Quantitative Analysis of Range Image Patches by NEB Method

In this paper we analyze sampled high dimensional data with the NEB method from a range image database. Select a large random sample of log-valued, high contrast, normalized, 8×8 range image patches from the Brown database. We make a density estimator and we establish 1-dimensional cell complexes from the range image patch data. We find topological properties of 8×8 range image patches, prove that there exist two types of subsets of 8×8 range image patches modelled as a circle. Keywords-NEB; Range Image patch; Morse theory; cell complex; k-cells


Introduction
Computational topology becomes an important method to analyze large sets of high-dimensional data for all branches of science [1], [2], [3], [4].In order to study multi-dimensional data, we usually make a series of simplicial complexes from data to produce a presentation.The constructed simplicial complexes usually compose of lots of simplexes, often they are hard to compute.The authors utilize the NEB method to construct cell complexes by density functions [5], they built models for nonlinear data sets, and effectively detected the homology of the data.
In this paper, we find a circle model for 8×8 range image patches with the methods in [5], and give some topological characteristic of 8×8 range image patches.The range image patches used in this paper are from the Brown database.

BACKGROUND
We shortly introduce three topics: CW complexes, Morse theory, and the NEB method.

CW complexes.
First, A CW complex is a type of cell complex.For k a nonnegative integer, a k-cell is the closed ball   See Figure 1 for an example and [13] for further details.

Morse theory.
The following introduction to Morse theory is informal; see [13] for thorough treatment.Suppose M is a compact manifold of dimension d and Morse function f:M→R is smooth with non-degenerate critical points m1, . .., mk∈M satisfying the index λi of critical point mi is the number of linearly independent directions around mi in which f decreases.So a minimum has index 0, a maximum has index d, and a saddle point has index between 1 and d−1.Let M α =f −1 ((-∞, α]) be the sublevel set corresponding to a∈ R. Morse theory tells us that each Mai is homotopy equivalent to a CW complex with one λi-cell for each critical point mi.In particular, Mai is homotopy equivalent to Mai-1with a single λi-cell attached.For instance, Ma1 is homotopy equivalent to a point and is obtained from Ma0, the empty set, by attaching a single 0-cell.

NEB
To find saddle points in a high-dimensional space we use the nudged elastic band method (NEB) from computational chemistry.The nudged elastic band method is used to find minimum energy paths.An initial piecewise linear path (called a band) connecting two local minima of energy E is chosen, and forces move the band towards a minimum energy path [5].
An elastic band with N+1 images can be defined by [U0, U1, ..., UN], U0 and UN are initial and final states.The N−1 middle images are modified by an optimization algorithm [10].The total force acting on each image is defined as following: the first part  is true force, and local tangent at image i.where E is the energy of the system.The nudged elastic band method applies an optimization algorithm to shift the images depending to the force in (1) for finding the minimum energy path.For more details of NEB, please refer to papers [8], [9].

DATA SET OF RANGE IMAGE PATCHES
The Brown range image database by Lee and Huang is a set of 197 range images from indoor and outdoor scenes, mostly 444 × 1440 pixels.The database can be found at the following webpage: http://www.dam.brown.edu/ptg/brid/index.html.
The operational range for the Brown scanner is typically 2-200 meters, and the distance values for the pixels are stored in units of 0.008 meters.From the Brown database we obtain a space of range image patches as in [5], see Figure2 for some examples of range image patches.
Let R be the resulting set of high-contrast, normalized, 8×8 range patches.Our data set is a random subset X⊂R of size 5,0000.

COMPUTING METHOD
In this part, the steps of calculation procedure are given.refer to the paper [5] for more details of the method.For X ⊂ R n from probability density function f: R n → [0, ∞).We get superlevel sets X α = f −1 ([α, ∞)) = {x∈R n | f(x) ≥ α}, important topological information is given by the high dense regions.We may make CW complex models Z α to estimate X α .
Only the 1-dimensional skeleton of the cell complex is given.We give a differentiable function to estimate the unknown density function.then we can get the partial maximum of the density estimate in order to make 0cells, we make original bands randomly, then discover the convergent bands through NEB [10], so we can get the 1-cells.

A. Density estimator
For a data set X⊂R n , let Φx,σ:R n →[0, ∞) be the probability density of a normal distribution centered at x ∈ X, we apply a differentiable density estimator g(y)=|X| −1 ∑x∈XΦx,σ(y) to approach the unknown density.

B. 0-cells
To find 0-cells, we randomly select an initial point y0∈X, and iteratively define a sequence {y0, y1, ...} with yn+1=m(yn), where m(y): R n →R n is the mean shift function given by the formula: The sequence {yn} converges to a local maximum of g [13].we use single-linkage clustering to cluster the convergent points, and choose the densest member from each cluster as a 0-cell.

EXPERIMENTAL RESULTS
Now we analyze subsets of X8 with two types subsets of Xn: (1) of Xn with 50000; (2) with k=200,300 and p=30.Table 1 is the detailed data of the subsets.For ,with the deviation σ =0.36, we obtain four 0-cells with densities 97.15,97.83,105.7 and 150.8 respectively, and four 1-cells with densities 84.85,88.99,94.48and 99.0, these cells make a loop.Therefore, for α = 84.85, the Z α is a circle (Figure 3).

Figure 3
and the circle Z 84.85 , projected to a plane.

Figure 4
and the circle Z 126.8 , projected to a plane.
For , we take the value of standard deviation σ = 0.34, we get four 0-cells having densities of 194.9,225.4,256.8 and 412.3, and four 1-cells having densities of 128.0,146.7,185.6 and 201.1, all these cells make a circle too (Figure 5).

Figure 5
and the circle Z 128.0 , projected to a plane.

CONCLUSIONS
We exploit the NEB technique to analyze 8×8 Range image patches, the results experimentally show that the spaces of 8×8 range image patches have different subsets modelled as a circle.We find four quarter-circular 1cells forming a loop.
We test the approach on range image data sets and find compact complexes revealing important nonlinear patterns.But It's difficult to create higher dimensional cells with this method yet.

Figure 1 A
Figure 1 A stick figure represented as a CW complex containing eight 0-cells, eight 1-cells, and one 2-cell.The 0skeleton is on the left, the 1-skeleton is in the center, and the full 2-skeleton is on the right.