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Combining Photometric and Geometric Constraints

Yael Moses

IDC, Herzliya

Joint work with Ilan Shimshoni and Michael Lindenbaum, the Technion

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Recover the 3D shape of a smooth bilaterally symmetric object from a single image.

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Shape Recovery

- Geometry: Stereo
- Photometry:
- Shape from shading
- Photometric stereo

Main problems:

Calibrations and Correspondence

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3D Shape Recovery

Photometry:

- Shape from shading
- Photometric stereo

Geometry:

- Stereo
- Structure from motion

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Photometric Stereo

- 3D shape recovery:

surface normals from two or more images taken from the same viewpoint

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Photometric Stereo

Main Limitation:

Correspondence is obtained by a fixed viewpoint

- 3D shape recovery (surface normals)

Two or more images taken from the same viewpoint

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Overview

- Combining photometric and geometric stereo:
- Symmetric surface, single image
- Non symmetric: 3 images
- Mono-Geometric stereo
- Mono-Photometric stereo
- Experimental results.

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- Smooth featureless surface
- Taken under different viewpoints
- Illuminated by different light sources

- The Problem:
- Recover the 3D shape from a set of calibrated images

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n

*

- Perspective projection

- Three or more images

- Given correspondence the normals can be computed (e.g., Lambertian, distant point light source …)

*

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Our method

Combines photometric and geometric stereo

We make use of:

- Given Correspondence:
- Can compute a normal
- Can compute the 3D point

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First Order Surface Approximation

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First Order Surface Approximation

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First Order Surface Approximation

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Basic Propagation

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Basic Propagation

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Basic method: First Order

- Given correspondence pi and L

Pand n

- Given P andn

T

- Given P, T andMi

a new correspondence qi

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Extensions

- Using more than three images
- Propagation:
- Using multi-neighbours
- Smart propagation
- Second error approximation
- Error correction:
- Based on local continuity
- Other assumptions on the surface

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Multi-neighbors Propagation

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Smart Propagation

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Using more than three images

- Reduce noise of the photometric stereo
- Avoid shadowed pixels
- Detect “bad pixels”
- Noise
- Shadows
- Violation of assumptions on the surface

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Smart Propagation

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Error correction

The compatibility of the local 3D shape can be used to correct errors of:

- Correspondence
- Camera parameters
- Illumination parameters

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Score

- Continuity:
- Shape
- Normals
- Albedo
- The consistency of 3D points locations and the computed normals:
- General case: full triangulation
- Local constraints

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Extensions

- Using more than three images
- Propagation:
- Using multi-neighbours
- Smart propagation
- Second error approximation
- Error correction:
- Based on local continuity
- Other assumptions on the surface

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Detected Correspondence

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Error correction + multi-neighbord

Multi-neighbors

Basic scheme (3 images)

Error correction no multi-neighbors

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Basic scheme

Multi-neighbors Error correction

Sec d

Ground truth

Basic scheme

Multi-neighbors approx.

Error correction

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Combining Photometry and Geometry

Yields a dense correspondence and

dense shape recovery of the object in a single path

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Assumptions

- Bilaterally Symmetric object
- Lambertian surface with constant albedo
- Orthographic projection
- Neither occlusions nor shadows
- Known “epipolar geometry”

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Geometric Stereo

- 2 different images
- Known camera parameters
- Known viewpoints
- Known correspondence

3D shape recovery

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Symmetry and Geometric Stereo

Non frontal view of a symmetric object

Two different images of the same object

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Symmetry and Geometric Stereo

Non frontal view of a symmetric object

Two different images of the same object

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Mono-Geometric Stereo

- 3D reconstruction:

given correspondence and ,

image

x object

known

z

unknown

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Photometric Stereo

- 2 images
- Lambertian reflectance
- Known illuminations
- Known correspondence

(same viewpoint)

3D shape recovery

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Symmetry and Photometric Stereo

Non-frontal illumination of

a symmetric object

Two different images of the same object

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Mono-Photometric Stereo

- 3D reconstruction given correspondence and E (up to a twofold ambiguity):

known

unknown

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Mono-Photometric Stereo

- 3D reconstruction E unknown but correspondence is given
- Frontal viewpoint with

non-frontal illumination.

- Use image first derivatives.

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Mono-Photometric Stereo Using image derivatives

- 3 global unknowns: E
- For each pair:
- 5 unknowns zxzy zxx zxy zyy
- 6 equations
- 3 pairs are sufficient

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Correspondence

- No correspondence => no stereo.
- Hard to define correspondence in images of smooth surfaces.
- Almost any correspondence is legal when:
- Only geometric constraints are considered.
- Only photometric constraints are considered.

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Combining Photometry and Geometry

- Yields a dense correspondence

(dense shape recovery of the object).

- Enables recovering of the global parameters.

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Dense Correspondence using Propagation

Assume correspondence between a pair of points, p0land p0r.

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Computing and

- Object coordinates:

Given computing and is trivial

- Moving from object to image coordinates depends on the viewing parameter

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Derivatives with respect to the object coordinates:

- Derivatives with respect to the image coordinates:

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General Idea

- Given a corresponding pair and E

n=(zx,zy,-1)T

- Given and n

cxand cy

- Given cxand cy

a new corresponding pair

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Finding Global Parameters

- Assume E and are unknown.
- Assume a pair of corresponding points is given.
- Two possibilities:
- Search for Eand directly.
- ComputeE andfrom the image second derivatives.

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Integration Constraint:Circular Tour

- All roads lead to Rome …
- Find and verify correct correspondence
- Recover global parameters, E and

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Finding Global Parameters

Consider image second derivatives

- Due to foreshortening effect:

and

- We can relate image and object derivatives by

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Testing E and : Image second derivatives

For each corresponding pair:

and

Plus 4 linear equations in 3 unknown.

Where

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Counting

- 5 unknowns for each pair:

zx zy,zxx zxy zyy

- 4 global unknowns: E,
- For each pair: 6 equations.
- For n pairs: 5n+4 unknowns

6n equations.

4 pairs are sufficient

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Recovering the Global Parameters

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Degenerate Case

- Close to frontal view: problems with geometric-stereo.

reconstruction problem

- Close to frontal illumination: problems with photometric-stereo.

correspondence problem

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Future work

- Perspective photometric stereo
- Use as a first approximation to global optimization methods
- Test on other reflection models
- Recovering of the global parameters:
- Light
- Cameras
- Detect the first pair of correspondence

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Future Work

- Extend to general 3 images under 3 viewpoints and 3 illuminations.
- Extend to non-lambertian surfaces.

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Thanks

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Integration Constraint

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Integration Constraint

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Searching for E

- Illumination must satisfy:
- E is further constrained by the image second derivatives.

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Image second derivatives

For each corresponding pair and E: 4 linear equations in 3 unknown.

Where

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Counting

- 5 unknowns for each pair: zx,zy,zxx,zxy,zyy
- 3 global unknowns: E
- For each pair: 6 equations.
- For n pairs: 5n+3 unknowns

6n equations.

3 pairs are sufficient

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Correspondence

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Variations

- Known/unknown distant light source
- Known/unknown viewpoint
- Symmetric/non-symmetric image
- Frontal/non-frontal viewpoint
- Frontal/non-frontal illumination

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Correspondence

- Epipolar geometry is the only geometric constraint on the correspondence.
- Weak photometric constraint on the correspondence.

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