jung-ha an: statement of research - csu stan

Jung-Ha An: Statement of Research

California State University, StanislausDepartment of MathematicsOne University CircleTurlock, CA 95382

Phone: (209) 417-9552Email: [email protected]: http://www.csustan.edu/math/anHome: 2660 Jubilee Drive, Turlock, CA 95380

1 Introduction

My general research interests include mathematical modeling, mathematical biology, medical imaging,image processing, image analysis, calculus of variations,partial differential equations, statistical shape anal-ysis, and pattern recognition and classification. Image segmentation and registration, and shape analysisproblems play an important role in image processing, image analysis, and computer vision. My researchdirection is focused on developing mathematical models which provide solutions to image segmentation andregistration and shape analysis problems with an applications to medical imaging.

My interest in this specific research area stems from the limitations inherent in current medical imagingtechniques resulting in poor image quality. In order to address these shortcomings, the creation of efficientmodels to improve image quality with emphasis on particulardata sets must be achieved.

My research goal is the creation of effective mathematical models in medical imaging ultimately resultingin improved patient care and treatment. My long-term research objectives are:1) to create software products which will be used by medical specialists to diagnose or to prevent diseasethrough the discovery of fast and accurate numerical computational methods of the proposed models;2) to connect the university research community, a medical specialist, and industry partners by collaborat-ing in joint research. Since my doctorate, my research effort has been focused on solving medical imagingproblems to achieve my research goal. Effective mathematical models are developed to solve medical imag-ing problems and those techniques are often center around optimization problems. Calculus of variationsand partial differential equations are used to solve those optimization problems. Numerical computations ofproposed models are achieved by a gradient descent method and finite difference methods.

2 Past Research; Dissertation Work and Post-Doctorate Work

2.1 Dissertation Work

My past research work involved computational work with medical image segmentation problems. Mydoctoral dissertation focused on the solution to image segmentation and registration, and shape analysisproblems with an emphasis on human heart ultrasound medicalimaging. In medical imaging, prior shapeinformation is often used to obtain better segmentation results. Statistical shape analysis was used to gener-ate an average shape, often represented as prior shape information. In particular, self-organizing maps werestudied. The Self-Organizing maps were used with the Procrustes method to classify and average separateheart contour clusters. This model requires specification of the number of clusters in advance, but does notdepend upon the choice of an initial contour. The Procrustesmethod [13] was used for the alignment tomeasure the closeness between two contours. This techniquehad the advantage of averaging each cluster

Jung-Ha An: Statement of Research 2

and classifying all clusters simultaneously. The effectiveness of the method was illustrated in my exper-imental results in relation to human heart cardiac borders [3]. In an extension of my work in [3], I havecombined the Self-Organizing map with Principal ComponentAnalysis to gain more precise average shapeand clustering data. The orthogonal Procrustes method is used for the alignments, since it is both simpleand easy to implement and can be applied to any matrix form of data.

My previous doctoral work also involved a new variational partial differential equation (PDE) based levelset model for a simultaneous image segmentation and registration. This was an edge based algorithm. Tosolve an edge based simultaneous segmentation and registration problem, the following model [1, 4] wassuggested using a geometric active contour [12, 14]. Let C∗(p) = (x(p), y(p))(p ∈ [0, 1]) be a differentiableparameterized curve, called shape prior, in an associated prior imageI∗. The domainN(C∗) is the neighbor-

hood of the prior shape. LetGσ(x) = 1σ∗ e−

|x|2

4σ andg(|∇I|) can be chosen asg(|∇I|) = 11+β∗|∇(Gσ∗I)|2

,

whereβ > 0 is a parameter. The model is aimed to find u, v, a,µ, R, T by minimizing the energy functional:

E(u, v, a, µ,R, T ) = λ1

∫

N(C∗)|∇(Gσ ∗ I∗)(x, y) − a∇(Gσ ∗ I)(u, v)|2dx + λ2

∫

N(C∗)(|u|2 + |v|2)dx

+λ3

∫

N(C∗)(|∇u|2 + |∇v|2)dx + λ4

∫ 1

0g(|∇I|)(u, v)(C∗(p))|(C∗)′(p)|dp,

and the vector

[

u(x, y)v(x, y)

]

= µ ∗R ∗

[

x

y

]

+T +

[

u(x, y)v(x, y)

]

