Body Reflection + Surface Reflection

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Example Surfaces Body Reflection: Diffuse Reflection Matte Appearance Non-Homogeneous Medium Clay, paper, etc Surface Reflection: Specular Reflection Glossy Appearance Highlights Dominant for Metals Many materials exhibit both Reflections:

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Diffuse Reflection and Lambertian BRDF viewing direction surface element normal incident direction Lambertian BRDF is simply a constant : albedo Surface appears equally bright from ALL directions! (independent of ) Surface Radiance : Commonly used in Vision and Graphics! source intensity source intensity I

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Diffuse Reflection and Lambertian BRDF

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White-out: Snow and Overcast Skies CAN’T perceive the shape of the snow covered terrain! CAN perceive shape in regions lit by the street lamp!! WHY?

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Diffuse Reflection from Uniform Sky Assume Lambertian Surface with Albedo = 1 (no absorption) Assume Sky radiance is constant Substituting in above Equation: Radiance of any patch is the same as Sky radiance !! (white-out condition)

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Specular Reflection and Mirror BRDF source intensity I viewing direction surface element normal incident direction specular/mirror direction Mirror BRDF is simply a double-delta function : Valid for very smooth surfaces. All incident light energy reflected in a SINGLE direction (only when = ). Surface Radiance : specular albedo

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Combing Specular and Diffuse: Dichromatic Reflection Observed Image Color = a x Body Color + b x Specular Reflection Color R G B Klinker-Shafer-Kanade 1988 Color of Source (Specular reflection) Color of Surface (Diffuse/Body Reflection) Does not specify any specific model for Diffuse/specular reflection

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Diffuse and Specular Reflection diffusespeculardiffuse+specular

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Photometric Stereo Lecture #12

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Image Intensity and 3D Geometry Shading as a cue for shape reconstruction What is the relation between intensity and shape? –Reflectance Map

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Surface Normal surface normal Equation of plane or Let Surface normal

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Surface Normal

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Gradient Space Normal vector Source vector plane is called the Gradient Space (pq plane) Every point on it corresponds to a particular surface orientation

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Reflectance Map Relates image irradiance I(x,y) to surface orientation (p,q) for given source direction and surface reflectance Lambertian case: : source brightness : surface albedo (reflectance) : constant (optical system) Image irradiance: Letthen

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Lambertian case Reflectance Map (Lambertian) cone of constant Iso-brightness contour Reflectance Map

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Lambertian case iso-brightness contour Note: is maximum when Reflectance Map

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Glossy surfaces (Torrance-Sparrow reflectance model) diffuse termspecular term Diffuse peak Specular peak Reflectance Map

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Shape from a Single Image? Given a single image of an object with known surface reflectance taken under a known light source, can we recover the shape of the object? Given R(p,q) ( (p S,q S ) and surface reflectance) can we determine (p,q) uniquely for each image point? NO

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Solution Take more images –Photometric stereo Add more constraints –Shape-from-shading (next class)

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Photometric Stereo

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We can write this in matrix form: Image irradiance: Lambertian case: Photometric Stereo

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Solving the Equations inverse

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More than Three Light Sources Get better results by using more lights Least squares solution: Solve for as before Moore-Penrose pseudo inverse

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Color Images The case of RGB images –get three sets of equations, one per color channel: –Simple solution: first solve for using one channel –Then substitute known into above equations to get –Or combine three channels and solve for

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Computing light source directions Trick: place a chrome sphere in the scene –the location of the highlight tells you the source direction

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For a perfect mirror, light is reflected about N Specular Reflection - Recap We see a highlight when Then is given as follows:

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Computing the Light Source Direction Can compute N by studying this figure –Hints: use this equation: can measure c, h, and r in the image N rNrN C H c h Chrome sphere that has a highlight at position h in the image image plane sphere in 3D

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Depth from Normals Get a similar equation for V 2 –Each normal gives us two linear constraints on z –compute z values by solving a matrix equation V1V1 V2V2 N

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Limitations Big problems –Doesn’t work for shiny things, semi-translucent things –Shadows, inter-reflections Smaller problems –Camera and lights have to be distant –Calibration requirements measure light source directions, intensities camera response function

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Trick for Handling Shadows Weight each equation by the pixel brightness: Gives weighted least-squares matrix equation: Solve for as before

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Original Images

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Results - Shape Shallow reconstruction (effect of interreflections) Accurate reconstruction (after removing interreflections)

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Results - Albedo No Shading Information

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Original Images

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Results - Shape

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Results - Albedo

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Results 1.Estimate light source directions 2.Compute surface normals 3.Compute albedo values 4.Estimate depth from surface normals 5.Relight the object (with original texture and uniform albedo)

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Next Class Shape from Shading Reading: Horn, Chapter 11.

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