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## 3D reconstruction Class 16

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Papers

http://www.unc.edu/courses/2004fall/comp/290b/089/papers/

3D modeling

- Aligning range images
- Pairwise
- Globally
- Surface reconstruction
- Single range image
- Merged

(some slides from S. Rusinkiewicz, J. Ponce,…)

Aligning 3D Data

- If correct correspondences are known,it is possible to find correct relative rotation/translation

Intermezzo: quaternions

q is aquaternion,

a 2R is itsrealpart, and

2R3 is itsimaginarypart.

q = a +

Operations on quaternions:

- Sum of quaternions:

- Multiplication by a scalar:

- Quaternion product:

- Conjugate: )

Note:

Intermezzo: quaternions and rotations

Let R denote the rotation of angle about the unit vector u.

Define

Then for any vector ,

Reciprocally, if q = a + ( b, c, d )T is a unit quaternion, the corresponding rotation matrix is:

n

E = | xi’ – R xi – t |2 .

E = | yi’ – R yi |2

i=1

i=1

n

( xi’ – R xi – t ) .

At a minimum: 0 = E/t = –2

Or.. t = x’ – R x.

i=1

If yi = xi –x and yi’ = xi’ –x’, the error is (at a minimum):

n

n

= | yi’ q – qyi |2

= | yi’ – q yi q|2|q|2

i=1

i=1

Linear least squares !!

Estimate rigid transformation

Problem: Find the rotation matrix R and the vector t that

minimize

Or E =

Aligning 3D Data

- How to find corresponding points?
- Previous systems based on user input,feature matching, surface signatures, etc.

Aligning 3D Data

Alternative: assume closest points correspond to each other, compute the best transform…

Aligning 3D Data

… and iterate to find alignment

Iterated Closest Points (ICP) [Besl & McKay 92]

Converges if starting position “close enough“

ICP Variants

- Classic ICP algorithm not real-time
- To improve speed: examine stages of ICP and evaluate proposed variants

[Rusinkiewicz & Levoy, 3DIM 2001]

- Selecting source points (from one or both meshes)
- Matching to points in the other mesh
- Weighting the correspondences
- Rejecting certain (outlier) point pairs
- Assigning an error metric to the current transform
- Minimizing the error metric

ICP Variant – Point-to-Plane Error Metric

- Using point-to-plane distance instead of point-to-point lets flat regions slide along each other more easily [Chen & Medioni 91]

Finding Corresponding Points

- Finding closest point is most expensive stage of ICP
- Brute force search – O(n)
- Spatial data structure (e.g., k-d tree) – O(log n)
- Voxel grid – O(1), but large constant, slow preprocessing

Finding Corresponding Points

- For range images, simply project point [Blais 95]
- Constant-time, fast
- Does not require precomputing a spatial data structure

High-Speed ICP Algorithm

- ICP algorithm with projection-based correspondences, point-to-plane matchingcan align meshes in a few tens of ms.(cf. over 1 sec. with closest-point)

- [Rusinkiewicz & Levoy, 3DIM 2001]

3D Global registration

The problem:

- Given: n scans around an object
- Goal: align them all
- First attempt: ICP each scan to one other

3D Global registration

- Want method for distributing accumulated error among all scans

Approach #1: Avoid the Problem

- In some cases have 1 scan that covers large part of surface (e.g., cylindrical scan)
- Align all other scans to this “anchor”
- Disadvantage: not always practical to obtain anchor scan

Approach #2: The Greedy Solution

- Align each new scan to all previous scans
- Disadvantages:
- Order dependent
- Doesn’t spread out error

Approach #3: Brute-Force Solution

- While not converged:
- For each scan:
- For each point:
- For every other scan
- Find closest point
- Minimize error w.r.t. transforms of all scans
- Disadvantage:
- Solve (np)(np) matrix equation, where n is number of scans and p is number of points per scan

Approach #3a: Slightly Less Brute-Force

- While not converged:
- For each scan:
- For each point:
- For every other scan
- Find closest point
- Minimize error w.r.t. transform of this scan
- Faster than previous method (matrices are pp)

Graph Methods

- Many globalreg algorithms create a graph of pairwise alignments between scans

Scan 3

Scan 5

Scan 1

Scan 2

Scan 4

Scan 6

Pulli’s Algorithm

- Perform pairwise ICPs, record sample (e.g. 200) of corresponding points
- For each scan, starting w. most connected
- Align scan to existing set
- While (change in error) > threshold
- Align each scan to others
- All alignments during globalreg phase use precomputed corresponding points

Sharp et al. Algorithm

- Perform pairwise ICPs, record only optimal rotation/translation for each
- Decompose alignment graph into cycles
- While (change in error) > tolerance
- For each cycle:
- Spread out error equally among all scans in the cycle
- For each scan belonging to more than 1 cycle:
- Assign average transform to scan

Lu and Milios Algorithm

- Perform pairwise ICPs, record optimal rotation/translation and covariance for each
- Least squares simultaneous minimization of all errors (covariance-weighted)
- Requires linearization of rotations
- Worse than the ICP case, since don’t converge to (incremental rotation) = 0

