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Appendix A

The Fourier Transform and3D

A?ne Deformations

In this section,we demonstrate the e?ect of a spatial3D a?ne deformation on the Fourier representation of an object.Suppose that f(x)=f(x,y,z)has a3D Fourier transform F(u)=F(u,v,w).The3D a?ne transformation of f can be de?ned for points x∈f by

A(x)=A.x+t.(A.1) where A is a3x3matrix,and t is a3x1translation vector(see Section2.1.1).If g(x) is the3D a?ne transformation of f(x)then

g(x)=f A?1(x?t) (A.2) The Fourier transform of g(x)can be written as

G(u)= ∞?∞f A?1(x?t) e?i2π(u·x)d x(A.3)

Making the substitution of variables?x=A?1(x?t),and since d?x=|det(A?1)|d x we obtain

G(u)=

1

|det(A?1)|

∞

?∞

f(?x)e?i2π(u·(A?x+t))d?x(A.4)

=

e?i2π(u·t)

|det(A?1)|

∞

?∞

f(?x)e?i2π(u·A?x)d?x(A.5)

=

e?i2π(u·t)

|det(A?1)|

∞

?∞

f(?x)e?i2π(A T u·?x)d?x(A.6)

=

e?i2π(u·t)

|det(A?1)|

F(A T u)(A.7)

To write the inverse mapping,make the variable substitution?u=A T u,

G(A?T?u)=e?i2π(A?T?u·t)

|det(A?1)|

F(?u),(A.8)

so that

F(?u)=e i2π(A?T?u·t)

|det(A)|

G(A?T?u).(A.9)

Vita

EDUCATION

Johns Hopkins University,Baltimore,Maryland USA

?Ph.D.,Biomedical Engineering,1996-2003

?B.S.,Biomedical Engineering,1993-1996

?B.A.,Computer Science,1993-1996

EXPERIENCE

?Research Engineer05/2001-01/2002 INRIA(National Research Institute for Computer Science and Control)

Sophia Antipolis,France

–Member of the Chir(Surgical Robotics)group

–Developed an algorithm for3D motion tracking of coronary arteries in

x–ray biplane cineangiograms

?Pre-doctoral Research Fellow2000-2003 Laboratory of Cardiac Energetics,NHLBI,NIH,DHHS

Bethesda,MD,USA

–“Respiratory Motion of the Heart:Implications for MR Imaging”

–Primary advisor:Elliot R.McVeigh

?Graduate Student and Teaching Assistant1996-2003 Department of Biomedical Engineering,Johns Hopkins School of Medicine

Baltimore,MD,USA

–Developed a visualization and interactive template-based segmentation

tool for medical images

–Conducted weekly review sessions as a teaching assistant for:Honors C

Programming(1995);Intermediate Programming in C++(1996);Mag-

netic Resonance in Medicine(1998);and Physiologic Foundations in Biomed-

ical Engineering(1999)

?Undergraduate Research Assistant1994-1996 Department of Biomedical Engineering,Johns Hopkins School of Medicine

Baltimore,MD,USA

–“Computer aided navigation through parameter space:A case study using

the incremental model for calcium release by the IP3receptor”

–Primary advisor:Robert D.Phair

HONORS AND A W ARDS

?NIH Fellows Award for Research Excellence,2003

?NIH Pre-IRTA Fellowship,2000-2003

?ISMRM Student Stipend Award,1999

?HHMI Predoctoral Fellowship,Honorable mention,1997

?Whitaker Foundation,Graduate Fellowship,1997-2000

?JHU Biomedical Engineering Undergraduate Research Award,1996

?JHU Computer Science Outstanding Double Major Award,1996

?Tau Beta Pi,Engineering Honor Society,1996-1997.

?JHU Provost’s Undergraduate Research Award,1995

PEER REVIEWED JOURNAL ARTICLES

1.G.Shechter,F.Devernay,E.Coste-Mani`e re,A.Quyyumi,and E.R.McVeigh.

Three-dimensional motion tracking of coronary arteries in biplane cineangiograms.

IEEE Transactions on Medical Imaging22(4):493–503,2003.

2.G.Shechter,C.Ozturk,J.R.Resar,and E.R.McVeigh.Respiratory motion

of the heart from free breathing coronary angiograms.IEEE Transactions on Medical Imaging,in press,2004.

CONFERENCE ABSTRACTS AND PAPERS

1.G.Shechter,J.R.Resar,and E.R.McVeigh.Translational motion correction

nearly doubles the rest period of the coronary arteries during tidal breathing, 11th Scienti?c Meeting of the International Society for Magnetic Resonance in Medicine,Kyoto,Japan,May2004.

2.G.Shechter,C.Ozturk,J.R.Resar,and E.R.McVeigh.Respiratory motion of

the heart:Translation,Rigid body,A?ne,or More?7th Annual Meeting of the Society for Cardiovascular Magnetic Resonance,Barcelona,Spain,Feb2004.

3.G.Shechter,C.Ozturk,J.R.Resar,and E.R.McVeigh.Free breathing respi-

ratory motion of the heart measured from x–ray coronary angiograms.SPIE Medical Imaging Conference,San Diego,CA,Feb2004.

4.G.Shechter and E.R.McVeigh.MR motion correction of3D a?ne deforma-

tions.11th Scienti?c Meeting of the International Society for Magnetic Reso-nance in Medicine,Toronto,ON,July2003.

5.G.Shechter and https://www.wendangku.net/doc/47362872.html,puter models of vascular trees from biplane

angiograms.28th Annual Scienti?c Meeting of the Society for Interventional Radiology,Salt Lake City,UT,Mar2003.