text:
WASHINGTON A surging Rick Santorum is running even with Mitt Romney atop the Republican presidential field, but neither candidate is faring well against President Barack Obama eight months before Americans vote, a new survey shows.
Obama tops 50 percent support when matched against each of the four GOP candidates and holds a significant lead over each of them, according to the Associated Press-GfK poll. Republicans, meanwhile, are divided on whether they'd rather see Romney or Santorum capture the nomination, with Newt Gingrich and Ron Paul lagging behind. It's a troubling sign for the better-funded Romney as the GOP race heads toward crucial votes in his home state of Michigan, in Arizona and in an array of states on Super Tuesday, March 6.
"I'd pick Santorum, because it seems Romney may be waffling on a few issues and I'm not sure I trust him," said Thomas Stehlin, 66, of St. Clair Shores, Mich. He thinks the Detroit-born son of a Michigan governor is facing a strong challenge from Santorum in his home state because of his tangled answers on the auto industry bailout.
Also, he says, there's this: Romney, the self-described can-do turnaround artist of the corporate world and the troubled Salt Lake City Olympics, with his millions of dollars, has been unable to vanquish his political opponents.
"That may be the reason right there," said Stehlin, a retired government worker and a Republican. "He spends lots of money and he doesn't get anywhere."
Nationally, Republicans are evenly split between Romney and Santorum. The poll found 33 percent would most like to see Santorum get the nomination, while 32 percent prefer Romney. Gingrich and Paul each had 15 percent support.
Romney's fall from presumed front-runner to struggling establishment favorite has given his opponents an opening as he tries to expand his support. His Republican rivals have stepped in claiming to be a more consistent conservative and viable opponent against Obama, and each of the last three AP-GfK polls has found a different contender battling Romney for the top spot. But Santorum, the former Pennsylvania senator and abortion foe, has hit his stride at a key moment in the nomination contest.
Santorum's spike comes as satisfaction with the field of candidates remains tepid and interest in the contest is cools. About 6 in 10 Republicans in the poll say they are satisfied with the people running for the nomination, stagnant since December and below the 66 percent that felt that way in October. Only 23 percent are strongly satisfied with the field and 4 in 10 said they are dissatisfied with the candidates running, the poll found. And deep interest in the race is slipping: Just 40 percent of Republicans say they have a great deal of interest in following the contest, compared with 48 percent in December.
"It seems like in the last month or so everything's just chilled out," said James Jackson of Fort Worth, Texas, a 40-year-old independent who leans Republican. "I just haven't been following it lately."


text:
Mar 25 - Homily: The Annunciation and Good Friday
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Fr. Elias gives he homily on Monday in Holy Week which this year falls on the 25th of March, the Annunciation (which will be celebrated after Easter). He uses this fact to highlightĀtheĀconnection between the Annunciation, and therefore Christmas and Good Friday and Easter.ĀIsaiah is a summary of the faith and foresees many of these events,ĀagainĀconnecting them all together.ĀThis will give us confidence that all the events in our lives are connected and fitting in with God's wise and holy plans and will reconcile and interconnect many things that seem likeĀoppositesĀlike charity and dogma as well as rich and poor and even love and duty to God with that which we owe to our neighbor. Ave Maria! Mass: Monday Holy Week - Wkdy - Form:ĀOF Readings: 1st: isa 42:1-7Ā Resp: psa 27:1, 2, 3, 13-14 Gsp: joh 12:1-11 To Download Audio go to http://airmaria.com?p=34457
Annunciation Good Friday Holy week Isaiah
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Google's not the only one to have drawn scrutiny for fake mortgage programs.
First published on Search Engine Watch
The fake mortgage programs that brought Google under recent scrutiny don't seem to be run by Google-exclusive con artists. Mortgage scams that claim to provide assistance from the U.S. government have also been located in Microsoft adCenter. Microsoft has cooperated in banning the associated advertisers from Bing and Yahoo search results.
According to the statement from SIGTARP (the Special Inspector General for the Troubled Asset Relief Program), at least 125 illicit companies of this nature were identified. The investigation, however, remains ongoing.
Microsoft, meanwhile, has extended its cooperation in much the same way Google did earlier on in the investigation. Microsoft proactively shut down the accounts of more than 400 advertisers that were connected to the 125 mortgage scam companies.
"Many homeowners who have fallen prey to these scams were enticed by web banner ads and online search advertisements that promised, for a fee, to help lower mortgage payments," according to Christy Romero, Deputy Special Inspector General for SIGTARP.
In addition to working with Google and Microsoft to shut down these ads (which Romero states will "dramatically decrease the scope and scale of these scams"), SIGTARP hopes to educate home owners.
According to the official SIGTARP statement, "Homeowners can protect themselves from becoming a victim of these scams by seeking a HAMP mortgage modification directly through their lender or mortgage servicer or through HUD-approved housing counselors." The HAMP program is free of charge and approved by the U.S. government.
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Winnipeg Free Press - PRINT EDITION
Overland flooding significant in areas of southern Manitoba
SOUTHERN areas of Manitoba are experiencing significant overland flooding. Cooler weather and forecasted snow and rain in southern Manitoba and the Fargo, N.D., area will slow melt rates and extend the length of time flooding occurs.
While the upper level of the forecast range has been reduced slightly, it's still above 2009 levels for all points on the Red River -- 2009 was the second-largest flood in the past 150 years.
There were 88 provincial roads affected by flooding, including 44 sections closed, as of Thursday afternoon, and hundreds of municipal roads.
There have been 669 evacuations related to flooding, primarily from the Peguis, Roseau River and Dauphin River first nations.
Red River Valley
Highway 75 continues to be monitored, but was still open Thursday. Residents of the personal-care home in St. Adolphe will be transferred to Grunthal early next week in a precautionary move.
Southwest Manitoba / Assiniboine River
A significant ice jam caused sudden high water levels at Spruce Woods Provincial Park and overtopped the dike on PTH 5.
Tracks along the main Canadian Pacific Railway line have been damaged by flood waters east of Brandon. Trains are being temporarily rerouted.
Patients in care homes in Wawanesa and Gladstone have been moved to alternate facilities.
Forecast summary
Significant snow is forecast for the upper portions of the Red, Souris, and Assiniboine river basins this weekend.
Red River
Water levels on the main stem of the Red River at Emerson increased 0.36 feet since Wednesday. Water levels at the Red River Floodway inlet have declined slightly since Wednesday.
Assiniboine River
Ice has moved out at various locations along the Assiniboine River and the main stem of the river remains within its banks at most locations.
Pembina River
A flood warning remains in effect for the Pembina River between Rock Lake and upstream of La Rivière. Flows on the main stem of the Pembina River and its tributaries are expected to crest for a longer period of time than in previous years.
Souris River
A flood warning remains in effect for the Souris River between Melita and Wawanesa and the river is over the banks at several locations. The water level on the Souris River near Melita has increased 1.5 feet since Wednesday, and 2.5 feet at Wawanesa.
Interlake Region
The Icelandic River has crested and flows are decreasing. The Icelandic River has reached bank-full in Arborg. Water levels upstream of Riverton have decreased. Water levels in the Fisher River have stabilized after rising more than three feet.