Open Questions in Global Registration

- Best way to do correctly-weighted globalreg without linearizing rotations?
- How to prevent bias (if many scans in one area, few scans in another)?
- Robust outlier detection
- For graph methods, pairwise ICP often automated
- One bad ICP can throw off the entire model

Bad ICP in Globalreg

One bad ICP can throw off the entire model

Correct Globalreg

Globalreg Including

Bad ICP

Huber’s Automatic Modeling System

- Start with unaligned cans
- Generate candidate pairwise alignments using spin images
- Incrementally add scans, checking consistency
- ICP error in overlapping regions
- Free space consistency
- Occupied space consistency

Spin Images

- Johnson and Hebert
- “Signature” that captures local shape
- Similar shapes similar spin images

Computing Spin Images

- Start with a point on a 3D model
- Find (averaged) surface normal at that point
- Define coordinate system centered at this point, oriented according to surface normal and two (arbitrary) tangents
- Express other points (within some distance) in terms of the new coordinates

Computing Spin Images

- Compute histogram of locations of other points, in new coordinate system, ignoring rotation around normal:

Spin Image Parameters

- Size of neighborhood
- Determines whether local or global shapeis captured
- Big neighborhood: more discriminatory power
- Small neighborhood: resistance to clutter
- Size of bins in histogram:
- Big bins: less sensitive to noise
- Small bins: captures more detail, less storage

Using Spin Images for Alignment

Not robust

Verification – Visibility Consistency

surfaces are close

S2 blocks S1

S1 is not observed

S1

S1

S2

S2

S1

S2

C1

C1

C2

C2

C1

C2

consistentsurfaces

free spaceviolation

occupied spaceviolation

15

13

10

9

3

6

6

12

11

top view

5

5

1

8

8

10

9

13

11

15

2

3

2

4

4

7

7

1

12

14

horizontal slice

Automatically Modeling the Angelinitial model(bad matches in red)

connectivity afterlocal verification

Surface Reconstruction from Range Images

- Often an easier problem than reconstruction from arbitrary point clouds
- Implicit information about adjacency, connectivity
- Roughly uniform spacing

Surface Reconstruction From Range Images

- First, construct surface from each range image
- Then, merge resulting surfaces
- Obtain average surface in overlapping regions
- Control point density

Range Image Tesselation

- Given a range image, connect up the neighbors

Range Image Tesselation

- Caveat #1: can’t be too aggressive
- Introduce distance threshold for tesselation

Range Image Tesselation

- Caveat #2: Which way to triangulate?
- Possible heuristics:
- Shorter diagonal
- Dihedral angle closer to 180
- Maximize smallest angle in both triangles
- Always the same way (best triangle strips)

Scan Merging Using Zippering

- Turk & Levoy, 1994
- Erode geometry in overlapping areas
- Stitch scans together along seam
- Re-introduce all data
- Weighted average

Point Weighting

- Higher weights to points facing the camera
- Favor higher sampling rates

Point Weighting

- Lower weights(tapering to 0)near boundaries
- Smooth blendsbetween views

Volumetric Reconstruction

- Implicit function defined volumetrically
- Usually stored sampled on a 3D grid
- Can be compressed (e.g., using RLE)
- Another possibility: hierarchical data structures
- Can extract isosurface (i.e., subset of space where implicit function = some constant)

Volumetric Reconstruction Overview

- Generate signed distance function (or something close to it) for each scan
- Compute average (possibly weighted)
- Extract isosurface

Volumetric integration

(Curless and Levoy, Siggraph´96)

range surfaces

signed distance

to surface

volume

weight

(~accuracy)

distance

depth

sensor

surface1

- use voxel space
- new surface as zero-crossing
- (find using marching cubes)
- least-squares estimate
- (zero derivative=minimum)

surface2

combined estimate

Volumetric Reconstruction Benefits

- Always generates a manifold surface
- Can control sampling density
- Averaging of signed distance functions corresponds to averaging the surfaces

Volumetric Reconstruction Drawbacks

- Represent a 3D entity rather than 2D
- Running time
- Storage
- Resampling step – bandlimits the function
- Generates consistent topology, but not always the topology you wanted
- Problems with very thin surfaces

From volume to mesh:Marching Cubes

First 2D, Marching Squares

“Marching Cubes: A High Resolution 3D Surface Construction Algorithm”,William E. Lorensen and Harvey E. Cline,Computer Graphics (Proceedings of SIGGRAPH '87), Vol. 21, No. 4, pp. 163-169.

From volume to mesh:Marching Cubes

“Marching Cubes: A High Resolution 3D Surface Construction Algorithm”,William E. Lorensen and Harvey E. Cline,Computer Graphics (Proceedings of SIGGRAPH '87), Vol. 21, No. 4, pp. 163-169.

From volume to mesh:Marching Cubes

Improvement

“Marching Cubes: A High Resolution 3D Surface Construction Algorithm”,William E. Lorensen and Harvey E. Cline,Computer Graphics (Proceedings of SIGGRAPH '87), Vol. 21, No. 4, pp. 163-169.

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