Pietilä y el pensamiento irracional: el lenguaje y la naturaleza

In order to connect the candidate points into complete curves, potential endpoints of (yet

unknown) sulcal curves need to be estimated. For

∀u∈U, thus, sulcal candidates points are

collected within a geodesic distance

r

defined by an indicator function:

R(u,v) =

        

1

ifL

u

(v)≤r ,

0

otherwise,

(3.4)

where

L

u

is a geodesic distance fromu

to

v∈Ω. The geodesic distance

L

u

can be computed

by the Eikonal equation (see Equation 2.14). Assignuto a source of the wavefront propagation

by letting the tensor matrix

M(v) =

I

for

∀v∈

Ω in Equation 2.14, where

I

is the 2×2

identity matrix. Thus, with a sufficient boundary condition

L

u

(u) = 0, the speed function in

Figure 3.5: Maximum curvature points (top) and detected sulcal points (bottom, where sulcal

endpoints are colored in

green

and

blue). The vertices with positive curvatures are selected

for candidate sulcal points. Since the candidate vertices are spread over a large portion of the

sulcal fundic regions, they are further filtered out by the proposed method that eventually

selects sulcal points (blue). The endpoints (green) are then selected from the selected sulcal

points.

Equation 2.15 becomes

F

x,

∇L

u

(x)

k∇L

u

(x)k

!

= 1.

(3.5)

ForM

=I, this simplifies the H-J PDE to the Eikonal PDE:

The solution provides a geodesic distance

L

u

for all locations of Ω. The geodesic distance

r

in

Equation 3.4 is chosen under the assumption that the sulcal regions are separated from each

other by at least

r. In this chapter, this quantity was empirically set

r= 4.0

mmbased on

the average width of the sulcal regions in the MNI-305 template [70]. This geodesic distance

can be adjusted depending on the target population.

The candidate points are determined as endpoints if every point holding

R(u,·) = 1 is

located within an octant centered at

u

as shown in Figure 3.6. This is easily achieved by

testing the sign of the inner products of all neighboring points holdingR(u,·) = 1, i.e., no

line between the neighboring points and the center point

u

has a separating angle above

90 degrees. Let

E

⊆

U

be the set of the endpoints determined by this way. To find the

neighboring sulcal point at

u∈E, the weighted shortest distance is employed from

u

to

s,

such that

s

holds

R(u,s) = 1. The distance weighting is based on the assumption that the

tangent direction of the sulcal curve changes smoothly along the curve. Thus, the weighted

distance is given by the following form:

C(s,u) =k(s−u)×T(u)k,

(3.7)

where

T(u) is the tangent vector at

u. Therefore, the neighboring sulcal point ˆs

at

u

is

obtained by

ˆ

s= argmin

s

C(s,u).

(3.8)

However,

T(u) is unknown due to no prior knowledge of the sulcal curve available. Instead

T(u) is estimated using the local principal direction,T

k2

(u). This is an incremental procedure;

ݑݑ

ݏ

_଴

ݏ

_ଵ

Figure 3.6: A schematic overview of the proposed endpoint detection. An example of the

octant of the sphere is determined by three orthogonal axes.

u

is determined as an endpoint

if its neighboring points in a geodesic kernel

S(u) belong to one of the octants.

us

0

and

us

1

form the maximum angle across every possible line starting from

u, such that

R(u,s

0

) = 1

and

R(u,s

1

) = 1.

R(ˆs,·) = 1 and it stops if ˆs∈E

or ˆs

is a part of the other already delineated curve (junction

point). Starting at an arbitrary endpoint, the curve estimation is finished if every element in

E

has been connected. Figure 3.7 shows the estimated curves from sulcal points. The sulcal

endpoints are located in the end of each sulcal curve.

3.6

Materials

I chose the Kirby reproducibility dataset [53] to evaluate my sulcal extraction method

for reproducibility. Briefly, the Kirby reproducibility dataset was aquired on 21 healthy

volunteers with no history of neurological disease and is publicly available on NITRC

2

_{. Scan-}

2

Figure 3.7: The estimated curves with endpoints in the lateral (left) and medial (right) views.

The sulcal curves are reconstructed from the detected sulcal points. Each curve is labeled

with a distinct color. End points are highlighted via larger dots. Sulcal regions with branches

are represented as several curves with junction points.

rescan imaging sessions withT

1

-weighted scans were acquired at the F.M. Kirby Research

Center (Baltimore, MD, USA), using the MP-RAGE sequence on 3T Philips Achieva scanners

at 1.0mm

×1.0mm

×1.2mm

resolution (204 slices with TR = 6.7ms, TE = 3.1ms, flip

angle = 8

◦

, matrix = 240×256) scans. The central cortical surfaces were created using the

standard Freesurfer v5.3 pipeline. For validation, the left hemispheres were resampled with

163,842 points via spherical icosahedron subdivision.