-- source: Government of Manitoba
Republished from the Winnipeg Free Press print edition April 15, 2011 A4
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Although the State Department is both a producer and a consumer of geospatial data, geo-analytics and maps, not all of its work is kept behind closed doors.
For instance, State's Humanitarian Information Unit collects, analyzes and disseminates unclassified information regarding humanitarian emergencies, and publishes high-quality maps that track relevant variables such as refugee migration and global health initiatives. The HIU team consists of roughly 20 analysts, researchers, geospatial analysts, cartographers and developers under the department's Office of the Geographer and Global Issues.
Josh Campbell, GIS architect at HIU, said the unit's maps can help people make sense of complex humanitarian emergencies and understand what is happening on the ground.
Main feature: Why maps matter
State Department: Mapping the humanitarian crisis in Syria
NGA: Seeking full immersion in geospatial data
EPA: Using maps to make sense of water pollution data
The team is known for its accuracy, and HIU's maps are widely cited by the media. But the work is intensive. In 2013, the team was particularly busy tracking the civil war in Syria. To document the migration of refugees, analysts pooled data from media reports, commercial satellite imagery and internal documents. Then subject-matter experts sifted through the data to ensure its legitimacy, technical staffers built corresponding datasets, GIS analysts compiled the data, and cartographers produced the finished products.
Some maps showed the displacement of refugees over time while others mapped the escape routes and destinations of the 1.8 million Syrian refugees, including the hot spots from which they fled, where they went and where the humanitarian resources were located. The maps gave decision-makers valuable insight into the conflict from a humanitarian standpoint.
"Maps unlock a great cognitive power," Campbell said. "Visualization, whether graphical or geographical, simplifies complex relationships."
About the Author
Frank Konkel is a former staff writer for FCW.
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New to Typophile? Accounts are free, and easy to set up.
i'm in a bind here on a rush job. i need to know the supporting serif used in the twilight materials here. i understand the twighlight title is set in p22's zephyr however i can't place the other type (saga, new moon etc…) . i'd look myself, but i'm already under the gun and need to start the layout. any help is very much appreciated. thanks in advance.
nvr mind. it's requiem fyi.
This is the kind of ID request we like. Good work.
- Mike Yanega


text:
District of Florida Real Property Cases
Cases filed
Cases 1 - 10 of 3,005
ORCHID QUAY LLC v. SUNCOR BRISTOL BAY LLC
as 2:2015cv14109
Plaintiff: ORCHID QUAY LLC
Defendant: SUNCOR BRISTOL BAY LLC
Cause Of Action: Diversity-Injunctive & Declaratory Relief
ETIENNE v. FLORIDA HOME IMPROVEMENT, LLC
as 4:2015cv00171
Cause Of Action: Diversity-Tort/Non-Motor Vehicle
Martin et al v. PNC Bank, National Association
as 6:2015cv00487
Plaintiff: Jade M. Martin , Michael A. Overbey , Randy E. Overbey
Defendant: PNC Bank, National Association
Cause Of Action: Notice of Removal-Petition to Quiet Title
Melia v. Wilmington Finance et al
as 2:2015cv00189
Plaintiff: Pamela A. Melia
Defendant: Wilmington Finance, Bank of New York Mellon, Greentree Servicing and others
Cause Of Action: (a) Fed. Question: Real Property
Wint et al v. BAC Home Loans Servicing, L.P.
as 9:2015cv80376
Plaintiff: Gail Wint, Oral Wint
Defendant: BAC Home Loans Servicing, L.P.
Cause Of Action: Fair Debt Collection Act
O'neil v. CitiMortgage, Inc.
as 1:2015cv21138
Plaintiff: Joseph H. O'Neil
Defendant: CitiMortgage, Inc.
Cause Of Action: Notice of Removal-Foreclosure
WASHINGTON et al v. THE WINE GROUP, INC. et al
as 4:2015cv00163
Plaintiff: TACARRA WASHINGTON , LATANJA JOHNSON , BERNARD JOHNSON and others
Defendant: THE WINE GROUP, INC., THE WINE GROUP, LLC, SUTTER HOME WINERY, INC., d/b/a TRINCHERO FAMILY ESTATES and others
Cause Of Action: Diversity-Tort/Non-Motor Vehicle
ECP Station I LLC v. JJS Chevron, Inc. et al
as 8:2015cv00589
Plaintiff: ECP Station I LLC
Defendant: JJS Chevron, Inc., S.K. Petroleum, Inc., Pasco Oil, Inc. and others
Cause Of Action: Foreclosure
Bryan v. Nationstar Mortgage LLC
as 1:2015cv21052
Cause Of Action: Diversity-(Citizenship)
Plevin v. US Bank National Association et al
as 6:2015cv00412
Plaintiff: Lawrence M. Plevin
Defendant: US Bank National Association, CitiMortgage, Inc.
Cause Of Action: (a) Fed. Question: Real Property


text:
Do you clean your oven before or after?
We'll be doing a 10# turkey tomorrow by high heat, and the oven needs cleaning. So, do you clean your oven before or after?
You're supposed to clean the oven? Huh. Mine is nine years old, and I never found out if the self-clean function works. I can no longer see throuh the glass on the door, but I do "sweep" out the crumbs that collect on the oven floor. Clean ovens are over-rated!
If a "dirty oven" has been used often enough, all the grease and spills will be burned off and there's no need to clean it before doing a high heat bird.
Someone once calculated that to run a full cycle clean it costs nearly $6.00 in power....think about it before you set the dials!
Linda C
How bad is your oven? I never clean mine. Okay, maybe once every few years. I don't like the self-cleaning cycle, I think the heat is bad for the electronics of an appliance that is already shoddily made in China and at 12 years old is well past its designed mean time before failure. And last time the self-cleaning door lock wouldn't disengage and I had to disassemble the oven to get it open. So I wouldn't run the self-cleaning before an important dinner. Maybe I'd use elbow grease to clean up the worst of the spills. Otherwise I'd just crank the vent fan, crack the windows, close the doors, unplug the smoke detector, and roast away.
Clean? I'm just praying the darned thing works. Bought it off Craig's List a few weeks ago and haven't installed it yet. On my to-do list for today as I'm hosting Christmas dinner. Please wish me luck. It's OLD!
YES, because my SIL might look to see if it's clean.
I did recently ruin the finish on my beautiful, much loved, Breville toaster oven by cleaning it with oven spray (I tried ammonia first).
FOAS, Good luck! The first time I cooked Christmas dinner (which was the first week in our new house), the turkey wouldn't fit! We were busy moving in and I didn't realize that the oven was an apartment size oven.
I don't know about you guys, but, if it came down to me cleaning the oven the old fashioned way or spending $6.00 worth of electricity--- Well, you can guess what I would do. (That's true even if I had hands and knees capable of doing so.) I also don't let it get that bad. If I get a bad spill, it gets cleaned right away. I set it for the shortest cycle I can, 2 hrs on my oven, and have the option of canceling sooner if need be.
Just to add my 2 cents, I would have cleaned before, for no other reason than the fact it was dirty, and if necessary, after. I have a self cleaning oven that does work well.