, whereµ is a scaling,R is a rotation matrix

with respect to an angleθ, andT is a translation. A prior imageI∗, a novel imageI, the prior shapeC∗, andthe neighborhoodN(C∗) of C∗ as a domain are given andλi > 0(i = 1, 2, 3, 4) are parameters balancingthe influences from four terms in the model. This technique incorporates both prior shape and intensityinformation. Since global rigid registration has limited applicability when non-rigid shapes are considered,I combine global registration with a non-rigid term. A simultaneous segmentation and non-rigid registrationis performed in this model. A gain setting is then applied to capture human heart motion and can be variedfrom image to image. Therefore, the variable scale factor ’a’ between the gradients of two smoothed imagesis also needed in the model. This model was applied to human heart endocardial two chamber end systoleultrasound images and was tested on images from thirteen patients.

Prior shapeC∗ and prior intensityI∗ of thirteen Human heart two chamber end systole ultra sound imageswhich have 0.62mm resolution in each pixel were generated by the method from [11]. Experimental resultsin Figure 1 indicate the model effectively detected the boundaries of the incompletely resolved objects whichwere plagued by noise, dropout, and artifact.

2.2 Post-Doctorate Work

During my post-doctorate, I worked to create an effective piecewise smooth region based mathematicalmodel in collaboration with Chenynag Xu and Mikael Rousson from Siemens Corporate Research. Thismodel was applied to human liver magnetic resonance imaging. Image segmentation is obtained using aΓ-Convergence approximation and multi-scale local statistics. Two phases are assumed for the simplicity ofour model. The model aims to find thephase fieldθ by minimizing the following energy [8]:

E(θ) = λ

∫

Ωf(θ)(I−uin(θ))2+(1−f(θ))(I−uout(θ))2dx+(1−λ)

∫

Ωε1|∇f(θ)|2+

f(θ)2(1 − f(θ))2

ε1dx,

(1)


100 150 200 250 300 350 400

100

150

200

250

300

350

(a)100 150 200 250 300 350 400

100

150

200

250

300

350

(b)

50 100 150 200 250 300

20

40

60

80

100

120

140

160

180

200

220

(c)50 100 150 200 250 300

20

40

60

80

100

120

140

160

180

200

220

(d)50 100 150 200 250 300

20

40

60

80

100

120

140

160

180

200

220

(e)

50 100 150 200 250 300

50

100

150

200

250

(f)50 100 150 200 250 300

50

100

150

200

250

(g)50 100 150 200 250 300

50

100

150

200

250

(h)

50 100 150 200 250 300

50

100

150

200

250

(i)50 100 150 200 250 300

50

100

150

200

250

(j)50 100 150 200 250 300

50

100

150

200

250

(k)

50 100 150 200 250 300

50

100

150

200

250

(l)50 100 150 200 250 300

50

100

150

200

250

(m)50 100 150 200 250 300

50

100

150

200

250

(n)

Figure 1: (a): An average image as a prior intensity information (b): An average curve with a solid line as aprior shape (c),(f),(i),(l): Solid line is an initial contour in the novel image (d),(g),(j),(m): The results fromthe model with non-rigid term (solid) and expert’s border (dotted) in an image (e),(h),(k),(n): The resultsfrom the model without non-rigid term (solid) and expert’s border (dotted) in an image


whereI is a given image,Ω ∈ R3 is its domain,f is a smooth version of the Heaviside function,ε1 is a

positive parameter, and0 < λ < 1 is a parameter balancing the influence of the two terms in the model.Following [15], uin and uout are expressed as local weighted intensity averages that canbe obtained byGaussian convolutions:

uin(θ) =gσ ∗ [f(θ)I]

gσ ∗ f(θ)and uout(θ) =

gσ ∗ [(1 − f(θ))I]

gσ ∗ (1 − f(θ)),

wheregσ is a Gaussian kernel with standard deviationσ, and “∗” stands for the convolution in the imagedomainΩ. The model considered so far is based on local intensity averages (uin, uout). The locality of theseterms is determined by the standard deviationσ of the Gaussian kernelgσ, which has been assumed to be thesame for all pixels. This may be a limitation, sinceσ should be large enough to attract the contour to edgesand small enough to detect weak edges. To resolve this problem, σ should be defined pixel-wise. However,this would highly increase the complexity of the algorithm.An intermediate step is to consider only twodifferent scalesσ1 andσ2, and combine them at each pixel with different weights. Letσ1 < σ2. We wish touse the region term withσ1 where the image gradient is high, and withσ2 where it is low. This leads us tothe following modification of the first term of Equation (1):

