HIERARCHICAL BRAIN MODEL FOR COREGISTRATION

A Physical Model for Analysis of Brain MRI Data

Terrence R. Oakes

Waisman Center Brain Imaging Lab, University of Wisconsin-Madison

1500 Highland Ave., Madison, Wisconsin, USA

Keywords: Physical Model, MRI.

Abstract: A ubiquitous problem in coregistration of brain images is that individual sulci and gyri vary considerably

between individuals, both with respect to location and shape as well as for simple existence of particular

sulci. The underlying assumption of most coregistration processes is that one structure can be smoothly

morphed to exactly resemble another structure if enough parameters are used. Although in a strict sense this

may be true for intersubject brain registration, due to differing structures the result may not be as

meaningful as desired. The proposed approach offers a groundbreaking alternative to the standard approach

of continuously deformable coregistration algorithms, introducing instead a hierarchical structure of related

nodes (a "nodetree") to model the brain structure using grey-matter and white-matter masks. Additionally, a

proposal is made for using the nodetree structure for coregistration, employing a novel locally discontinuous

but focused registration to more accurately align and compare corresponding features. This approach can

provide a framework for identifying structural differences, with a goal of relating them to functional

differences. Although this method uses the brain as an example, it is quite general and not limited to the

brain, or even to medical images.

1 INTRODUCTION

Current mainstream coregistration packages for

medical images use variations of one of two basic

approaches: a) whole-brain voxelwise or volumetric

registration which minimizes a cost function

summarizing the average difference between two 3D

volumes (Woods et al, 1998), or b) registration of

discrete points or features, with the transforms for

nearby unmarked voxels determined by interpolation

(Pelizzari et al., 1989). A feature common to both of

these approaches is to treat the structure as a

continuous 3-dimensional object, so that although

voxels may get stretched or distorted, neighboring

voxels remain neighbors. One result of this

assumption is that missing or extra structures in

either the target or object volume determines the

ultimate accuracy of the procedure. To date,

increases in accuracy have been achieved by using a

larger number of parameters to improve local fits,

but the fundamental assumption is that one structure

can be smoothly morphed to become identical to

another. In brain imaging, frequently there is not a 1-

to-1 intersubject correspondence of structures, so

even if an algorithm is able to smoothly morph one

object to achieve a good pixel-intensity match with

another, the result is not always desirable (Fig. 1).

Current approaches to brain coregistration are

limited to a large extent by the underlying similarity

of the structures to be registered, in that there is no

acknowledgment or allowance for missing or extra

structures. For example, a common but difficult

problem is to accurately register a brain containing a

tumor to a standard brain template. A more subtle

but also more ubiquitous problem is the fact that

individual gyri and sulci vary considerably between

individuals, both with respect to location and shape

as well as for simple existence of particular gyri.

Current coregistration approaches can reshape a

gyrus with some success, but do not address the fact

that a gyrus present in one individual may be

missing in another. The sensitivity of a multi-subject

functional activation study [e.g. functional Magnetic

Resonance Imaging (fMRI) or Positron Emission

Tomography (PET)] is directly related to the

accuracy of combining and examining the same

functional signal from a group of subjects, and

anatomical variation is emerging as one of the

103

R. Oakes T. (2007).

HIERARCHICAL BRAIN MODEL FOR COREGISTRATION - A Physical Model for Analysis of Brain MRI Data.

In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - GM/R, pages 103-108

DOI: 10.5220/0002073201030108

 SciTePress

Figure 1: Cartoon showing two different cortical-like structures (1a, 1b) that cannot be meaningfully morphed to achieve a

similar shape. Cartoons 1c, 1d demonstrate typical results expected from a continuously deformable coregistration model.

1c: a fit using fewer parameters might result in a single gyrus from the object image (blue) spanning two gyri in the target

image (orange). 1d: a fit using a large number of parameters could deform the single gyrus to achieve an accurate pixel

intensity match, but there may not be a physiological justification for splitting a single gyrus into two. 1e: an example of a

more physiologically plausible scenario, where two gyri should match but there is an “extra” or unassigned gyrus. This

result is quite difficult to obtain for current smoothly morphing algorithms.

limiting factors in functional comparisons (Juch et

al., 2005). This paper seeks to establish a framework

for applying established skeleton-based, hierarchical

registration techniques to anatomical medical

images, and subsequently to related functional data.