After! I had the same experience as Johnliu. Set the oven for the clean cycle and after cleaning, the lock wouldn't unlock. Something of an ordeal to get the dang oven to open. DH had to do some disassembly and, after that, the clean cycle won't work. It's a Kenmore rebadged General Electric hunk of junk. We don't use the oven a whole lot anyway so I just keep up w/ the spills w/ Comet or an SOS pad.


text:
Companion for Sunflowers?
greenwood85(6b)December 25, 2008
I'm going to plant a large variety of sunflower. I imagine pole beans might like to climb the fast-growing stalk. Am I right? Any other suggestions?
I hope it is not to late to post a reply. I have had success with vine cucumbers, morning glory, and sweet peas. Just to name a few vines. Zinnias look nice in front of sunflowers. And also I use clover as a ground cover/green manure for about everything. good luck:)
According to Liouse Riotte in her book, "Carrots Love Tomatoes", pole beans and sunflowers do not get along. She recommends planting sunflowers with cucumbers or melons.
Here is a link that might be useful: Carrots Love Tomatoes


text:
Possible Answer
A lipoma is a benign tumor composed of adipose tissue (body fat). It is the most common benign form of soft tissue tumor. Lipomas are soft to the touch, usually ... - read more
Authentic Puppia Soft Dog Harness made out of 100% Polyester; Adjustable chest belt and quick-release buckle; Comfortable padded neck opening; Machine washable, but ... - read more
Please vote if the answer you were given helped you or not, thats the best way to improve our algorithm. You can also submit an answer or search documents about soft lump on top of dogs head.


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Currently in graduate school.
In May 2011, I graduated with a MA degree in rhetoric and public discourse from the University of Montana. During my time in Missoula, I...
Indiana Universty Purdue Universty Indianapolis (IUPUI) GPA:3.7 Masters of Science in Electrical and Computer Engineering
2008 - Associate degree in English language, literature and civilization 2008- Participation in the NESA UGRAD which is Program is a...
Working on Master's degree in secondary education with a license to teach math at Indiana University
Senior Sociology and Communication Major at Marian University GPA:3.67 Dean's List: Fall 2010, Fall 2011, Spring 2011
Education major at Ivy Tech Community College
Junior at Indiana University -Major: Human Biology -Minor: Chemistry -Pre-Med
I will be a Junior at IU Bloomington, pursuing Studio Art, Outdoor Rec, and Gender Studies.
My Name is Dr. Stephen Jones, I received my PhD from University of North Carolina Chapel Hill in Aug 2013 in cell biology and physiology....
Graduated with B.S. in Psychology and a Minor in Music from Indiana University. 3.6 GPA in psychology and 4.0 GPA in music.
Completed my Undergraduate degrees in Accounting and Operations, along with a Minor in Spanish, in Spring '14. Member of Kelley's MSA...
Freshman Journalism Major and a French Minor at Indiana University.
I am currently a chemistry and biology major at Indiana University. I also have a degree in economics from Penn State University.
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text:
CNBC's Rick Santelli reports on February's personal income and consumer spending data.» Read More
It's no secret that the global economy is in trouble, so if you're bored of the world of stocks and bonds and feel the tedium of a slowing market, the time may have come to explore whole new world of investment: space travel.
Breaking in is difficult due to a lack of investment vehicles, limited access to buyers and sellers and varying pricing models.
If your long-term plan is to keep working, investing and living where you are until the day you retire, and then change your lifestyle and financial life, you are practicing what I call "cliff living"—living one way until you abruptly leave the world of work, then leaping to a whole new level of life and finances.
Vintage fashion investment is an increasingly lucrative area, and with substantial public demand for luxury clothes, fashion retail businesses have seen impressive returns – so is there still room to get money for old rope?
Federal Reserve chairman Ben Bernanke went to the well of make-believe wealth once again, setting the stage for more bond buying in last week's announcement. With a sullen economy supported by 36 months of smoke and mirrors, the punch bowl was never far away.
With around 140 countries considered frontier markets—regions trading at a 25 percent discount to more popular emerging ones—CNBC explores the best opportunities for prospective investors.
The third in a special series of investors' clinics on CNBC, Chief Economist at Exotix, Stuart Culverhouse, answered your questions on the opportunities in frontier market investment on Tuesday 31 January.
What's preventing you from having the life you want, the money you want? Could it all be in your head?
For reliable investment-grade wine, look only to France, principally select wines from Bordeaux, and a few from Burgundy and the Rhone Valley.
In an age of celebrity and fame, Hollywood might seem like a great place to make money fast, but while the payoffs in major film production can be massive, so can the losses.
CNBC.com ranks the 10 richest people who have sought the U.S. presidency since 1992. Check out the list!
A couple of years ago, the words "American consumer" cast a shadow over global markets. No wonder. Back in the days of the credit bubble, American consumer borrowing helped to create a crazy debt binge, the Financial Times reports.
It's a vast, valuable and diverse category of art , and taking a narrow religious view could hurt you as an investor.
Every lap around the track, each sit up and all of those salads are just one more way to save for retirement.
With uncertainty and volatility big issues much like last year, money managers say go for high dividend-paying stocks and sectors such as healthcare, consumer staples and utilities.
In the low-interest rate environment of today, new strategies are needed to make the most of your savings and avoid the inevitable bite of inflation.
Amid a plethora of worries ranging from political unrest in the Middle East to slowing growth in China, the hope normally associated with the start of a new investment year has been trumped by fear. Rather than playing down risk, investors are obsessing over it.
Our special report focuses on the financial decisions that will pay off throughout the year — saving, preserving, allocating and investing money.
2011 was a volatile year for the financial markets, and many experts expect 2012 to be the same. What's an investor to do? Cast your vote.
Savings rates are near zero, stocks are expected to be volatile, and the job market remains weak.


text:
Dr. Vesselina S. Mateva, MD
- 265 Griffin St E
Amery, WI 54001
- 2 Locations
- (715) 268-8000
Dr. Vesselina S. Mateva graduated from the Plovdiv Med Academy, Fac of Med, Plovdiv, Bulgaria in 1984. She works in Amery, WI and 1 other location and specializes in Neurology. Dr. Mateva is affiliated with Hudson Hospital & Clinic, Regions Hospital and Westfields Hospital & Clinic. She speaks English and Spanish.
Gender: Female
Languages Spoken: English, Spanish
Medical School: Plovdiv Med Academy, Fac of Med, Plovdiv, Bulgaria; Graduated 1984
*This physician has multiple office locations. See Office Information for details.


text:
Page 1
DT-REFinD: Diffusion Tensor Registration With Exact
Finite-Strain Differential
Yeo, B.T.T. et al. "DT-REFinD: Diffusion Tensor Registration
With Exact Finite-Strain Differential." Medical Imaging, IEEE
Transactions on 28.12 (2009): 1914-1928. © 2010 Institute of
Electrical and Electronics Engineers.