Eregion(θ) = λ

∫

Ωf(θ)

[

g(|∇I |)(I − uσ1,in(θ))2 + (1 − g(|∇I |))(I − uσ2,in(θ))2]

dx

+λ

∫

Ω(1 − f(θ))

[

g(|∇I |)(I − uσ1,out(θ))2 + (1 − g(|∇I |))(I − uσ2,out(θ))2]

dx,

whereg(|∇I |) acts as an edge detector,|∇I| stands for the magnitude of the gradient of a smoothed versionof the image that is normalized between 0 and 1, and the function g is an increasing function from[0, 1] to[0, 1]. This permits us to assign the lower sigma to low gradient edges, and enables the model to captureweak boundaries. Numerical results in Figure 2 showed the effectiveness of the proposed model.

3 Current Research; Ongoing Research Projects

My recent efforts have been focused on image segmentation byutilizing various partial differential equa-tion based (PDE) models. A simultaneous segmentation and registration model using a region intensityvalues is created with an application to simulated magneticresonance brain images. This is generalizedand extended result from my previous work [5] by combining two segmentation models into a simultaneoussegmentation and registration model. This work is in collaboration with Yunmei Chen.

A new variational region based model for a simultaneous image segmentation and a rigid registrationis proposed. The purpose of the model is to segment and register novel images simultaneously using amodified piecewise constant Mumford-Shah functional and region intensity values. The segmentation isobtained by minimizing a modified piecewise constant Mumford-Shah functional. A registration is assistedby the segmentation information and region intensity values. The model’s purpose is to findθ, c1, c2, Υ, d1,andd2 by minimizing the energy functional [6]:

E(θ, c1, c2,Υ, d1, d2) = λ1

∫

Ω1

2(1 +

2

πarctan(

θ

ǫ))2(I1(x) − c1)

2+

1

2(1 −

2

πarctan(

θ

ǫ)2(I1(x) − c2)

2dx + λ2

∫

Ω

ǫ1|∇(θǫ)|2

π2(1 + (θǫ)2)2

+(π2 − 4 arctan2(θ

ǫ))2

ǫ1π4dx+


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 2:Liver segmentation in an MR volume - Segmentations obtained on 2-dimensional slices of anMR volume ((a)σ = 4, (b) σ = 12, (c) σ1 = 4 andσ2 = 12, (d)∼(l) σ1 = 4 andσ2 = 8)


Synthetic Data Original Image

10 20 30 40 50 60

10

20

30

40

50

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

Segmented Contour as a Solid Line

10 20 30 40 50 60

10

20

30

40

50

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

20 40 60 80 100 120

20

40

60

80

100

120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c)20 40 60 80 100 120

20

40

60

80

100

120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d)20 40 60 80 100 120

20

40

60

80

100

120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(e)

20 40 60 80 100 120

20

40

60

80

100

1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)20 40 60 80 100 120

20

40

60

80

100

120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(g)20 40 60 80 100 120

20

40

60

80

100

1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(h)

Figure 3: (a): A synthetic imageI1 (b): The segmented synthetic imageI2 with noise, loss of information,and rotation with a solid line (c): A novel noisy imageI1 (f): A novel rotated noisy imageI2, (d): Thesegmented simulated brain image ofI1 (e): The segmented simulated brain image as a solid line ofI1 (g):The segmented simulated brain image ofI2 (h): The segmented simulated brain image as a solid line ofI2

λ3

∫

Ω1

2(1 +

2

πarctan(

θ

ǫ))2(I2(Υ(x)) − d1)

2 + 1

2(1 −

2

πarctan(

θ

ǫ)2(I2(Υ(x)) − d2)

2)dx,

whereI1 andI2 are novel images,Ω is domain,λi > 0 (i = 1, 2, 3), ǫ, andǫ1 are parameters balancing theinfluences from the four terms in the model. Even though the model is not limited to a rigid registration,only rigid transformation is considered in our numerical simulations. Therefore the rigid transformationvectorΥ(x) is equal toµRx + T, whereµ is a scaling,R is a rotation matrix with respect to an angleΘ,andT is a translation.

