Gradient-Oriented Boundary Profiles for Shape Analysis Using Medial Features
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Transcript of Gradient-Oriented Boundary Profiles for Shape Analysis Using Medial Features
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Gradient-Oriented Boundary Profiles for Shape Analysis Using Medial Features
Robert J. Tamburo, BS
Bioengineering
University of Pittsburgh
Under the Advisement of:
George D. Stetten, MD, PhD
U. Pitt. Bioengineering
CMU Robotics Institute
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Overview
Background Part I Gradient-Oriented Boundary Profiles Validation of Boundary Profiles Background Part II Boundary Profiles and Shape Analysis Results on Synthetic and RT3D Ultrasound Data Future Work Conclusion
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Clinical Motivation
In 1999:– Cardiovascular Disease (CVD) contributed to one-
third of worldwide deaths– CVD ranks as the leading cause of death in the U.S.
responsible for 40% of deaths per year– 62 million Americans live with some form of
cardiovascular disease Americans were expected to pay about $330
billion in CVD-related medical costs this year
*CDC/NCHS and the American Heart Association Causes of Death for All Americans in the United States, 1999 Final Data
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Image Analysis
Left ventricular (LV) and myocardial volume to calculate cardiac function parameters:
- cardiac output- stroke volume
- ejection fraction Myocardial thickness and motion can be monitored Diagnoses of CVD, including cardiomyopathy,
arrhythmia, ischemia, valve disease, myocardial infarction, and congestive heart failure
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Medical Imaging
2D ultrasound 3D ultrasound
– Gating to the electrocardiogram– Mechanically scanned
Cine-CT– 50 ms/slice (400 ms for full volume)
Real-time three-dimensional (RT3D) ultrasound– 22 frames/sec (45 ms)
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Goals
Automatically identify and measure structures RT3D ultrasound data
Develop “intelligent” boundary points: Gradient-Oriented Boundary Profiles
Apply to Profiles to a shape analysis routine
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Boundary Detection
First step in most Image Analysis routines Convolution with kernel in spatial domain High-pass frequency filters in frequency
domain
Spatial domain detection:– is computationally less expensive– often yields better results
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Gradient Based Detectors
Gradient magnitude is rotationally insensitive Gradient magnitude sensitive to:
– object intensity– background intensity– overall image contrast
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Common Gradient Based Detectors
Roberts Cross– 2x2 kernel– Very sensitive to noise– Very fast
Sobel– 3x3 kernel– Somewhat sensitive to noise– Slower than Roberts Cross
Both amplify high-frequency noise (derivative)
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Gradient Based Boundary Detectors With Smoothing
Marr– Gaussian Smoothing– Laplacian of Gaussian
Canny– Gaussian smoothing – Ridge tracking
Both require multiple applications Some fine detail lost
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Algorithm for Classifying Boundaries
1. Find candidate boundary points
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Difference of Gaussian (DoG) Detector
Gradient magnitude Gaussian smoothing Difference between 3 same-scale Gaussian
kernels Measures gradient direction components in 3D
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Finding Candidate Boundary Points
Over sample with small sampling interval Apply gradient detector (DoG) Accept those above pre-determined threshold
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Algorithm for Classifying Boundaries
1. Find boundary candidates
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Generating an Intensity Profile
Sample voxels in a neighborhood Partition sampling region Project voxels (splat) to the major axis
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Sampling Voxels
Ellipsoidal or cylindrical Centered at boundary point Major axis in direction of gradient Reduces the effect of image noise
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Splatting Voxel Intensity
Triangular or Gaussian footprint Store weights of contribution Profile of average voxel intensity
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The Intensity Profile
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Algorithm for Classifying Boundaries
1. Find boundary candidates
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Fitting the Profile
Choice of function– Should parameterize boundary– Should be intuitive
Image acquisition blurs boundaries Convolution with a Gaussian kernel Step function becomes a cumulative
Gaussian
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Fitting the Profile cont.’d
Image Acquisition
Real Boundary
Image Boundary
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Derivation of Cumulative Gaussian
2
2
2
2
1)(
x
exG
2
)(
xerf
x
xdvvG
2
12
121
xerf
IIIxC
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Cumulative Gaussian
A function of 4 parameters
1. Mean, 2. Standard deviation, 3. Asymptotic value for one side, I1
4. Asymptotic value for other side, I2
2
12
121
xerf
IIIxC
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Boundary Parameterization
• - boundary location • - boundary width• I1 - intensity far inside boundary
• I2 - intensity far outside boundary
d i s t a n c e a l o n g g r a d i e n t
d d
p 1 p 2
s a m p l e d r e g i o n o f p r o f i l e
1 2 i n t e n s i t y
I1
I2
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Curve Fitter
