Computer Vision

Spring 2010 15-385,-685

Instructor: S. Narasimhan

PH A18B

T-R 10:30am – 11:50am

Lecture #13

Mechanisms of Reflectionsource

surfacereflection

surface

incidentdirection

bodyreflection

• Body Reflection:

Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc

• Surface Reflection:

Specular ReflectionGlossy AppearanceHighlightsDominant for Metals

Image Intensity = Body Reflection + Surface Reflection

Example Surfaces

Body Reflection:

Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc

Surface Reflection:

Specular ReflectionGlossy AppearanceHighlightsDominant for Metals

Many materials exhibitboth Reflections:

Diffuse Reflection and Lambertian BRDF

Diffuse Reflection and Lambertian BRDF

viewingdirection

surfaceelement

normalincidentdirection

in

v

s

d

rriif ),;,(• Lambertian BRDF is simply a constant :

albedo

• Surface appears equally bright from ALL directions! (independent of )

• Surface Radiance :

v

• Commonly used in Vision and Graphics!

snIIL di

d .cos

source intensity

source intensity I

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?

Specular Reflection and Mirror BRDFsource intensity I

viewingdirectionsurface

element

normal

incidentdirection n

v

s

rspecular/mirror direction

),( ii ),( vv

),( rr

• 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 : )()( vivisIL

v r

)()(),;,( vivisvviif specular albedo

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 forDiffuse/specular reflection

Diffuse and Specular Reflection

diffuse specular diffuse+specular

Photometric Stereo

Lecture #9

Image Intensity and 3D Geometry

• Shading as a cue for shape reconstruction• What is the relation between intensity and shape?

– Reflectance Map

Surface Normal

surface normal Equation of plane

or

Let

Surface normal

Gradient Space

Normal vector

Source vector

plane is called the Gradient Space (pq plane)

• Every point on it corresponds to a particular surface orientation

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:

Let then

• Lambertian case

Reflectance Map(Lambertian)

cone of constant

Iso-brightness contour

Reflectance Map

• Lambertian caseiso-brightnesscontour

Note: is maximum when

Reflectance Map

• Glossy surfaces (Torrance-Sparrow reflectance model)

diffuse term specular term

Diffuse peak

Specular peak

Reflectance Map

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) ( (pS,qS) and surface reflectance) can we determine (p,q) uniquely for each image point?

NO

Solution

• Take more images– Photometric stereo

• Add more constraints– Shape-from-shading (next class)

Photometric Stereo

• We can write this in matrix form:

Image irradiance:

Lambertian case:

Photometric Stereo

Solving the Equations

inverse

More than Three Light Sources

• Get better results by using more lights

• Least squares solution:

• Solve for as beforeMoore-Penrose pseudo inverse

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

Computing light source directions

• Trick: place a chrome sphere in the scene

– the location of the highlight tells you the source direction

• For a perfect mirror, light is reflected about N

Specular Reflection - Recap

• We see a highlight when • Then is given as follows:

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

rN

C

H

c

h

Chrome sphere that has a highlight at position h in the image

image plane

sphere in 3D

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

Trick for Handling Shadows

• Weight each equation by the pixel brightness:

• Gives weighted least-squares matrix equation:

• Solve for as before

Results: Lambertian Sphere

Input Images

Estimated AlbedoEstimated Surface Normals

Needles are projectionsof surface normals on

image plane

Lambertain Mask

Results – Albedo and Surface Normal

Results: Lambertian Toy

Input Images

Estimated Surface Normals Estimated Albedo

I.2

Depth from Normals

• Get a similar equation for V2

– Each normal gives us two linear constraints on z

– compute z values by solving a matrix equation

V1

V2

N

Results – Shape of Mask

Results

1. Estimate light source directions2. Compute surface normals3. Compute albedo values4. Estimate depth from surface normals5. Relight the object (with original texture and uniform albedo)

Original Images

Results - Albedo

No Shading Information

Results - Shape

Shallow reconstruction (effect of interreflections)

Accurate reconstruction (after removing interreflections)

Next Class

• Shape from Shading

• Reading: Horn, Chapter 11.