The numerical experiments of the proposed model are tested against synthetic data and simulated nor-mal noisy human-brain magnetic resonance (MR) images. The preliminary experimental results show theeffectiveness of the model in detecting the boundaries of the given objects and registering novel imagessimultaneously in Figure 3. We are currently working on extending this model to globally non-rigid regis-tration.

Another direction of my present research has been concernedwith a nerve ultrasound image segmentationproblem. Ultrasound scans have many important clinical applications in medical imaging. One major


clinical application is nerve location. One of the skills necessary to conduct ultrasound guided nerve blocksis the ability to recognize the nerves, vessels, muscles andbones in sagittal and axial cross sections. In fithealthy patients, these structures are reasonably easy to recognize but in obese patients, extra adipose tissueattenuates the ultrasound beam. This work is in collaboration with Paul Bigeleisen and Steven Damelin.

A new region based variational model is proposed using a modified piecewise constant Mumford-Shahfunctional and prior information. The region of interests are extracted by usingΓ-approximation to a piece-wise constant Mumford-Shah functional. However, this method alone, is not able to accommodate all typesof imaging difficulties including noise, artifacts, and loss of information. Therefore, the prior knowledgeis necessary to obtain an efficient image segmentation result. The prior information is incorporated withthe distance function. And this function consists of the global rigid transformation and local non-rigiddeformation. The model is aimed to findφ, u, µ, R, andT by minimizing the energy functional [7, 2]:

E(φ, u, µ,R, T ) = λ1

∫

ΩH2

ε (φ)(I(x) − c1)2 + (1 − Hε(φ))2(I(x) − c2)

2dx +

∫

Ωε1|∇Hε(φ)|2+

λ2Hε(φ)2(1 − Hε(φ))2

ε1dx +

λ3

2

∫

Ωδε(φ)d2(µRx + T + u)|∇(

φ

ε)|dx + λ4

∫

Ω|∇u|2dx + λ5

∫

Ωu2dx,

whereI is a given image,Ω is domain,λi > 0, i = 1, 2, 3, 4, 5 are parameters balancing the influencesfrom the five terms in the model,d is the distance function from the given prior shape,µ is a scaling,R isa rotation matrix with respect to an angleθ, T is a translation, andε andε1 are positive parameters. Thepreliminary numerical results in Figure 4 show the effectiveness of the suggested model and is comparedto an existing piecewise constant Mumford-Shah model and expert results. Currently, we are developingvalidation methods to measure the distance between a true solution from an anesthesiologist and proposedmodel’s results.

I am also actively engaged in medical imaging research with undergraduate students. I have been mentor-ing six undergraduate students since the fall of 2010. During the 2010-2011 academic year, I worked withfour students on the topics of vector calculus with applications to classical mechanics and with an empha-sis on Green’s theorem as background knowledge. Those research efforts were submitted for publications[9, 17]. Currently, four undergraduate students are focusing on three research areas including calculus ofvariations with emphasis on derivations of Euler-Lagrangeequations, applying gradient descent and finitedifferences schemes in contour extraction, and the alignment of arbitrary contours using area differencedistance measurements.

4 Conclusions and Future Research Directions

Through creative and collaborative research efforts, I have made important contributions to medical imag-ing research in various technical areas including ultrasound and magnetic resonance. My future researchwill be focused on the three major areas: 1) deepening my research by developing better faster numericalalgorithms to obtain more accurate segmentation results; 2) exploring further brain, cancer and cell imag-ing while also creating effective mathematical models to trace the region of interests in those images; 3)developing image segmentation models which are robust to any type of image noise.


Initial Contour as a White Solid Line

50 100 150 200 250 300 350

50

100

150

200

250

300

350

(a)

Segmented Image by Proposed Model

50 100 150 200 250 300 350

50

100

150

200

250

300

350

(b)

Segmented Contour as a White Solid Line by Proposed Model

50 100 150 200 250 300 350

50

100

150

200

250

300

350

(c)Initial Contour as a White Solid Line

50 100 150 200 250

50

100

150

200

250

300

350

(d)

Segmented Image by Proposed Model

50 100 150 200 250

50

100

150

200

250

300

350

(e)

Segmented Contour as a White Solid Line by Proposed Model

50 100 150 200 250

50

100

150

200

250

300

350

(f)Segmented Result by Chan−Vese Model

50 100 150 200 250

50

100

150

200

250

300

350

(g)