As Published
Institute of Electrical and Electronics Engineers
Final published version
Thu May 19 04:13:19 EDT 2011
Citable Link
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
Page 2
DT-REFinD: Diffusion Tensor Registration
With Exact Finite-Strain Differential
B.T. Thomas Yeo*, Tom Vercauteren, Pierre Fillard, Jean-Marc Peyrat, Xavier Pennec, Polina Golland,
Nicholas Ayache, and Olivier Clatz
Abstract—In this paper, we propose the DT-REFinD algorithm
for the diffeomorphic nonlinear registration of diffusion tensor
images. Unlike scalar images, deforming tensor images requires
choosing both a reorientation strategy and an interpolation
scheme. Current diffusion tensor registration algorithms that
use full tensor information face difficulties in computing the
differential of the tensor reorientation strategy and consequently,
these methods often approximate the gradient of the objective
function. In the case of the finite-strain (FS) reorientation strategy,
we borrow results from the pose estimation literature in computer
vision to derive an analytical gradient of the registration objective
function. By utilizing the closed-form gradient and the velocity
field representation of one parameter subgroups of diffeomor-
phisms, the resulting registration algorithm is diffeomorphic and
fast. We contrast the algorithm with a traditional FS alternative
that ignores the reorientation in the gradient computation. We
show that the exact gradient leads to significantly better reg-
istration at the cost of computation time. Independently of the
choice of Euclidean or Log-Euclidean interpolation and sum of
squared differences dissimilarity measure, the exact gradient
achieves better alignment over an entire spectrum of deformation
penalties. Alignment quality is assessed with a battery of metrics
and closeness to synthetic warps. The improvements persist even
when a different reorientation scheme, preservation of principal
directions, is used to apply the final deformations.
Index Terms—Diffeomorphisms, diffusion tensor imaging, fi-
nite-strain (FS), finite-strain differential, preservation of principal
directions, registration, tensor reorientation.
Manuscript received April 20, 2009; revised June 09, 2009. First published
June 23, 2009; current version published November 25, 2009. This work
was supported by the INRIA Associated Teams Program Compu-Tumor
(http://www-sop.inria.fr/asclepios/projects/boston/), by the NAMIC under
Grant NIH NIBIB NAMIC U54-EB005149, and by the NAC under Grant
NIH NCRR NAC P41-RR13218). The work of T. Yeo was supported by the
Agency for Science, Technology, and Research, Singapore. Asterisk indicates
corresponding author.
*B.T. T. Yeo is with the Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(e-mail: firstname.lastname@example.com).
P. Golland is with the Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(e-mail: email@example.com).
T. Vercauteren is with the Mauna Kea Technologies, 75010 Paris, France
(e-mail: firstname.lastname@example.com).
P. Fillard, J.-M. Peyrat, X. Pennec, N. Ayache, and O. Clatz are
with the Asclepios Group, INRIA, 06902 Sophia Antipolis, France
email@example.com; firstname.lastname@example.com;
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMI.2009.2025654
diffusion is anisotropic in tissues such as cerebral white matter.
matter structures in the brain. The rate and anisotropy of diffu-
sion at each voxel of a diffusion tensor image is summarized
by an order 2 symmetric positive definite tensor, i.e., a posi-
tive definite 3
3 matrix. This is in contrast to scalar values
in traditional magnetic resonance images. The eigenvectors of
the tensor correspond to the three principal directions of diffu-
sion while the eigenvalues measure the rate of diffusion in these
To study the variability or similarity of white matter struc-
tures across a population or to track white matter changes of a
single subject through time, registration is necessary to estab-
lish correspondences across different diffusion tensor (DT) im-
ages. Registration can be simplistically thought of as warping
one image to match another. For scalar images, such a warp can
be defined by a deformation field and an interpolation scheme.
For DT images however, one also needs to define a tensor reori-
entation scheme. Reorientation of tensors is necessary to warp
a tensor image consistently with the anatomy [3]. There are two
orientation and the preservation of principal directions (PPD)
reorientation. In this paper, we derive an exact differential of
FS reorientation strategy and show that incorporating the exact
differential into the registration algorithm leads to significantly
better registration than the common practice of ignoring the re-
orientation when computing the gradient [3]. Their empirical
performance is similar [23], [34], [49].
Many DTI registration algorithms have been proposed [2],
[15], [24], [28], [31], [36], [49], [50]. Because the reorientation
strategies greatly complicate the computation of the gradient of
the registration objective function [49], many of these registra-
tion techniques use scalar values or features that are invariant
to image transformations. This includes the use of fractional
anisotropy [31] and fibers extracted through tractography [50].
Leemans et al. [28] use mutual information to affinely align
the diffusion weighted images from which the DT images are
estimated. Nonlinear fluid registration of DT images based
on information theoretic measures has since been introduced
[16], [39].
Instead of using deformation invariant features, Alexander
and Gee [2] perform elastic registration of tensor images by re-
orienting the tensors after each iteration using PPD reorienta-
thediffusionof water ininvivo biologicaltissues[8].The
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Page 3
YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1915
the gradient of the objective function. Cao et al. [15] propose a
diffeomorphic registration of tensor images using PPD reorien-
tation. The diffeomorphism is parameterized by a nonstationary
velocity field under the large deformation diffeomorphic metric
mapping (LDDMM) framework [11]. An exact gradient of the
PPD reorientation is computed by a clever analytical reformu-
lation of the PPD reorientation strategy. In this paper, we com-
plement the work in [15] by computing the exact gradient of the
FS reorientation.
For a general transformation, such as defined by B-splines or
nonparametric free form displacement field, the FS reorienta-
tion [3] is defined through the rotation component of the defor-
mation field. This rotation is estimated by the polar decomposi-
tion of the Jacobian of the deformation field using principles of
continuum mechanics. The rotation produced by the polar de-
composition of the Jacobian is the closest orthogonal operator
to the Jacobian under any unitary invariant norm [26]. However,
the polar decomposition requires computing the square root of
a positive definite matrix, which replaces the eigenvalues of the
original matrix with their square roots. The dependence of the
plex and the gradient of any objective function that involves re-
orientation is hard to compute.
Zhang et al. [48], [49] propose and demonstrate a piecewise
FS reorientation. The tensor image is divided into uniform re-
gions and the optimal affine transformation is then estimated
for each such region. The rotation component of the deforma-
tion need notbe estimated as a separatestep.Instead, sincerota-
tion is already explicitly optimized in the affine registration, the
gradient due to FS reorientation can be easily computed. These
piecewise affine transformations are fused together to generate
a smooth warp field. The algorithm is iterated in a multiscale
fashion with smaller uniform regions. Unfortunately, it is un-
clear how much of the optimality is lost in fusing these locally
optimal piecewise affine transformations.
In this paper, we borrow results from the pose estimation lit-
erature in computer vision [20] to compute the analytical differ-
ential of the rotation matrix with respect to the Jacobian of the
displacement field. We propose a diffeomorphic DTI registra-
tion algorithm DT-REFinD, which extends the recently intro-
duced diffeomorphic Demons registration of scalar images [42]
to registration of tensor images. The availability of the exact an-
alytical gradient allows us to utilize the Gauss–Newton method
for optimization. Implemented within the Insight Toolkit (ITK)
framework, registration of a pair of 128
tensor volumes takes 15 min on a Xeon 3.2 GHz single pro-
cessor machine. This is comparable to the nonlinear registration
of scalar images whose runtime might range from a couple of
minutes to hours. DT-REFinD has been incorporated into the
freely available MedINRIA software.1
The diffeomorphic Demons registration algorithm [42] is
an extension of the popular Demons algorithm [37]. It guar-
antees that the transformation is diffeomorphic. The space of
transformations is parameterized by a composition of deforma-
12860 diffusion
1MedINRIA can be downloaded at http://www-sop.inria.fr/asclepios/soft-
tions, each of which is parametrized by a stationary velocity
field. Such a representation is similar to that used by the large
deformation diffeomorphic metric mapping (LDDMM) frame-
work [11], [38]. However, unlike LDDMM, the diffeomorphic
Demons algorithm does not seek a geodesic of the Lie group of
diffeomorphism. At each iteration, the diffeomorphic Demons
algorithm seeks the best diffeomorphism to be composed with
the current transformation. Restricting each deformation up-
date to belong to a one parameter subgroup of diffeomorphism
results in a faster algorithm than the typical algorithm based on
the LDDMM framework or algorithms that parameterize the
entire diffeomorphic transformation by a stationary velocity
field [7], [25].
