Singularities in a Tensor Field

Singularities in a tensor field are locations where the tensor has equal eigenvalues. Equivalently, singularities in a field of orientations are the points where the orien- tation is undefined.

If the smoothing parameter h is sufficiently small, many singularities appear due to the variation in initial tensor estimates of the local orientations. Compare Fig- ure 4.7 and Figure 4.8, the fields of orientations from tensor fields with smoothing parameters 10 and 60 respectively. The first has many singularities whereas the larger smoothing parameter of Figure 4.8 has ‘smoothed out’ the anomalous singu- larities, leaving the key singularity which defines the overall shape of the field of orientations.

Assume for now that there exists an underlying field of orientations that follows the true ridge-line orientation and which we are trying to estimate. The underlying orientation field may have singularities; in the case of the fingerprint in Figure 4.1, there appears to be one central singularity at the loop of the fingerprint. Applying a

Figure 4.8: Principal eigenvector field of the tensor field empirically estimated from the extracted pore data from fingerprint a005-05 (from the NIST database Watson, 2001) calculated with h = 60. Lines indicate the orientations of principal eigen- vectors over a regular grid. The circled point denotes a singularity in the tensor field.

kernel smoothing to the initial tensors causes a bias on the location of the singular- ities. This bias is evident in Figure 4.8 where the central singularity is significantly displaced.

Through comparison with the application of kernel smoothing to a tensor field over the window W, we estimate the extent of the bias arising from applying kernel smoothing to a set of tensors sparsely located across the windowW.

We begin with an arch-model tensor field based on fingerprint a002-05 (Watson, 2001) in Figure 4.1, and show in the subsequent section how a similar model, the parabola, gives very different results.

Estimation of Singularity Location Bias: Basic Arch Model

Theorem 1 (Singularity Location Bias: Arch Model). Let T₀arch : R2 → [0, π) be

a tensor field with constant eigenvalues λ1 > λ2 > 0, and principal eigenvectors

Figure 4.9: A basic fingerprint structure of concentric arches (not to be confused with the arch fingerprint pattern). The singularity is located at the origin with concentric circles above the horizontal axis and parallel vertical lines below.

coordinate system (x1, x2)∈R2 with origin(0,0)at the singularity, eigenvectors are

tangent to a circle centred at (0,0) if x2 >0 and equal to (0,1) if x2 ≤ 0. Denote

by T_harch the result of applying a kernel smoothing in the log-Euclidean metric to

Tarch

0 , using a Gaussian kernel with parameter h. Then Tharch has a singularity at

(x1, x2) = (0, hc). The constant c is the solution of

Z c 0   Z 3₂π−cos−1 c r0x cos−1c r0_x −π 2 cos2θ0−sin2θ0dθx + 2 cos−1 c r0 x exp(−(r_x0)2/2) 2π dr 0 x + Z ∞ c Z 2π 0 cos2θ0−sin2θ0 dθx exp(−(r_x0)2/2) 2π dr 0 x= 0 (4.18) where θ0 =θ0(rx, θx) = tan−1 tanθx+ β rcosθx . (4.19)

The proof of this theorem is given in Appendix B.1.

based on Riemann integrals givesc ≈0.772 to 3 decimal places. This implies that the result of smoothing the tensor field is that the singularity is displaced vertically upwards to a distance of 0.772×h.

Theorem 1 describes the extent of the singularity location bias arising from the application of kernel smoothing to a tensor field. In our tensor field construction, the same kernel smoothing is applied but to a discrete set of tensors sparsely located overW, rather than an infinite tensor field. Approximating the sparse set of tensors by an infinite tensor field is necessary to simplify calculations. Noise, or variation from the arch model orientation, in the point estimates of the tensor field will likely affect the extent of the bias, but with no further information on the nature of the noise it is assumed that the infinite tensor field provides a reasonable approximation to the discrete collection of tensors. The effect of noise on the initial tensors is discussed in Section 7.3, in order not to distract from the current discussion. Approximating the tensor calculation by integrating over the whole of R2, rather

than justW, will only affect the estimation of the extent of the bias if the distance from the singularity to the boundary of W is small in comparison to h. Heuristic evidence shows that singularities are ‘pulled’ to the boundary ofW as the smoothing parameter is increased beyond a certain threshold. This is believed to be a result of edge effects in kernel smoothing.

The assumption that eigenvalues are constant and therefore that tensors are all equal except for orientation is unlikely to be true in practice as variations in the point pattern intensity and local anisotropy directly affect the eigenvalues. However, it is a reasonable approximation, particularly as the variation in eigenvalues across the set of tensors is not easily predictable.

Applying the result of Theorem 1 to the example of Figure 4.5, we find that increas- ingh from 30 to 60 displaces the singularity by a further 36 length units which is of the same order as the estimate 0.772×30 = 23.2. The difference is likely due to the inaccurate assumptions, in particular the fingerprint seems to have higher curvature than the arch model. Increasing h to 90 displaces the singularity by a further 41 length units in approximately the same direction. This appears to support the claim that the displacement is approximately linear inh.

Estimation of Singularity Location Bias: Parabolic Model

In the previous section it was shown that smoothing an arch-shaped field of tensors will displace the singularity by a distance proportional to the smoothing param- eter h. We now consider a second model based on parabolas which, although it

Figure 4.10: The parabolic tensor field, illustrated by integral fibres of the associated field of orientations.

appears similar in shape to the arch model gives quite different results. For the parabolic model, the smoothing causes no bias on the location of the singularity of the parabolic tensor field.

Let (x1, x2) denote a Cartestian coordinate system over R2. The parabolic tensor

field T₀para(x1, x2) is defined at (x1, x2) to be the tensor with constant eigenvalues λ1 > λ2 > 0 and principal eigenvector tangent to the parabola x2 = ₄1_a −ax21

for x1 6= 0, where a > 0 takes different values depending on x1, x2. For x1 = 0,

the principal eigenvector is proportional to (1,0) if x2 > 0 and (0,1) if x2 < 0.

The corresponding orientation field of the parabolic tensor field is continuous over

R2\{(0,0)} ( verified by considering the derivatives ddxx12 and dx2

dx1) and has a single

singularity located at the origin (0,0). Any integral curve of the associated field of orientations is a parabola with a focus at (0,0) and directrix given by x2 = c for

somec >0, hence the term, parabolic tensor field. These parabolic integral curves are illustrated in Figure 4.10.

Theorem 2 (Singularity Location Bias: Parabolic Model). Let T₀para(x1, x2) be a

parabolic tensor field with constant eigenvalues λ1 > λ2, and let T_hpara(x1, x2) be

the result of applying a convolution in the log-Euclidean metric to this tensor field

with a Gaussian kernel and smoothing parameter h. Then T_hpara(x1, x2) contains a

singularity located at the origin (0,0).

The proof of this result is a consequence of the following lemma

Lemma 1. Let e(x1, x2) = (e1(x1, x2), e2(x1, x2)) denote the principal eigenvector

of the tensor T₀para(x1, x2). Then e(x1, x2) is perpendicular to e(−x1,−x2) for all

The proofs of both lemma and theorem are given in Appendix B.2.

These two plausible models - the arch and parabola, make it clear that when estimat-