Segmented Result without Prior Information

50 100 150 200 250

50

100

150

200

250

300

350

(h)

White Solid Line by Expert and Red Solid Line by Proposed Model

50 100 150 200 250

50

100

150

200

250

300

350

(i)

Figure 4: (a),(d): The given image with an initial contour (b),(e): The segmented image result by a heavisidefunction (c),(f): The segmented contour result as a white solid line (g): The segmented result by Chan-Vesemodel [10] as a red solid line and comparison to an expert result as a white solid line (h): The segmentedresult without prior information as a red solid line and comparison to an expert result as a white solid line(i): The segmented result of the proposed model as a red solidline and comparison to an expert result as awhite solid line


References

[1] An, J. : Various methods in shape analysis and image segmentation and registration. Dissertation,University of Florida, (2005).

[2] An, J., Bigeleisen, P., and Damelin, S. : Identification of nerves in ultrasound scans using a modifiedMumford-Shah functional and prior information, International Conference on Computer Science andApplications 2011 at World Congress on Engineering and Computer Science 2011, Volume I, SanFrancisco, California, USA, (2011) 13–16.

[3] An, J., Chen, Y., Chang, M., Wilson, D., and Geiser, E. : Generating geometric models through Self-Organizing maps. Multiscale optimization methods and applications, Nonconvex Optim. Appl.,82,Springer, New York, USA, (2006) 241–250.

[4] An, J., Chen, Y., Huang, F., Wilson, D., and Geiser, E. : A variational PDE based level set methodfor a simultaneous segmentation and non-rigid registration. Medical Image Computing and Computer-Assisted Intervention (MICCAI), California, USA, (2005) 286–293.

[5] An, J. and Chen, Y. : Region based image segmentation using a modified Mumford-Shah algorithm,Scale Space Variational Methods in Computer Vision, Ischia, Italy, (2007) 733–742.

[6] An, J. and Chen, Y. : A piecewise constant region based simultaneous image segmentation and registra-tion, International Conference on Signal Processing and Imaging Engineering 2011 at World Congresson Engineering and Computer Science 2011, Volume I, San Francisco, California, USA, (2011), 491–494.

[7] An, J., Damelin. S., and Bigeleisen, P. : Medical image segmentation using modified Mumford seg-mentation methods, Section VII The Future of Ultrasound in the Ultrasound-Guided Regional Anes-thesia and Pain Medicine book, (2009) 289–294.

[8] An, J., Rousson, M., and Xu, C. : Gamma-Convergence approximation to piecewise smooth medi-cal image segmentation, Medical Image Computing and Computer-Assisted Intervention (MICCAI),Brisbane, Australia, (2007) 495-502.

[9] Barnett, J., Bishop, W., and An, J. : George Green: His Theorem and Its Application, Submitted(Research with Undergraduate Students).

[10] Chan, T. and Vese, L. : Active contours without edges, IEEE Trans. Image Proc, vol. 10 no. 2 (2001)266–277.

[11] Chen, Y., Huang, F., Tagare, H., Rao, M., Wilson, D., andGeiser, E. : Using prior shapes and intensityprofiles in medical image segmentation. Int. Conf. Comp. Vis. Nice France (2003) 1117–1124.

[12] Caselles, V., Kimmel, R., and Sapiro, G. : Geodesic active contours. Int. J. Comp. Vis. Vol.22 (1)(1997) 61–79.

[13] Chen, Y., Wilson, D., and Huang, F. : A new procrustes methods for generating geometric models.Proc. World Multiconference on Systems, Cybernetics and Informatics, (2001) 227–232.


[14] Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., and Yezzi, A. : Gradient flows and geometricactive contour models. Proc. ICCV’95 (1995) 810–815.

[15] Piovano, J., Rousson, M., and Papadopoulo, T. : Efficient segmentation of piecewise smooth images.In Proc. Scale Space Varional Methods in Computer Vision. (2007) 709–720.

[16] Shen. J.: A stochastic-variational model for soft Mumford-Shah segmentation . Int’l J. BiomedicalImaging, special issue on ”Recent Advances in MathematicalMethods for the Processing of Biomedi-cal Images”, vol. (2006) 1–14.

[17] Soomalan, M., Aguilar, D., and An, J. : Applications of Line Integrals in Classical Mechanics, Sub-mitted (Research with Undergraduate Students).

jung-ha an: statement of research - csu stan

Documents