3.7

Results

All experiments were performed on a PC equipped with an Intel Core (TM) i5-3570K 3.4

GHz CPU with 12.00 GB memory, and only a single core was used. It took 1-2 minutes to

obtain a full set of complete curves on a cortical surface on average. Quantitative verification

on clinical data is extremely difficult as there is no ground truth for such sulcal curve data.

Qualitatively I compared the results to standards in the field such as BrainVisa. The following

sections mainly focus on the reliability and reproducibility of the proposed method versus

other methods.

3.7.1

Noise Sensitivity

To evaluate noise sensitivity, synthetic surfaces were generated by vertex-wise perturbation

of an existing brain surface model. Perturbations were simulated via uniformly distributed

independent displacements of the vertices. The left hemisphere of the MNI-305 average

healthy control template surface was used for this evaluation purpose, also employed in [70],

as the original brain surface model. Figure 3.8 shows the detected sulcal curves with different

levels of perturbation. Visually, there is no significant difference from the original surface.

For quantitative evaluation of the robustness to noise, for each sulcal point on the original

surface, the Euclidean distance was measured to the closest point on the perturbed surface

as no ground truth is available. Quantitatively, the experimental result showed that for each

level of noise, the average distance was 0.505±0.616, 0.776±0.629, and 0.982±0.702

mm,

respectively. This indicates that the average distances are reasonable given the original MR

image resolution (1.0

mm).

3.7.2

Reproducibility

From the KIRBY dataset, the BrainVISA pipeline

3

_{[107] was used to generate surfaces from}

the dataset with sulcal ribbon extraction, and the extracted sulcal ribbons then were projected

onto the sulcal fundi to have sulcal curves for comparison. Figure 3.9 shows an example of

Original Surface

Δ

max

=2.27 mm

Δ

max

=4.54 mm

Δ

max

=6.81 mm

Figure 3.8: Robustness to noise. Sulcal points are detected on the surface perturbed by a

random noise. ∆

max

indicates the maximum displacement of the vertex. The detected sulcal

points (blue) and the estimated curves (red) are stable across levels of noise. The quantitative

evaluation also shows that the average distance to the corresponding points is less than 1.0

mm.

the sulcal curves extracted by using the proposed method. Again, the closest distances were

measured at sulcal points between two corresponding surfaces and then averaged along each

sulcus. Since the closest distance computation is asymmetric (one-way), it took the maximum

of the two possible distances from both corresponding surfaces. Figure 3.10 illustrates a

statistics on the maximum average distances of the corresponding surfaces. Importantly,

the sulcal extraction in the BrainVISA pipeline generates only the set of the labeled major

(a)

(b)

(c)

Figure 3.9: Sulcal curve extraction from the different scans for the same subject. (a)-(b) Two

sets of curves are labeled with respective colors (red

and

blue). (c) The two sets are well

aligned in the same space.

curves as its output (unlabeled curves provided in this pipeline). On the other hand, the

proposed method extracted both major and minor curves (major curve labeling is out of

scope in this chapter). Thus, it could be a unfair evaluation for the proposed method without

a major curve labeling procedure as one would expect higher reproducibility errors in those

smaller curves. Despite that disadvantage, the experimental results show that the proposed

method has less average distance than BrainVISA. The proposed method achieved 2.01

±

0.33

mm

(average±

standard deviation across all 21 cases), whereas BrainVISA showed an

average distance of 2.89

±

0.73

mm. To obtain statistical significance, I applied a paired

t-test between the maximum average distances obtained from BrainVISA and the proposed

method (total 21 maximum average distances for each method as shown in Figure 3.10). This

reveals that the proposed method achieves a statistically significant smaller average distance

(p

0.0001).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Subject # A v er age Distance (mm) 0 1 2 3 4 5

Proposed Method BrainVISA

Figure 3.10: The maximum average distances of the extracted sulcal curves on the 21 subjects

in the Kirby reproducibility dataset. The proposed method achieves consistent results over

those obtained in the BrainVISA pipeline.

3.8

Summary

This chapter presented a fast and accurate automatic sulcal curve extraction method to

provide cortical geometric landmarks based on the observation that sulcal fundic regions

are well defined as local extrema. A set of candidate sulcal points is chosen by the line

simplification method. The extracted sulcal points are further connected to form a set of sulcal

curves on the cortex. The resulting landmarks showed a high reliability in the multi-scan

dataset as well as robustness to a high level of noise.

The experimental results showed that the proposed method captures sulcal curves robustly

in the presence of noise and shows high computational efficiency. In comparison to Brain-

VISA, a standard neuroimaging tool, the proposed method showed significantly improved

reproducibility. The proposed method has several advantages: First, the parameter tuning is

quite simple as there is a small set of parameters and the results are robust to reasonable

CHAPTER 4: ROBUST ESTIMATION OF SURFACE

In document TítuloSverre Fehn : desde el dibujo (página 36-50)