In addition to DT-REFinD, we also propose a simpler and
faster algorithm that ignores the reorientation during the gra-
dient computation. Instead, reorientation is performed after
variant of the method proposed by Alexander and Gee [2] with
Gauss–Newton optimization. We compare the two algorithms
and show that using the exact gradient results in significantly
better registration at the cost of computation time.
While many methods for interpolating and comparing tensor
images exist [27], [32], we use Euclidean interpolation and
sum-of-squares difference (EUC-SSD) [2], [3], [49], as well
as Log-Euclidean interpolation and sum-of-squares difference
(LOG-SSD) [6], [21]. Regardless of the choice of interpo-
lation and dissimilarity metric, we find the exact gradient
achieves better alignment over an entire range of deforma-
tion regularization. Alignment quality is assessed with a set
of seventeen different metrics including tensor overlap, frac-
tional anisotropy and inverse consistency of the warps. We
also find that the exact gradient method recovers synthetically
generated warps with higher accuracy. Finally, we show that
the improvements persist even when PPD is used to apply the
final deformations.
the true gradient will lead to a better solution. After all, the reg-
istration problem is nonconvex and any solution we find is a
local optimum. In practice however, the experiments show that
taking reorientation into account does significantly improve the
registration results. We believe that the reorientation provides
an additional constraint. The registration algorithm cannot arbi-
the reorientation of tensors in other regions (cf. the famous "C"
example in large deformation fluid registration [18]). This ad-
ditional constraint acts as a further regularization, leading to a
better solution.
This paper extends a previously presented conference article
[47] and contains detailed derivations, experiments and discus-
sions left out in the conference version. The paper is organized
as follows. The Section II describes the computation of the FS
differential. We then present an overview of the diffeomorphic
and numerical limitations of representing diffeomorphic trans-
formations. We extend the diffeomorphic Demons to tensor im-
ages in Section IV using the exact FS differential. We also pro-
during the gradient computation. In Section V, we compare the
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Page 4
two algorithms on a set of 10 DT brain images. Further discus-
sion is provided in Section VI.
To summarize, our contributions are as follows.
1) We derive the exact FS differential.
2) WeincorporatetheFSdifferentialintoafastdiffeomorphic
DT image registration algorithm. We emphasize that the
FS differential is useful, even if one were to use a different
registration scheme with a different model of deformation
or dissimilarity metric.
3) We demonstrate that the use of the exact gradient leads
to better registration. In particular, we show that using the
exact gradient leads to better tensor alignment over an en-
tire range of deformation, regardless of whether we use
LOG-SSD or EUC-SSD in the objective function. We also
show that the exact gradient recovers synthetically gener-
an approximate gradient that ignores reorientation.
4) OurimplementationallowsforEuclideaninterpolationand
EUC-SSD metric, as well as Log-Euclidean interpolation
and LOG-SSD metric.
Deforming a tensor image by a transformation
tensor interpolation followed by tensor reorientation [3]. To
compute a deformed tensor at a voxel
the tensor to get the interpolated tensor
schemes include Euclidean interpolation [3], Log-Euclidean
interpolation [6], affine-invariant framework [10], [22], [29],
[30], [32], Geodesic–Loxodromes [27], or other methods. In
this work, we focus on Euclidean and Log-Euclidean interpola-
tion since they are commonly used and computationally simple.
The FS differential we compute in this section characterizes
tensor reorientation. The following discussion is therefore
independent of the interpolation strategy.
Supposethetransformation mapsapoint tothepoint
be the displacement field associated with the
transformation . Then
, one first interpolates
. Interpolation
Similarly, we denote
representation of transformations, such as splines, one can al-
ways derive the equivalent displacement field representation.
According to the FS tensor reorientation strategy [3] for non-
linear deformation, one first computes the rotation component
of the deformation at the th voxel
. Note that even for parametric
is the Jacobian of the spatial transformation at the
inthe , and directions.
of the matrix
ment field
in the neighborhood of . Under the identity trans-
formation, i.e., zero displacements,
Because of the matrix inverse in (2), to maintain numerical sta-
bility of the computations, the invertibility of the deformation
(corresponding to
) is important.
The interpolated tensor
the final tensor
are the components of the displacement field
and is therefore a function of the displace-
is then reoriented, resulting in
For registration based on the FS strategy, it is therefore neces-
sary to compute the differential of rotation
transformation . Using chain rule, this reduces to computing
the differential of rotation
with respect to the Jacobian . Let
betheinfinitesimalchangeintheJacobian .Then,asshown
in Appendix A, the infinitesimal change in the rotation matrix
is computed as follows:
with respect to the
, denotes the 3-D vector cross product,
and is the operator defineddenotes the th column of
This skew-symmetric operator is actually the matrix represen-
tation of cross-product, so that for two vectors
plicated (5).
[20] is presented in Appendix A. Let
of .Equation(5)tellsusthevariationoftherotation
of the components of the Jacobian
computed by setting the matrix
which is set to 1.
and ,
beth component
. In particular,
in (5) to 0, except for
In this section, we briefly review the diffeomorphic exten-
sion [42] of Thirion's Demons algorithm [37]. We also discuss
numerical issues related to representing diffeomorphism by ve-
locity fields and optimization methods we use in this paper.
A. Diffeomorphic Demons for Scalar Images
Weconsider themodified Demons objectivefunction [14]for
registering a moving scalar image
to a fixed scalar image
where is the dense spatial transformation to be optimized,
an auxiliary spatial transformation,
denotes the -norm of a vector (or vector field, depending
on the context). We can think of the fixed image
moving image
as 1-D vectors of length
diagonal matrix that defines the variability observed at a
denotes composition and
and warped
voxels.is a
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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1917
particular voxel.
The instantiation of these parameters are further discussed in
Section V-B.
This formulation enables a fast and simple optimization that
alternately minimizes the first two terms and the last two terms
of (7). Typically,
to be close and
model. We note that
preted probabilistically as a hierarchical prior on the deforma-
For the classical Demons algorithm and its variants, the
objective function is optimized over the complete space of
nonparametric spatial transformations [14], [35], [37], [43],
typically represented as displacement fields. Unfortunately,
the resulting deformation might not be diffeomorphic. Instead,
Vercauteren et al. [42] optimize over compositions of diffeo-
morphic deformations, each of which is parametrized by a
stationary velocity field. At each iteration, the diffeomorphic
Demons algorithm seeks the best diffeomorphism parame-
terized by the stationary velocity , to be composed with the
current transformation.