The nodetree method is based on a variant of the

Medial Axis Transform (MAT), which seeks to

define a skeleton representative of the major features

of an image. The medial axis (MA) (Blum 1967) is

described as the locus of the centers of all bi-tangent

circles contained within a shape. Here, the term

"MA" is used as shorthand for the MAT skeleton. A

useful feature of the MA is that the skeletal pixels

are connected, so shape features such as length are

easily computed. Pixels in a MA skeleton can be

ranked post-hoc according to how many branches

radiate from them, so that node points can be easily

identified. However, one of the lingering problems

with the MA and related approaches is a lack of

robustness (see e.g. Parker 1997). Small changes in

the overall structure can lead to large changes in the

final MA structure, which makes it difficult to apply

this approach to a variety of situations. A number of

variants have been proposed to address this

shortcoming, such as a recent method using a

Bayesian probabalistic approach to estimate a

skeleton shape (Feldman & Singh, 2006), and which

seems to be more robust to noise and minor

perturbations. However, this and similar approaches

have not yet been widely tested on medical images.

A major advance in the MA with respect to

medical imaging was a generalization to 3

dimensions (Sherbrooke et al. 1996). Further

refinements were added by Amenta and Kolluri

(2001) to develop a 3D medial axis from a union of

overlapping balls. In this case, the medial axis is not

a series of lines, as in the 2D case, but rather a group

of vertices that define a closed surface in space.

Ranjan and Fournier (1996) proposed using a union

of balls to describe a volume, and furthermore they

developed a method to coregister two similar

structures by finding the closest spatial match

between corresponding pairs of balls.

One of the major benefits of a hierarchical

skeletal model is that the various branches may be

moved independently of one another. This concept

underlies much of the computer animation field,

where a skeletal model is wrapped with an outer

surface so changes in the orientation and shape of

the skeleton can be propagated to the surface

structures (see e.g. Gagvani et al. 1998). This

process has an innate hierarchy, since movements to

one element of the skeleton (e.g. the forearm) lead to

predictable changes in subservient elements (e.g. the

hand).

The nodetree approach combines many of these

aspects, including ideas from the MA, the union-of-

balls, and the hierarchical skeleton. Unlike a MA,

the nodetree does not need to include all pixels

connecting the nodes. The goal is to produce a

hierarchical skeleton to which volumetric data can

be associated in a robust and logical manner. The

nodetree is a collection of nodes with essential

properties of location, spatial domain, and lineage.

All other properties can be derived from these,

including internode distance, object distance, and

included pixels.

The novelty of the nodetree is in the integration

of the MA and union-of-balls ideas to create a

skeleton. The particularly innovative aspect is the

subsequent coregistration approach it will enable

which is not spatially continuous, but rather which

recognizes that different structures (e.g. brains) may

have different spatial structures performing the same

function. Current alternatives which attempt to

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match such disparate structures using a continuous

model reduce the achievable accuracy of

coregistrations and also reduce the sensitivity of

related functional activation analysis. This paper

seeks to develop a framework for addressing and

facilitating the comparison of objects with

topologies and/or morphologies which are not

mutually deformable.

2 METHODS

2.1 Nodetree Properties

Construction of a nodetree for the brain starts with a

binary image of the White Matter (WM) tissue,

which can be readily obtained using standard

neuroimaging software , e.g. BET (Smith 2002). The

nodetree attempts to model the WM projections

(gyri) to the cortical Gray Matter (GM) surface. To

be useful as a medium for identifying specific

cortical structures and for intersubject registration,

there are several fundamental properties that each

nodetree should have:

1) Each gyrus should be principally

represented by one major node at the distal

end (apex);

2) There should be a node at every gyral

opening and branch point;

3) Lines connecting nodes must stay within

the tissue type;

4) The size of each node should be relative to

the volume of surrounding WM;

5) Node importance is directly related to node

size;

6) Each node must have only one parent.

This semantic description will be used as the

guiding principal behind the nodetree algorithms.

Current standard skeletal approaches were unable to

yield the desired characteristics, leading to the

development of the current nodetree approach.

2.2 Nodetree Algorithm

A prototype software program has been developed to

create and prune a 2D nodetree. Ultimately, a fully

3D implementation is desired, but the software and

examples presented in this paper are for the 2D case,

in order to simplify initial algorithm development

and display of results.

The algorithm starts with a seed-point that all

subjects can be expected to have, such as the center

of the largest WM region. The largest possible circle

is drawn within this region (Fig. 2a) and assigned a

rank of 1. This circle defines a "node" whose

properties include the location, radius, rank, and a

unique identification number. In the second step, the

edge pixels of this circle are used as seed-points for

Figure 2: Creation of a nodetree in the left hemisphere of a coronal section. (2a): The initial node is created by filling the

region near a seed point with the largest possible circle. (2b): Children nodes are added to each node until the structure is

filled. The brown color shows pixels that are included inside a node. The color of each node indicates its rank or generation

number, repeating in order over red, orange, yellow, light green, green, light blue, blue, purple, and magenta. (2c): The

nodetree is pruned to remove small and/or redundant nodes, leaving behind only nodes needed to define the WM structure.

