Automatic Localization of Anatomical Point Landmarks for Brain Image Processing Algorithms

Abstract

Many brain image processing algorithms require one or more well-chosen seed points because they need to be initialized close to an optimal solution. Anatomical point landmarks are useful for constructing initial conditions for these algorithms because they tend to be highly-visible and predictably-located points in brain image scans. We introduce an empirical training procedure that locates user-selected anatomical point landmarks within well-defined precisions using image data with different resolutions and MRI weightings. Our approach makes no assumptions on the structural or intensity characteristics of the images and produces results that have no tunable run-time parameters. We demonstrate the procedure using a Java GUI application (LONI ICE) to determine the MRI weighting of brain scans and to locate features in T1-weighted and T2-weighted scans.

Appendix: Relation to Cross-Correlation

Cross-correlation (Russ 2006) is a well-known method in image processing that is commonly used to find features within images. The shift required to align a target image with another image is determined by sliding the target image along the latter image and summing the products of all overlaid pixel values at each location. The sum is a maximum where the images are best matched.

In this Appendix we relate cross-correlation to the method described in this paper in one-dimension; it is straight-forward to extend this relation to two and three dimensions. If I is a one-dimensional image intensity function and is displaced a distance x relative to another function A, the cross-correlation is defined as

$$\int A(g)I(g+x) dg.$$

(8)

In the limit of small x, we approximate I(g + x) as I(g) + x I′(g) and note that Eq. 8 has the form

$$\int A(g) [ I(g) + x I'(g) ] dg \sim D + x C,$$

(9)

where D and C are constants. In particular, if we choose C = − 1, then in this limit the integral is equal to the distance between I and A (with D defined as the distance between them at x = 0).

In our case, we are choosing a function for the cross-correlation integral and solving for A (as opposed to being given I and A and computing the integral). We are then interested in finding a function A that makes

$$\int A(g)I(g+x) dg = D - x$$

(10)

approximately true. The integral can be approximated by evaluating the integrand at n locations that are a distance δ apart

$$\sum_{j=1}^{n} A(j\delta)I(j\delta+x) = D - x,$$

(11)

and if the range of x is evaluated in m steps of length Δ, we have the system of equations given in Eq. 1 where A _j  = A(jδ), I _ij  = I(jδ + iΔ), and D _i  = D − iΔ. The coefficients A _j are determined using least-squares to minimize the differences between both sides of Eq. 10.

One advantage of using least-squares is that the solution is insensitive to constant changes in intensity. For example, let I ₁(x) and I ₂(x) be image intensities where I ₂(x) = I ₁(x) + I ₀ and I ₀ is a constant. Then the function A that minimizes the differences between both sides of the equations

$$\begin{array}{ccc} {\kern-3pt} \int A(g)I_{1}(g+x) dg & = & D - x, \\ {\kern-3pt} \int A(g)I_{2}(g+x) dg & = & D - x, \end{array}$$

(12)

is also the function that minimizes the differences between both sides of the equations

$$\int A(g)I_{1}(g+x) dg = D - x,$$

(13)

$$\int A(g)I_{0} dg = 0,$$

(14)

by subtracting the two equations. Equation 14 states that the contributions of constant intensity terms are minimized.