In this case, the velocity field
andis the diffeomorphism associated with .
The operator
is the group exponential relating the Lie
to its associated Lie algebra
be the solution at time of the following stationary or-
dinary differential equation (ODE):
andare parameters of the cost function.
, encouraging
, encouraging
to be
can together be inter-
is an element of the Lie al-
. More formally, let
We define
An image
obtained by transforming the coordinate system of
: a point in the deformed coordinate system corre-
sponds to a point
in the old coordinate system.
The above formulation of the Demons objective function fa-
cilitates a fast iterative two-step optimization. We summarize
the diffeomorphic Demons algorithm [42] in Algorithm 1 (see
algorithm at the top of the page). Steps 2(ii) to 2(iv) essentially
optimize the last two terms of (7). We refer the reader to [13],
[14] for a detailed discussion of using convolution kernels to
achieve elastic and fluid regularization. We also note that the
is therefore a deformed version of image
Demons algorithm can be extended to non-Euclidean domains,
such as the sphere [46].
B. Numerical Details in Velocity Field Representations
defined on the entire continuous image domain, in practice,
andare represented by vector fields on a discrete grid of
image points, such as voxels [37], [42] or control points [7],
[11]. From the theories of ODEs, we know that the integral
(or trajectories) of a velocity field
exist and are unique ifis Lipschitz continuous in
continuous in [12]. Uniqueness means that the trajectories do
not cross, implying that the deformation is invertible. Further-
more, we know from the theories of ODEs that a
uous velocity field
produces a
. Therefore, a sufficiently smooth velocity field re-
sults in a diffeomorphic transformation.
Since the velocity field
is stationary in the case of the one
parameter subgroup of diffeomorphism [5],
uous (and in fact
) in . A smooth interpolation of
tinuous in the spatial domain and is Lipschitz continuous if we
images that are closed and bounded.
To compute the final deformation of an image, we have to
at least at the set of image grid points. For ex-
ample, we can compute
smoothly interpolated velocity field
In this case, the estimate becomes arbitrarily close to the true
as the number of integration time steps increases. With
a sufficiently large number of integration steps, we expect the
estimate to be invertible and the resulting transformation to be
The parameterization of diffeomorphism by stationary ve-
locity field is made popular by the use of the fast "scaling and
squaring" approach to computing
integration, the "scaling and squaring" method works by mul-
tiple composition of displacement fields
are technically
continuous deformation
is clearly contin-
is con-
by numerically integrating the
with Euler integration.
[5]. Instead of Euler
While this method is correct in the continuous case, in the
discrete case, composition of the displacement fields requires
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Page 6
interpolation of displacement fields, introducing errors in
the process. In particular, suppose
the true trajectories found by performing an accurate Euler
integration up to time
does not exist a trivial interpolation scheme that guarantees
. In practice however, it is widely
reported that "scaling and squaring" tends to preserve invert-
ibility even with rather large deformation [5], [7], [42]. In this
work, we employ trilinear interpolation because it is fast. We
Technically speaking, since we use linear interpolation for the
displacement field, the transformation is only homeomorphic
rather than diffeomorphic. However, we will follow the con-
vention of [5], [7], [42] which call the resulting homeomorphic
transformations diffeomorphisms.
and are
respectively. Then, there
C. Gauss–Newton Nonlinear Least-Squares Optimization
We now focus on the optimization of step 2(i) of the diffeo-
morphic Demons algorithm. We choose
. We "subtract" the identity transformation from the resulting
deformation field so that the identity transformation carries no
penalty. The objective function in step 2(i) can then be written
in a nonlinear least-squares form
where we define
. Using Taylor series expansion around
we can write (13) as
for 3-Dimages
block corresponds to a 1
is ablock diagonal matrix, whose
3 matrix
is the transformation of
and is the identity transformation when the velocity
. In (17), we utilize the fact that the differential of the
exponential map at
is the identity. is the
spatial derivativeof theimage intensityat voxel
moving image
is aidentity matrix. The Gauss–Newton optimiza-
tion method ignores the
leading to the classical linear least-squares problem. In partic-
ular, (14) can then be rewritten as
of thewarped
term within the norm in (14),
which is a linear least-squares problem. Independently of the
since the equations for each voxel can be decoupled
from all other voxels. With the help of the Sherman-Morrison
matrix inversion lemma, no matrix inversion is even needed
to invert the resulting small system of linear equations at each
voxel [42].
We note that the original Demons algorithm [37] replaced
by . This is justified by the fact that at the op-
timum, the gradient of the warped moving image should be al-
most equal to the gradient of the fixed image.
A. Diffeomorphic Demons for Vector Images
Before incorporating the FS differential for tensor regis-
tration, let us extend the diffeomorphic Demons algorithm to
vector images. In addition to helping us explain our complete
algorithm, the derivation will also be useful for computing up-
date steps when ignoring tensor reorientation in Section IV-D.
We define a vector image to be an image with a vector of inten-
sities at each voxel. We can treat a vector image like a scalar
image in the sense that each vector component is independent
of the other components. Deformation of a vector image works
just like a scalar image, by treating each component of the
vector separately.
It is fairly straightforward to re-derive the results from the
previous section for vector images. Let
the intensity vector at each voxel. For convenience, we define
to be the intensity vector of the
, andto be the
image intensities
Demons algorithm from the previous section applies exactly to
vector images except that in (20),
blockdiagonalmatrix, where each blockis
particular,the thblockof
at voxel
be the dimension of
th voxel,
vector of all
. Then the diffeomorphic
is now a sparse
. In
... ... ...
The resulting least-squares linear system
harder to solve than before. However, for each voxel , we only
is slightly
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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL 1919
. The solution of the system is also more stable as there are
more constraints.
B. DT-REFinD: Diffusion Tensor Registration With Exact FS
We will now extend the Demons algorithm to DT images.
A DT image is different from a vector image because of the
additional structure present in a tensor. In particular, the space
of symmetric positive definite matrices (tensors) is not a vector
space. When deforming a DT image, reorientation is also nec-
essary. We extend the diffeomorphic Demons registration of
vector images to tensor images.
In this work, we use the modified Demons objective function
(7) and the FS reorientation strategy in our registration. The ob-
jective function in step 2(i) of the Demons algorithm, shown in
(11) for scalars, becomes
squares difference (EUC-SSD) between the tensor images. In
can be seen as a
the 3
3 order 2 tensor at each voxel into a column vector.
should be interpreted as the interpolated
tensor image. In practice, since the tensors are symmetric,
we can work with
vectors to represent tensors and
increase the weights of the entries of
nondiagonal entries of the tensors by
tensor is then reoriented using the rotation matrix
voxel and "rasterized" into a column vector. Note that
implicitly dependent on the transformation
is the Euclidean sum of
vector by "rasterizing"
corresponding to the
. Each interpolated
of each
. The term
computes the SSD between
each tensor of the fixed image and the corresponding reoriented
and interpolated tensorin thewarped movingimage, bytreating
each tensor as a vector and adding the SSD for all voxels.
