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Volume enhancement: tube-enhancing filter, coherence-enhancing
diffusion and surface-enhancing filter
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After wavelet enhancement, the microtubules are still embedded in
the low contrast volume. Further enhancement is still needed. We propose to
enhance the volume with a set of spatial domain 3D filters by exploiting
the local geometric properties of tubular structures.
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Shape
estimation from curvatures of hyper-surface
3D image is
treated as 4D hyper-surface
Local shape
is estimated using the curvatures of the hyper-surface. Shape filters
designed in this way provide better feature localization and can handle
contrast variations.
First
fundamental form is derived as
Surface
normal is derived as
First
fundamental form is derived as
Weingarten
matrix
The eigenvalues of W are the principal
curvatures, order them as
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Tube-enhancing
filter
For tubular object
Roundness criterion
Elongation criterion
A filter is constructed to produce strong
response on tubular object
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Tube-enhancing
filtering on synthetic data
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(a)
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(b)
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(c)
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(d)
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Enhancement with
tube-enhancing filter on synthetic data.(a) the original image, (b) noise
added, (c) enhanced image in (b) with our method, (d) enhanced image in (b)
with the Hessian matrix based method. Our filter not only accentuates the
object centerline significantly but also attenuates other morphology more
effectively
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Modified coherence-enhancing diffusion
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The modified coherence-enhancing diffusion
completes the interruptions along microtubules. The diffusion tensor is constructed based on the response of the
tube-enhancing filter and the eigenvectors of the Weingarten matrix.
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Tube-enhancing filter and modified
coherence-enhancing filter on tomography image
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(left) enhanced with anisotropic invariant
wavelet transform, (middle) further enhanced with tube-enhancing filter,
(right) further enhanced image in (middle) with the modified
coherence-enhancing filter
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Microtubules enhanced and localized in 3D volume
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(top left) the original volume, (top right)
the enhanced microtubules with 3D rendering, (bottom) extracted centerlines
from different views
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Surface-enhancing
filter
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For ideal cylinder surface features
In practice, only the first and the third
eigenvalues are used
A surface-enhancing filter is constructed to
produce strong response for voxels on surface features
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Enhanced
surface features
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(a)
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(b)
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(c)
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(a) a
longitudinal original slice, (b) enhanced image, (c) cross-sectional
slices. top row: original slices, middle row: enhanced with our surface-enhancing filter, bottom row: enhanced
with Hessian based filter
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