Recognizing Objects in Range Data Using Regional Point Descriptors

A. Frome, D. Huber, R. Kolluri, T. Bulow, and J. Malik. Proceedings of the European Conference on Computer Vision, May, 2004.

a.k.a. 3D Shape Contexts

Talk prepared by Nat Duca, duca@jhu.edu

Motivation

Find instances of known shapes in 2.5D range scans

Image source: Frome04

2D Shape Contexts

Take a random point on the shape

Image source: Belongie02

2D Shape Contexts

Compute the offset vectors to all other samples

2D Shape Contexts

Histogram the vectors against sectors and shells

Perform this for a large sampling of points

Extension to 3D

Step 1: pick random points on surface

Image source: Koertgen03

Extension to 3D

For each point, compute and histogram offsets

Image source: Koertgen03

Extension to 3D

For each point, compute offsets

Image source: Koertgen03

Extension to 3D

Now we histogram the offset vectors. The 3D histogram of looks like:

Image source: Frome04

Extension to 3D

Shells are spaced logarithmically apart Histogram votes are weighted by the volume of the bin Some Ln difference of the histogram vector can be

used to compare two contextsImage source: Frome04, Koertgen03

Challenges

1. How do we orient the histogram “spheres”

2. How do we compute distance between a model and one of its subsets?

3. Speed

Initial histogram orientation

Align the object’s north-pole to the surface normal

Problems:1. One degree of freedom remains2. Histogram values depend on the

precision of the surface normals

The paper solves both problems using:

– Brute force rotation– spherical harmonics

Harmonic shape context

Each shell’s histogram is a spherical function Convert each shell to a harmonic representation and

store the amplitude coefficients only

Initial histogram placement doesn’t matter, Noise in surface normals doesn’t affect descriptor

Image source: Weisstein04

the big picture: Partial Shape Matching

For a query shape Sq and a stored model Si, their nearness is defined as:

A shape context placed randomly on the query surface Sq

A precomputed shape context for model Si query surface

Experiment 1: resilience to noise

(a) model with 5cm gaussian noise (b) model with 10cm gaussian noise (c) reference (databased) model

Image source: Frome04

Experiment 2: partial matching

Input:

or

Output:

Image source: Frome04

Evaluating the results

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor approach

Is their presentation fair?

Results for noise

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor

Is their presentation fair?

Comments: Recognition rate: across 100

trials, how many times did we get the correct answer back the first time?

All three techniques are equivalent in absence of noise

Image source: Frome04

Results for 5cm noise

Results for noise

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor

Is their presentation fair?

Comments: Why is the harmonic

approach doing worse? We expect it to be doing as well or better than the basic approach

Image source: Frome04

10cm noise, 55cm normal window

Results for noise

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor

Is their presentation fair?

Comments: Notice how, when the

normals are better filtered, the harmonics do better! How can this be so?

Image source: Frome04

10cm noise, 105cm normal window

Results for partial matching

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor

Is their presentation fair?

Comments: Rank depth of R means that

the correct answer appeared in the top R results.

Clearly, the harmonics are throwing away too much

Or is the fact that the shells are rotationally independent to blame?

Image source: Frome04

View 1

Results for partial matching

Where does the blame lie:– Spherical histogram– Harmonics representation– Point choice– Representative descriptor

Is their presentation fair?

Comments: Rank depth of R means that

the correct answer appeared in the top R results.

The authors claim that the ground is setting off the match

Image source: Frome04

View 2

Speed considerations

We use a spherical hash with J sectors, and KxL latitudinal and longitudinal divisions

The basic vector is (roughly) J x K x L in size The harmonic representation is roughly the same size Without harmonics, they must store L extra rotations in

order: J x K x L2

They use Locality Sensitive Hashing to reduce the amount of effort required here:

Speed considerations: LSH results

Image source: Frome04

Without hashing

Summary

What was introduced:– 3D histogram extension of 2D shape contexts– A poorly-performing spherical harmonic decomposition of the

3D histogram– The representative decriptor method works pretty well

What would have been nice:– Precision of query when the shells are logarithmically or

linearly separated– Is the representative descriptor approach the limiting factor?

We need more data to confirm or deny!

Image sources

Frome04: A. Frome, D. Huber, R. Kolluri, T. Bulow, and J. Malik. Proceedings of the European Conference on Computer Vision, May, 2004

Belongie02: S. Belongie et al. Shape matching and object recognition using shape contexts. IEEE Trans on Pattern Analysis and Machine Intelligence. 24(4):509-522, April 2002.

Koertgen03: M. Körtgen, G.-J. Park, M. Novotni, R. Klein "3D Shape Matching with 3D Shape Contexts", in proceedings of The 7th Central European Seminar on Computer Graphics, April 2003

Weisstein: Eric W. Weisstein. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SphericalHarmonic.html