Equation (23) can also be interpreted as the LOG-SSD be-
tween tensors if
andare the Log-Euclidean transforms of
the original tensor images, obtained by converting each tensor
in the original image to a log-tensor
is simply a symmetric matrix [6].
terpolated log-tensor image.
terpolated and reoriented log-tensor image, since
. Note that
is then the in-
is the in-
for any rotation matrix
followed by Log-Euclidean transformation is the same as
Log-Euclidean transformation followed by reorientation. This
is convenient since we can perform a one time Log-Euclidean
transformation of the tensor images to log-tensor images before
registration and convert the final warped log-tensor images to
tensor images at the end of the registration.
In this case,
is a sparse
can interpret
In particular, the
th block
. Therefore, tensor reorientation
matrix. One
3 matrices.
is equal to
blocks of 9
, where we remind the readers that
and ,are also voxel indices. Using the
chain rule, the product rule and the fact that the differential of
the exponential map at
is the identity, we get
pression of
finite central difference as shown in (31) at the bottom of the
page where
of voxel
in the ,and directions, respectively. Therefore
denotes the -coordinate of
and denotes the
ings in the ,
anddirections respectively. Using the differ-
ential of
(5) and the expression of
using the chain rule. Appendix B provides the
detailed derivation. This definition of
is the identity transformation when the velocity
at the voxeland that (3) gives an analytical ex-
. In practice, is defined numerically using
are the neighbors
after transformation
-coordinate of
and are the voxel spac-
(31), we can compute
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Page 8
and for neighborof voxel , we get
Note that the first and second terms in the above expression are
transpose of each other. Therefore, for
is zeros if voxels and
As before,
computed the full gradient of our objective function
, theth block
are not neighbors.
. In summary, we have
a sparse
is a constant diagonal matrix, while
C. Gauss–Newton Nonlinear Least-Squares Optimization
From the previous sections, we can now write
The resulting least-squares problem is harder to solve than be-
fore, since the linear systems of equations cannot be separated
into voxel-specificset of equations. However, the sparsity of the
matrix makes the problem tractable. In practice, we solve the
linear systems of equations using Gmm++, a free generic C++
template library for solving linear sparse systems.2At the finest
resolution, solving the sparse linear system requires about 60 s.
This is the bottleneck of the algorithm. However, due to the fast
convergence of Gauss–Newton method, we typically only need
to solve the linear systems 10 times per multiresolution level.
single processor machine.
The efficiency of the Demons algorithm for scalar images
comes from separating the optimization into two phases: op-
timization of the dissimilarity measure and optimization of the
system of linear equations when considering the two phases to-
gether. Because of the reorientation in tensor registration, we
have to solve a sparse system of linear equations anyway. In
this case, we could have incorporated the optimization of the
keep the two phases separate to allow for fair comparison with
the case of ignoring the reorientation of tensors in the gradient
computation (see Section IV-D) by using almost the same im-
plementation. Any improvement must then clearly come from
the use of the true gradient and not from using a one-phase op-
timization scheme versus a two-phase optimization scheme.
D. Classical Alternative: Ignoring the Reorientation of
Previous work [2] performs tensor registration by not in-
cluding the reorientation in the gradient computation, but
reorienting the tensors after each iteration using the current
estimated displacement field. To evaluate the utility of the true
gradient, we modify our algorithm to ignore the reorientation
part of the objective function in the gradient computation.
In particular, we can simplify the Gauss–Newton optimiza-
tion in the previous section by setting
, effectively
ignoring the effects of the displacement field of a voxel on the
reorientation of its neighbors. Note that
different from before because we directly use the gradient of
the warped and reoriented image. In each iteration, we treat
the tensor like a vector, except when deforming the moving
image. The resulting least-squares problem degenerates to that
in Section IV-A. The algorithm is thus much faster since we
only need to invert a 3
3 matrix per voxel at each iteration.
Registration only takes a few minutes on a Xeon 3.2 GHz single
processor machine.
is slightly
We now compare the DT-REFinD algorithm that uses the
exact FS differential with the classical alternative that uses an
approximate gradient and a basic Demons algorithm that uses
the fixed image gradient.
A. Data and Preprocessing
We use 10 DT images acquired on a Siemens 1.5T scanner
using an EPI sequence, consisting of healthy volunteers with
the following acquisition parameters: echo time
diffusiongradients, imagedimensions
are kindly contributed by Dr. Ducreux, Bicêtre Hospital, Paris,
We first use morphological operations to extract a foreground
mask from the diffusion weighted (DW) images of each of the
10 DT images. This involves an automatic thresholding of any
single DW image, except the baseline, so that the skull and the
eyes do not interfere in the mask calculation. The threshold
is chosen so that the mask contains the entire brain. This in-
evitably contains some outliers in the background. Then, a se-
quence of erosions with a ball of radius 1 voxel is performed
to remove outliers (3 to 4 iterations are sufficient), which re-
sults in a set of connected components ensured to lie within
the brain. Finally, we use conditional reconstruction to create
thefinal mask.Thisinvolves dilatingtheconnectedcomponents
while intersecting the result with the initial mask, and repeating
this process until convergence. Doing so allows the connected
components to grows within the brain while ensuring the back-
ground outliers are canceled. If holes are still present in the
mask, a hole filling algorithm can be applied (a simple morpho-
logical closing is generally sufficient).
; 25
. These images
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YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1921
B. Implementation Details
multiresolution optimization, by smoothing and downsampling
the data for initial registration and using the resulting registra-
tion from a coarser resolution to initialize the registration of a
finer resolution. We find that 10 iterations per multiresolution
foreground are included.
There are three main parameters in the algorithm: the diag-
onal variability matrix
, and the tradeoff parameters
. could be, in principle, estimated from a set of diffusion
tensor images via coregistration. Since we deal with pairwise
registration, we set
to be a constant diagonal matrix. Con-
sequently, because the local optimum is determined by the rel-
ative weighting of
identity matrix.
determines the step-size taken at each iter-
ation, which affects the stability of the registration algorithm
[42]. Therefore, we empirically set
iteration is about 2 voxels. The relative values of
termine the width of the kernel used to smooth the deformation
field. Once
is determined, the value of
warp smoothness.
As previously shown [14], it does not make sense to com-
pare two registration algorithms with a fixed tradeoff between
the dissimilarity measure and regularization, especially when
the two algorithms use different dissimilarity measures and/or
regularizations. Furthermore, one needs to be careful with the
tradeoff selection for optimal performance in a given applica-
tion [44].
In this work, we compare the algorithms over a broad range
of kernel sizes. We note that larger kernel sizes lead to more
smoothing and thus smoother warps. Because kernel sizes are
not comparable across the different algorithms we consider, we
use harmonic energy as a more direct measure of warp smooth-
ness. We define the harmonic energy to be the average over all
voxels of the squared Frobenius norm of the Jacobian of the dis-
placement field.Note thattheJacobianofthedisplacementfield
corresponds to the Jacobian of the transformation defined in (3)
without the identity. Therefore lower harmonic energy corre-
sponds to smoother deformation.
, we simply setto be the
so that the update at each
determines the
C. Evaluation Metrics
To assess the alignment quality of two registered DT images,
we use a variety of tensor metrics [2], [9], [21]. Let
fusion tensor. We let
,,be its eigenvalues in descending
order with corresponding eigenvectors
the average eigenvalues. Similarly, let
diffusion tensor with corresponding eigenvalues
. The following measures are averaged
over the foreground voxels of the fixed image. This in turn al-
lows us to average results across different registration trials by
normalizing for brain sizes.