(2d): Six additional nodes (green and dark green) have been added manually using a semiautomated GUI in order to define

the structure more accurately.

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a new set of circles or "child nodes". For each node,

the goal is to create a family of children nodes

centered on the parent’s edge pixels, and to retain

only children nodes which are large and do not

overlap other sibling nodes. Proceeding from the

largest child to the smallest, siblings within each

child node are eliminated, leaving a few larger

children surrounding the parent. The children nodes

are assigned a parent node as an additional innate

property, and several convenience properties, such

as an "arm length" or distance to the parent node.

This process proceeds iteratively until the entire

object is filled with nodes (Fig. 2b). The full node-

set can be saved for later use.

An important property of nodetree growth is that

at each iteration, growth only occurs for nodes

created in the previous iteration, and this growth is

limited to previously unclaimed regions. We

hypothesize that this will produce a natural growth

pattern that is reproducible across similar branching

structures. This property also helps to ensure that the

initial full nodetree has a closed surface.

2.3 Nodetree Pruning

To emphasize its basic shape, the nodetree must be

pruned so that only the important nodes remain. In

principle, it is desirable for the nodetree algorithm to

yield a description of the object which needs little

post-processing; however in practice, some level of

post-processing (pruning) is required to better

emphasize the overall WM structure. Pruning is not

a single step, but rather is a series of algorithms

which can be varied ad infinitum to emphasize

various characteristics of the underlying structure.

In the current MRI example, the goal is to

represent the overall shape of the WM structures

with the fewest number of nodes. It is not necessary

(and unlikely) that all WM pixels be contained

within a node after this step. An "important" node

meets one of the following criteria: it is a) large and

in the center of a WM space; b) at the end of a WM

gyri; c) at a fulcrum (bend) in a gyral projection or

d) at the mouth (opening) of the gyrus into a larger

WM region. Specific parameters for each of these

criteria can be varied for different effects; for

instance, decreasing the minimal acceptable size for

a terminal node [criteria b) above] can more

accurately model the full extent of a WM gyrus, but

perhaps at the expense of indicating the importance

of the node terminus based on its size.

A series of automated pruning algorithms were

developed to remove small nodes, similar

neighboring nodes, and redundant nodes along a

straight path (Fig. 2c). Some pruning steps may

result in a node changing position and/or radius.

Although the goal is for full automation, the

nodetree can be adjusted manually to make sure that

all arms are filled in and that the nodes are located

properly (Fig. 2d). Either manual or automated

adjustment of individual nodes is simple, due to the

hierarchical compostion of the nodetree. After

pruning, the arm-lengths are recalculated and ranks

are re-assigned to minimize the number of ranks. A

variety of algorithms were developed for the pruning

stages, including functions such as: remove dead-

end nodes; remove nodes below a specified size;

consolidate long runs of nodes by removing nodes

that have only a single child; consolidate ranks to

remove gaps; remove nodes that are too close to

their parents.

3 RESULTS

A nodetree was created for each of 7 normal subjects

using a coronal slice of the left hemisphere from the

same location after the image data were coregistered

to the MNI T1 template (Evans et al., 1993) using

registration software from SPM2

(http://www.fil.ion.ucl.ac.uk/spm/) and skull-

stripped with BET (Smith 2002). Tissue segment

maps were produced using FAST (Zhang et al.,

2001). An 8th nodetree was created using the sum-

image of the segment maps. Similarity of the

nodetrees in Fig. 3 indicates the robustness of this

technique across individuals. The differences help to

highlight the variation in anatomic structure between

individuals.

While there is clearly room for improvement, all

of the nodetrees show similarities, and all are able to

define the overall shape of the WM, including most

of the larger arms.

The nodetree represents a significant data

reduction technique. Figure 2a shows a typical 2D

image of WM with 1018 WM pixels. The full

nodetree (Fig. 2b) contains ~92% of the WM pixels

yet is represented by only 126 nodes. The final

pruned nodetree (Fig. 2d) contains only 21 nodes,

but still captures the shape of the WM structure.

This savings is expected to be proportionally even

greater for 3D data using a nodetree comprised of

spheres. Furthermore, since the pruned structure is

represented by so few points, it is very efficient to

manipulate the structure.

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Figure 4: Cartoon showing a 2D scheme for arranging

Gray Matter (GM) nodes within cortical GM and for

associating them to the White Matter (WM) nodetree. The

color of the line connecting each GM node indicates the

line segment of the WM it belongs to. The GM nodes have

a radius designed to span the cortex at each node's

location. Although the GM nodes are depicted as disks in

this cartoon, they could be irregularly shaped in order to

cover all of the GM yet prevent overlap of node interiors.

Figure 3: Comparison of nodetrees from 7 different

subjects. The nodetree at the upper left was derived from

the thresholded WM segment from a sum-image of the

individual segment maps, and can be considered as a basis

for comparison.

