Fast Tractography Streamline Search

Abstract

In this work, a new hierarchical approach is proposed to efficiently search for similar tractography streamlines. The proposed streamline representation enables the use of binary search trees to increase the tractography clustering speed without reducing its accuracy. This hierarchical framework offers an upper bound and a lower bound for the point-wise distance between two streamlines, which guarantees the validity of a proximity search. The resulting approach can be used for fast and accurate clustering of tractography streamlines.

Appendices

A Sum of Norm Properties with Detailed Equations

Remark #1. The dist$_{L^1}(U,W)$ is equivalent to computing the $L^1$ distance between two $m \times n$ dimensional points (${{\mathbf {\vec {u}}}},{\mathbf {\vec {w}}} \in \mathbb {R}^{m \times n}$).

$$\begin{aligned} \text {dist}_{L^1}(U,W)&= \sum _{i=1}^m || \mathbf{u}_i - \mathbf{w}_i ||_1 \\&=\sum _{i=1}^m \sum _{j=1}^n |u_{i,j}-w_{i,j}| \\&= || {{\mathbf {\vec {u}}}} - {\mathbf {\vec {w}}} ||_1 \end{aligned}$$

Remark #2. The $L^1$ distance in n-dimensions can be used as an upper and a lower bound the $L^2$ distance, from Hölder’s inequality ($\mathbf{x} \in \mathbb {R}^n$).

$$\begin{aligned} \left\| \mathbf{x} \right\| _p \le&\left\| \mathbf{x} \right\| _r \le n^{ (1/r - 1/p) } \left\| \mathbf{x} \right\| _p&0< r < p \\ \left\| \mathbf{x} \right\| _2 \le&\left\| \mathbf{x} \right\| _1 \le \sqrt{n} \left\| \mathbf{x} \right\| _2&r=1,\, p=2 \\ \frac{1}{\sqrt{n}}\left\| \mathbf{x} \right\| _2 \le&\frac{1}{\sqrt{n}}\left\| \mathbf{x} \right\| _1 \le \left\| \mathbf{x} \right\| _2&\text {division by } \frac{1}{\sqrt{n}} \end{aligned}$$

Remark #3. The distance between the mean position ($\overline{\mathbf{u}} = \frac{1}{m} \sum _{i=1}^m \mathbf{u}_i$) of two streamlines is always smaller or equal to the average point-wise distance. This can be obtained from $L^p$-norm properties ($1 \le p < \infty$ $\mathbf{x},\mathbf{y} \in \mathbb {R}^n \,,\; \lambda \in \mathbb {R}$).

$$\begin{aligned} \left\| \mathbf{x} + \mathbf{y} \right\| _p \,\le&\, \left\| \mathbf{x} \right\| _p + \left\| \mathbf{y} \right\| _p&\text {triangle inequality} \end{aligned}$$

(11)

$$\begin{aligned} \left\| \lambda \mathbf{x} \right\| _p \,=&\, |\lambda | \, \left\| \mathbf{y} \right\| _p&\text {positive homogeneity} \end{aligned}$$

(12)

Using $\mathbf{u}_i,\mathbf{w}_i \in \mathbb {R}^n ,\, i \in \{1, ..., m\}$, such that $\mathbf{d}_i = \mathbf{u}_i - \mathbf{w}_i$

$$\begin{aligned} \big \Vert \overline{\mathbf{u}} - \overline{\mathbf{w}} \big \Vert _p&= \bigg \Vert \, \frac{1}{m}\sum _{i=1}^m \mathbf{u}_i - \frac{1}{m} \sum _{i=1}^m\mathbf{w}_i \,\bigg \Vert _p \\&= \frac{1}{m}\bigg \Vert \, \sum _{i=1}^m (\mathbf{u}_i - \mathbf{w}_i) \,\bigg \Vert _p&\text {from (12)}\\&\le \frac{1}{m} \sum _{i=1}^m \big \Vert \mathbf{u}_i - \mathbf{w}_i \big \Vert _p&\text {from (11)} \end{aligned}$$

This can be generalized to curves using the triangle inequality with a Lebesgue integrable function $\Vert \int _0^1 f(t) \, dt \,\Vert _p \,\le \, \int _0^1\Vert f(t)\Vert _p \, dt \,,\; 1 \le p < \infty \,,\; f \in \mathcal {L}^1(\mathbb {R})$ [6].

B Streamline Search Comparison

Fig. 8.

Results of the proximity search for the left Arcuate Fasciculus (AF): a) the bundle atlas from [13, 37], b) RecoBundles result, c) RecoBundles result (in green) showing in red a few streamlines missing in RecoBundles but present in the proposed technique with a 4 mm search, and in purple, streamlines missing in RecoBundles but present in the proposed technique with a 6 mm search. The proposed proximity search, mdist$_{L^2}(\cdot ,\cdot ) \le r$, using a radius of: d) 4 mm, e) 6 mm, and f) 8 mm. (Color figure online)

Fig. 9.

Results of the proximity search for the central portion of the Corpus Callosum (CC_3): a) the bundle atlas from [13, 37], b) RecoBundles result, c) RecoBundles result (in green) showing in red a few streamlines missing in RecoBundles but present in the proposed technique with a 4 mm search, and in purple, streamlines missing in RecoBundles but present in the proposed technique with a 6 mm search. The proposed proximity search, mdist$_{L^2}(\cdot ,\cdot ) \le r$, using a radius of: d) 4 mm, e) 6 mm, and f) 8 mm. (Color figure online)

Fig. 10.

Results of the proximity search for the left Uncinate Fasciculus (UF): a) the bundle atlas from [13, 37], b) RecoBundles result, c) RecoBundles result (in green) showing in red a few streamlines missing in RecoBundles but present in the proposed technique with a 4 mm search, and in purple, streamlines missing in RecoBundles but present in the proposed technique with a 6 mm search. The proposed proximity search, mdist$_{L^2}(\cdot ,\cdot ) \le r$, using a radius of: d) 4 mm, e) 6 mm, and f) 8 mm. (Color figure online)