Transcript Parameter Controlled Volume Thinning Nikhil Gagvani Deborah Silver

Parameter Controlled
Volume Thinning
Nikhil Gagvani
Deborah Silver
Introduction
Algorithm for the Distance Transform
 Skeleton Extraction
 Algorithm for the Centerline
 Reconstruction

The Neighbors of a Voxel

A voxel is the smallset unique element of
the volume.
F-Neighbors
E-Neighbors
V-Neighbors
Voxels
object-voxels
 background-voxels
 boundary-voxels 000000…

0000…
00…
S
_
S
BV
Distance Transform in 2D

(1/2)
The distance transform of a simple shape.
Distance Transform in 2D
(2/2)
Distance Transform in 3D

(1/6)
The distance transform at a voxel p = {x,y,z} is
defined as
DT p  min {dt (( x, y, z ), (i, j, k )) : (i, j, k )  BV }
( i , j ,k )

dt
: the distance from voxel(x,y,z) to voxel(i,j,k)
 BV : the set of boundary voxels
Distance Transform in 3D

We compute the distance
distance metric.



F-neighbor : 3
E-neighbor : 4
V-neighbor : 5
(2/6)
d t using a <3,4,5> weighted
Distance Transform in 3D

(3/6)
Algorithm
 Init
 First

pass
Calculate the Distance transform of boundary-voxels.
 Second

pass
Propagate the boundary inward.
Distance Transform in 3D

(4/6)
Init

For all voxels p  S , assign a distance transform DT p  
S
_
S
Distance Transform in 3D

(5/6)
First pass
 Calculate
the Distance transform of boundary-voxels
For all voxels p  S that have a (F/E/V) neighbor q  S
{
DT p  (3 for face/ 4 for edge/ 5 for vertex);
Add p to BV ;
}
S
_
S
BV
Distance Transform in 3D

(6/6)
Second pass
 Propagate
the boundary inward
Repeat for all p  BV
Find all voxels r  S which are (F/E/V) neighbors of p ;
Assign DTr  min{ DTr , DT p  (3 or 4 or 5) } ;
Remove p from BV ;
Add r to BV ;
until no DTr is modified
S
_
S
BV
Skeleton Extraction

(1/12)
Definition 1
a voxel p has a distance transform DT p , the Ball
B(p) associated with p is the set of object voxels q
such that the distance from p to q is less then DT p .
 If
DT p
p
S
_
S
Skeleton Extraction

(2/12)
Definition 2
 The
ball for an object voxel is maximal if it is not
contained in the ball of any other voxel.

Observation
 The
set of voxels whose balls are maximal is sufficient
to reconstruct the object.
Skeleton Extraction

(3/12)
Minimal set for reconstruction
 The
ball of a voxel may not be completely contained
in that of another, but may be contained in the union
of the balls of several other voxels.
not essential
Black voxel
Gray voxel
a
b
c
Skeleton Extraction

(4/12)
Definition 3
 The
witness for a voxel p is any 26-neighbor q such that
the distance transform of q, DTq  DT p  ( 3 or 4 or 5 ).
Thus, if a voxel q is a witness for a voxel p, B ( q)  B ( p )
DT p
p
S
_
S
Skeleton Extraction

(5/12)
Claim 1
 The
ball of a voxel p must be contained in the ball of
one of its 26 neighbor if it is to be contained in the ball
of any other voxel in the object.
p
S
_
S
Skeleton Extraction
(6/12)

To find non-witness voxels, the 26-neighbors of all object
voxels need to be scanned. (brute force approach)

Rather than grow the ball for every neighbor and scan
for containment in the union of balls, a simple approach
is to average the DT value of the neighbors qi of p.
Skeleton Extraction

(7/12)
Definition 4
 the
mean of the neighbors’ distance transform (MNT)


26
MNTp
i 1
DTqi
26
, p, qi  S , qi is a 26-neighbor of p.
If MNTp < DTp , add p to the skeleton.
MNTp = 4
Skeleton Extraction

(8/12)
Thinness parameter (TP)
 We
introduce the TP, that allows control over the
removal of non-witness voxels yielding skeletons of
varying density.

Condition 1
 If
MNTp < DTp – TP , add p to the skeleton.
Low value of TP
Thick
High value of TP
Thin
Skeleton Extraction

A maple leaf and it’s skeleton
TP = 0


(9/12)
TP = 2
TP = 4
TP = 6
The Complexity of this algorithm is therefore O(N) ,
where N is the number of object-voxels.
These skeletal voxels are not generally connected.
(Spanning Tree)
Skeleton Extraction

Trachea
a. The segmented trachea
b. Skeleton with a thinness = 2.5
c. The minimum spanning tree of the skeletal voxels
(10/12)
Skeleton Extraction

Effact of the Thinness Parameter
(11/12)
Skeleton Extraction

Skeleton of S shape
(12/12)
Algorithm for the Centerline
(1/4)

A centerline is a curve that is centered with
respect to the object boundaries.

It can serve as the path for a virtual camera in
surgical path planning.

It is a semi-automatic algorithm in which the user
specifies end-points for centerline generation.
Algorithm for the Centerline

SK : the set of skeleton voxels.
p1 and p2 : the end-points of the centerline , p1,p2  SK
F : subdivision parameter (“ fineness ”)
(2/4)
Algorithm for the Centerline

Centerlines for Surgical Navigation
(3/4)
Algorithm for the Centerline

Centerlines for Surgical Navigation
(4/4)
Reconstruction (1/3)

In order to reconstruct the object from the
skeletal voxels, balls of radius equal to the
distance transform value have to be constructed.

A path terminates when its value is less than or
equal to 3 but greater than zero.

The quality of reconstruction depends upon the
thinness of the skeleton.
Reconstruction (2/3)

Lossless reconstruction
Reconstruction (3/3)

Lossy reconstruction