AD Model Builder from Otter Research, Inc.*
Quasi-Newton non-linear optimization Auto-differentiation Rapid and robust
*http://otter-rsch.com/admodel.htm
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Algorithm for Classifying Boundaries
1. Find boundary candidates
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Eliminating “Bad” Profiles
“Bad” – profile for which parameters are unacceptible– I1 or I2 is outside range for the imaging modality
– is outside of the ellipsoidal sample region
These profiles are rejected and no longer considered
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Algorithm for Classifying Boundaries
1. Find boundary candidates
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Establishing Intrinsic Measures of Confidence
Based on location and width of boundary within sampling region
Place thresholds on measures of confidence Accept high-confidence parameters
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Measures of Confidence for I1 and I2
1
1d
z
22
dz and
d i s t a n c e a l o n g g r a d i e n t
d d
p 1 p 2
s a m p l e d r e g i o n o f p r o f i l e
1 2 i n t e n s i t y
I1
I2
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Measure of Confidence for
zmin = min(z1, z2)
Sufficient samples exist on both sides of
d i s t a n c e a l o n g g r a d i e n t
d d
p 1 p 2
s a m p l e d r e g i o n o f p r o f i l e
1 2 i n t e n s i t y
I1
I2
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Algorithm for Classifying Boundaries
1. Find boundary candidates
2. Create an intensity profile
3. Fit a cumulative Gaussian to the intensity profile
4. Eliminate blatantly “bad” profiles
5. Calculate measures of confidence
6. Classify the boundary
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Classify the Boundary
Classify boundary with high-confidence parameters
Boundary is classified by:– Intensity on both sides of boundary– Estimate of true boundary location
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Application to Test Data
3D data set– 8-bit voxels– 100x100x100
Generated sphere– radius of 30 voxels– interior value of 32– exterior value of 64
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Validation on Sphere
Ellipsoidal vs. Cylindrical sampling regions Triangle vs. Gaussian footprints Measures of confidence determined Validation of improved boundary location
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Radius RMS Errors
n
ierrorR
nRMS
1
21
Neighborhood Type Splat Type RMS
Cylindrical Gaussian 0.092
Cylindrical Triangle 0.104
Ellipsoidal Gaussian 0.086
Ellipsoidal Triangle 0.078
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Radius Error from Estimated Ellipsoidal Neighborhood and Triangle Splat
0
50
100
150
200
250
300
350
0.05 0.
10.
15 0.2
0.25 0.
30.
35 0.4
0.45 0.
50.
60.
80.
850.
95 1.8
1.85 1.
91.
95 23.
8
Radius Error (voxels)
Fre
qu
ency
95% of profiles estimate radius to less than 1 voxel
estimatetrueerror RRR
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Radius Error From DoG Kernel
0
50
100
150
200
250
300
1 2 3 4 5 6 7 8 9
Radius Error (voxels)
Fre
qu
ency
23% of points estimate radius to less than 1 voxel
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Boundary Points and Profiles
DoG boundary points Boundary profiles
90 secs
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The distribution of error in estimating the intensity values on either side of the boundary as a function of
Intensity Errors vs. relative
0
5
10
15
20
25
30
35
40
45
-4 -3 -2 -1 0 1 2 3 4
relative (voxels)
Inte
nsi
ty E
rro
r (v
oxe
ls)
I1 error
I2 error
interior exterior
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minz > 1.5 results in error < 1
Error vs min(z1,z2)Ellipsoidal Neighborhood and Triangle Splat
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5
min(z1,z2) (voxels)
E
rro
r (v
oxe
ls)
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I1 Error vs. z1
0
5
10
15
20
25
30
1 1.5 2 2.5 3 3.5 4 4.5 5
z1 (voxels)
I 1 E
rro
r (i
nte
ns
ity
0-2
55)
0.21 z 10errorIA threshold of guarantees
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I2 Error vs. z2
0
10
20
30
40
50
0 1 2 3 4 5 6
z2 (voxels)
I 2 E
rro
r (i
nte
ns
ity
0-2
55
)
5.12 z 10errorIA threshold of guarantees
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Radius Error From Estimated
0
50
100
150
200
250
300
0.05 0.1 0.15 0.2
Radius Error (voxels)
Fre
qu
enc
y
Boundary profiles with high-confidence estimates
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Medial-Based Shape Analysis
Medial axis by Blum Medialness by Pizer Robust against image noise and shape
variation* Stetten automatically identified LV and
measured volume
*Morse, B.S., et al., Zoom-Invariant vision of figural shape: Effect on cores of image disturbances. Computer Vision and Image Understanding, 1998. 69: p. 72-86
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Core Atom
1b 2b
center
Computationally efficient Statistically analyzed to extract medial
properties of the core Require a priori knowledge of object intensity Can not differentiate between objects of
different intensity
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Core Profiles
2I
1b 2b
2,1s
11I 22I
12I
1I
center
2I
I21
•Form independent of background intensity
•Multiple objects of differing intensities can be found
•Better boundary location
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Medial Requirements
•Face-to-faceness is close to 1
2
1,2
1,21
2,1
2,121, n
s
sn
s
sbbF
in is the orientation of the ith boundary profile
•Distance between boundary profiles within range
122,1 bbs
max2,1min sss
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Medial Requirements
•Boundary profiles have high-confidence estimates)( 1111 zthresholdz
)( 1212 zthresholdz
1211 II
where is an intensity tolerance
•1.
•2.
•3.