Although a nodetree can model a fairly complete

representation of an object, it should be emphasized

that the nodetree is not required to exist in isolation.

For detailed analysis of a shape, the original object

and its underlying data values may be interrogated

as long as nodetree-related spatial transforms are

recorded.

Figs. 2b-d illustrate a potential problem for the

nodetree: one of the terminal gyri remains unfilled

Fig. 2b-c) and has been manually filled in (Fig. 2d).

This is a result of the minimal acceptable node size,

which in this example is a 5-pixel cross-shape.

Using a smaller node (single pixel) or permitting the

search to proceed via diagonal pixels (i.e. pixels

touching at only a corner) solves this problem, but

must be balanced against the increased complexity

of the nodetree shape. The non-minimal node size is

used in Fig. 2 to highlight this tradeoff, in which a

more complex initial nodetree would require

additional pruning. In the current implementation,

the pruning is insufficiently developed to yield

robust results for a very complex nodetree.

Initial attempts to characterize the robustness of

the nodetree indicate that it can be quite sensitive to

noise in the binary WM representation. For example,

a single non-WM pixel in the center of a large WM

space will yield a number of small nodes

surrounding the non-WM island, rather than the

expected single large node. This is really more of a

problem related to creation of the initial binary

image, and isolated non-WM pixels can easily be

removed by standard filtering techniques.

An additional observation is that, while the

nodetree is not overly sensitive to minor changes in

the edge of a structure, the position of nodes at the

end of a gyrus can be sensitive to the width of the

gyrus in relation to the minimal acceptable node

size. Currently, a dedicated pruning step is needed to

minimize this, but further investigation of the

growth pattern with respect to this bias is needed.

In order to be useful for anatomical coregistration, a

systematic identification of important nodes is

required. For example, a template based on a large

number of individuals could label those nodes which

occur most frequently. Once a WM nodetree has

been created, the gray matter cortex could be

modeled as an additional layer, as represented in

Fig.4.

4 CONCLUSION

The nodetree algorithm can yield a reasonably

similar model of the brain white matter structure

across individuals. Further advances, particularly

with respect to pruning, are expected to yield

improved similarity. The hierarchical structure is

well suited as a framework for investigating non-

continuous spatial registration approaches.

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REFERENCES

Amenta N and Kolluri RK, 2001. The medial axis of a

union of balls, Computational Geometry (20):25-37.

Blum H, 1967. A Transformation for Extracting New

Descriptors of Shape, Symposium Models for Speech

and Visual Form, Weiant Whaten-Dunn (Ed).

Cambridge, MA: MIT Press.

Evans AC, Collins DL, Mills SR, Brown ED, Kelly RL,

Peters TM, 1993. 3-D statistical neuroanatomical

models from 305 MRI volumes, Proc IEEE Nucl Sci

Symp Med Imaging, 95:1813-1817.

Feldman J and Singh M, 2006. Bayesian estimation of the

shape skeleton, PNAS, 103(47):18014-18019.

Gagvani N, Kenchammana-Hosekote D, Silver D, 1998.

Volume Animation Using the Skeleton Tree, Proc.

IEEE Symposium on Volume Visualization.

Juch H, Zimine I, Seghier ML, Lazeyras F, Fasel JHD,

2005. Anatomical variability of the lateral frontal lobe

surface: implication for intersubject variability in

language neuroimaging, NeuroImage, 24:504-514.

Parker JR, 1997. Algorithms for Image Processing and

Computer Vision, John Wiley & Sons.

Pelizzari CA, Chen GTY, Spelbring DR, Weichselbaum

RR, Chen CT, 1989. Accurate three-dimensional

registration of CT, PET, and/or MR images of the

brain, J. Comput. Assist. Tomogr., 13(1):20-26.

Ranjan V and Fournier A, 1996. Matching and

Interpolation of Shapes using Unions of Circles,

Computer Graphics Forum, 15(3):129-142.

Sherbrooke EC, Patrikalakis NM, Brisson E, March 1996.

An Algorithm for the Medial Axis Transform of 3D

Polyhedral Solids, IEEE Trans. on Visualization and

Computer Graphics, v.2 n.1, p.44-61.

Smith SM, 2002. Fast robust automated brain extraction.

Human Brain Mapping, 17(3):143-155.

Woods RP, Grafton ST, Watson JDG, Sicotte NL,

Mazziotta JC, 1998. Automated image registration I:

Intersubject validation of linear and nonlinear models,

J. Comput. Assist. Tomogr., 22(1):153-165.

Zhang Y, Brady M, Smith S, 2001. Segmentation of brain

MR images through a hidden Markov random field

model and the expectation maximization algorithm,

IEEE Trans. on Medical Imaging, 20(1):45-57.

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

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