Gradient-Oriented Boundary Profiles for Shape Analysis Using Medial Features Robert J. Tamburo, BS Bioengineering University of Pittsburgh Under the Advisement

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

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

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

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

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)

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

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

Gradient Based Detectors

Gradient magnitude is rotationally insensitive Gradient magnitude sensitive to:

– object intensity– background intensity– overall image contrast

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)

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

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

Difference of Gaussian (DoG) Detector

Gradient magnitude Gaussian smoothing Difference between 3 same-scale Gaussian

kernels Measures gradient direction components in 3D

Finding Candidate Boundary Points

Over sample with small sampling interval Apply gradient detector (DoG) Accept those above pre-determined threshold


1. Find boundary candidates






Generating an Intensity Profile

Sample voxels in a neighborhood Partition sampling region Project voxels (splat) to the major axis

Sampling Voxels

Ellipsoidal or cylindrical Centered at boundary point Major axis in direction of gradient Reduces the effect of image noise

Splatting Voxel Intensity

Triangular or Gaussian footprint Store weights of contribution Profile of average voxel intensity

The Intensity Profile








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

Fitting the Profile cont.’d

Image Acquisition

Real Boundary

Image Boundary

Derivation of Cumulative Gaussian

2

2

2

2

1)(

x

exG

2

)(

xerf

x

xdvvG

2

12

121

xerf

IIIxC

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

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

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








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








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

Measures of Confidence for I1 and I2

1

1d

z

22

dz and


d d

p 1 p 2



I1

I2

Measure of Confidence for

zmin = min(z1, z2)

Sufficient samples exist on both sides of


d d

p 1 p 2



I1

I2








Classify the Boundary

Classify boundary with high-confidence parameters

Boundary is classified by:– Intensity on both sides of boundary– Estimate of true boundary location

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

Validation on Sphere

Ellipsoidal vs. Cylindrical sampling regions Triangle vs. Gaussian footprints Measures of confidence determined Validation of improved boundary location

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

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

Radius Error From DoG Kernel

0

50

100

150

200

250

300

1 2 3 4 5 6 7 8 9


Fre

qu

ency

23% of points estimate radius to less than 1 voxel

Boundary Points and Profiles

DoG boundary points Boundary profiles

90 secs

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

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)

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

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

Radius Error From Estimated

0

50

100

150

200

250

300

0.05 0.1 0.15 0.2


Fre

qu

enc

y

Boundary profiles with high-confidence estimates

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

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

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

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

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

Medial Requirements

1b

2b 3b

4b

2,1s

3,2s

4,3s

4,1s

•Solid lines are homogeneous

•Dashed lines are heterogeneous exhibiting lateralness

Basic Core Configurations

Measuring Medial Properties

•Population of core profiles analyzed

•Eigenvalues define dimensionality of the core

•Eigenvectors define population orientation

321

Lambda Triangle

sphere slabcylinder

12

13

sphere

slab

21

1321 Constraints:1.

2.

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

Hollow Sphere - Boundaries

as

Boundary ProfilesDoG Boundary Points

Hollow Sphere – Core Profiles

Hollow Sphere - Medialness

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

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

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

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%)

Concentric Ellipsoids

Models RT3D phantom Determines expected

medialness Illustrate non-parametric

volume measure

techniques

Concentric Ellipsoids – Profiles

Homogeneous Boundary Profiles

Concentric Ellipsoids – Medialness

Cylindricalness and slabness of concentric ellipsoids

Concentric Ellipsoids – Volume

•2 proposed techniques

•Rely on dense core profiles or medial node population

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

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

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

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


Homogeneous boundary profiles Population of core profiles


Slabness found from short core profiles

Medial nodes found from long core profiles

Two passes


Resulting medial nodes Applying constraints

Single pass

Future Work

Improve computational speed of profiles Construct models from medial nodes Compute volumes from models

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

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

Acknowledgments

Dr. Stetten Aaron Cois Damion Shelton Wilson Chang Dr. Sclabassi Dr. Li And….

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Documents

Gradient-Oriented Boundary Profiles for Shape Analysis Using Medial Features Robert J. Tamburo, BS Bioengineering University of Pittsburgh Under the Advisement