•Constraint 3 is for homogeneous core profiles
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Medial Requirements
1b
2b 3b
4b
2,1s
3,2s
4,3s
4,1s
•Solid lines are homogeneous
•Dashed lines are heterogeneous exhibiting lateralness
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Basic Core Configurations
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Measuring Medial Properties
•Population of core profiles analyzed
•Eigenvalues define dimensionality of the core
•Eigenvectors define population orientation
321
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Lambda Triangle
sphere slabcylinder
12
13
sphere
slab
21
1321 Constraints:1.
2.
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Hollow Sphere
Left Ventricle
MyocardiumEpicardium
Endocardium
Models cardiac data To calculate volumes 3D data set
– 8-bit voxels– 100x100x100
Hollow sphere– inner radius of 15 voxels (intensity of 32)– outer radius of 30 voxels (intensity of 128)– background of intensity 64
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Hollow Sphere - Boundaries
as
Boundary ProfilesDoG Boundary Points
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Hollow Sphere – Core Profiles
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Hollow Sphere - Medialness
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Hollow Sphere – Core Profile Radii
Distribution of Core Profile Radius
0
1000
2000
3000
4000
5000
6000
0 11.
5 22.
5 33.
5 44.
5 55.
5 66.
57.
5 11 1616
.5 1717
.5 18 2222
.5 23
Core Profile Radius (voxels)
Nu
mb
er o
f C
ore
Pro
file
s
The center of the sphere is at 0 and the center of the slab between the spheres is at 22.5
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Hollow Sphere – Radius Errors Error
DoG vs. Profiles
0
100
200
300
400
500
600
700
800
900
1000
0.4 0.7 1 1.3 1.6 4.6077
Error (voxels)
Fre
qu
ency
Error From DoG
Error From Profiles
96% of the total profiles vs. 29% of the total DoG points estimated a boundary location within one voxel
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Hollow Sphere – Core Profile Scale
Distribution of Core Profile Scale
0
500
1000
1500
2000
2500
14.5
15.5
20.5
21.5 24 29 30
43.5
45.5
48.5
55.5
56.5
57.5
58.5
59.5
60.5
64.5
68.5
Core Profile Scale (voxels)
Nu
mb
er o
f C
ore
Pro
file
s
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Hollow Sphere – Volume Measures
•Core atoms applied twice
•Volume measures are both fairly accurate
•Standard deviation of scales shows consistency
Method of Calculation LV Volume (voxels) Heart Volume (voxels) Myocardium Volume (voxels)
Known Parameters of Data 14,137 113,097 98,960
Average Core Atom Scale 13,158 (PE = 7%, 2.7) 114, 082 (PE = 1%, 5.4) 100,924 (PE = 2%)
Average Core Profile Scale 13,215 (PE = 6%, 2.1) 111,002 (PE = 2%, 2.3) 97,787 (PE = 1%)
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Concentric Ellipsoids
Models RT3D phantom Determines expected
medialness Illustrate non-parametric
volume measure
techniques
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Concentric Ellipsoids – Profiles
Homogeneous Boundary Profiles
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Concentric Ellipsoids – Medialness
Cylindricalness and slabness of concentric ellipsoids
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Concentric Ellipsoids – Volume
•2 proposed techniques
•Rely on dense core profiles or medial node population
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Search and Count Method
•Construct ellipsoids around core profiles
•Average intensity of core profile
•Add voxel to volume count if within tolerance of average
•Requires dense core profile population
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Medial Region Fill
•Construct spheres around each medial node
•Deform sphere to an ellipsoid in direction orthogonal to pop.
•Expand ellipsoid until they collide with object boundaries
•Count voxels within ellipsoid for volume measure
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Real-Time 3D Ultrasound
•Developed in the early 90’s at Duke University
•Matrix array of transducer elements
•Captures pyramid of data at approximately 22 frames per second
•Rapid enough to acquire cardiac data throughout its cycle
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RT3D Cardiac Phantom
Phantom from OHSU Two latex balloons Ultrasound Gel solution
between balloons Water in inner balloon
B-mode slices
C-mode slice
Myocardium
Left Ventricle
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RT3D Cardiac Phantom
Homogeneous boundary profiles Population of core profiles
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RT3D Cardiac Phantom
Slabness found from short core profiles
Medial nodes found from long core profiles
Two passes
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RT3D Cardiac Phantom
Resulting medial nodes Applying constraints
Single pass
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Future Work
Improve computational speed of profiles Construct models from medial nodes Compute volumes from models
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Insight Toolkit (ITK)
Sponsored by National Library of Medicine Open-source registration and segmentation
toolkit Architecture for large datasets Generic programming Boundary profiles have been contributed http://www.itk.org
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Conclusions
Gradient-Oriented Boundary profiles:– accurately parameterize boundaries – improve the results of core atoms– can locate boundaries in noisy data– computationally expensive
Measures of confidence shown to eliminate low-confidence parameters
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Acknowledgments
Dr. Stetten Aaron Cois Damion Shelton Wilson Chang Dr. Sclabassi Dr. Li And….
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YOU!YOU!