1) Euclidean Mean Squared Errors (EUC-MSE): squared
Frobenius norm of
2) LogEuclideanMeanSquaredErrors(LOG-MSE):squared
Frobenius norm of
be a dif-
,, . We denote
denote another
, and
3) 1—Overlap:
Note that the EUC-MSE and LOG-MSE correspond to the SSD
and LOG-SSD dissimilarity metrics we employ during registra-
tion. In additional to these tensor metrics, we also consider the
following scalar measures [1], [9]. The dissimilarity between
two tensors is defined to be the sum of squared differences
between these scalar measures, averaged over the foreground
2) LFA (Logarithmic Anisotropy): FA computed for
3) ADC(ApparentDiffusionCoefficient):
4) VOL (Volume):
5) CL (Linear Anisotropic Diffusion):
6) CP (Planar Ansiotropic Diffusion):
7) CS (Spherical Anisotropic Diffusion):
8) RA (Relative Anisotropy):
9) VR (Volume Ratio):
10) DISP (Dispersion):
A question then arises over whether these scalar measures
should be computed after deforming the moving image or
computed on the unwarped moving image and then interpolated
with the deformation field. The latter is attractive because the
result is independent of the tensor reorientation and interpola-
tion strategies. On the other hand, since registration is almost
never an end-goal—the deformed tensor images are presum-
ably used for other tasks, one could argue that it is important
to measure the quality of the deformed tensors. Therefore, in
this work, we consider both strategies. Finally, we define the
average distance between two deformation fields
average difference in the deformation fields obtained by the dif-
ferent methods to evaluate how different the deformation fields
are.Wefind thattheaveragedifferenceinthedeformationfields
between the exact gradient method and the approximate gra-
at high harmonic energy (cf. image resolution
). The average difference in the deformation
fields between the fixed image gradient method and the other
twomethodsrangesfrom3mmatlowharmonicenergyto8 mm
at high harmonic energy.
sistency. Without the availability of ground truth deformation,
deformation quality. In particular, given a deformation
subject to subject
and from subject
is the number of foreground voxels. We compute the
to subject , inverse
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Page 10
Fig. 1. Qualitative comparison between the exact FS gradient and the approximated gradient for registering a pair of subjects using the Log-Euclidean framework
and the same parameters in the registration. (a) Moving image. (b) Fixed image. (c) Registration using the approximated moving image gradient. (d) Registration
using the exact FS gradient. Volumes were slightly cropped for better display. Exact gradient achieves better alignment of fiber tracts with a smoother displacement
field. Tensors in the anterior limb of the internal capsule, highlighted in (b) and (d) are coherently oriented in a north-east direction. However, in (c), the directions
of the tensors are more scattered. Furthermore, the volume of the tensors in (c) is swollen relative to (b) and (d). Numerically, the exact FS gradient has lower SSD
with a smoother deformation field (not shown).
consistency is defined to be the average distance between the
displacement field associated with the composed warp
and a zero displacement field.
D. Qualitative Evaluation
data set. Visually, DT-REFinD results in better tract alignment,
such as the anterior limb of the internal capsule highlighted in
the figure. See figure caption for more discussion. In this partic-
mean-square-error (LOG-MSE) and smoother deformation as
measured by the harmonic energy.
E. Quantitative Evaluation I
To quantitatively compare the performance of the exact FS
we consider pairwise registration of the 10 DT images. Since
our registration is not symmetric between the fixed and moving
images, there are 90 possible pairwise registration experiments.
We randomly select 20 pairs of images for pairwise registra-
tion. By swapping the roles of the fixed and moving images,
we obtain 40 pairs of image registration. From our experiments,
we find that the statistics we compute appear to converge after
about 30 pairwise registrations, hence 40 pairwise registrations
are sufficient for our purpose.
Even though we are considering algorithms with the same
dissimilarity measure and regularization (and effectively the
same implementation) but different optimization schemes, we
find that for a fixed-size smoothing kernel, using the exact FS
differential tends to converge to a solution of lower harmonic
energy, i.e., a smoother displacement field. Smaller harmonic
energy implies a smoother deformation, providing evidence
that the reorientation acts as an additional constraint for the
registration problem. To properly compare the algorithms,
we consider smoothing kernels of sizes from 0.5 to 2.0 in
increments of 0.1. In particular, we perform the following
For each pair of subjects and for each kernel size
i) Run the diffeomorphic Demons registration algorithm
using Euclidean interpolation and EUC-SSD using:
(a) Exact FS gradient (DT-REFinD).
(b) Approximate gradient by ignoring reorientation.
(c) Fixed image gradient. This is the gradient proposed
in Thirion's original Demons algorithm [37].
ii) Repeat (i) using Log-Euclidean interpolation and LOG-
iii) Use the estimated deformation fields to compute the
tensor and scalar measures discussed in Section V-C
using FS reorientation or PPD reorientation.
iv) Compute the inverse consistency of the deformations
from subject to subject
For a given smoothing kernel and registration strategy,
registering different pairs of images leads to a set of error
metric values corresponding to different harmonic energies. To
average the error metric values across different pairs of images
and to compare registration results among different strategies,
for each registration, we linearly interpolate the dissimilarity
metric (EUC-MSE, LOG-MSE, and so on) over a fixed set of
harmonic energies sampled between 0.03 to 0.3. This allows us
to average the error metric across different pairs of images and
compare different strategies at a given harmonic energy.
1) Tensor Alignment: Fig. 2 shows the error metrics (av-
eraged over 40 pairwise registrations) with respect to the har-
monic energies when using the dissimilarity metric EUC-SSD,
euclidean interpolation and FS reorientation for registration.
The final deformations were applied using FS reorientation.
and from subject to subject .
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Page 11
YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL 1923
Fig. 2. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD,
Euclidean interpolation and FS reorientation for registration. The final deformations are applied using FS reorientation. We find that the exact FS gradient method
achieves the best performance.
Fig. 3 shows the corresponding plot when applying the final
warps using PPD reorientation. In both cases, we find that
at all harmonic energy levels, the exact FS gradient method
achieves the lowest errors. The approximate gradient method
outperforms the fixed image gradient method.
As mentioned earlier, the scalar measures, such as FA, can
be computed after deforming the moving tensor image or
computed on the unwarped moving image and then deformed.
Figs. 2 and 3 show results based on the former strategy. We
obtain similar results using the latter strategy, but omit them
here for brevity.
The amount of improvement increases as the harmonic ener-
gies increase. In our experiments, a harmonic energy of 0.3 cor-
responds to severe distortion (pushing the limits of the numer-
ical stability of scaling and squaring), while a harmonic energy
of0.03correspondstoverysmoothwarps.Inpreviouswork, we
showed in the context of image segmentation that extreme dis-
tortion causes overfitting, while extremely smooth warps might
inform us of the optimal amount of distortion and is the sub-
ject of future studies. For now, we assume a "safe" range for
assessing the algorithm's behavior to be between harmonic en-
that the exact FS gradient provides an improvement of between
5% to 10% over the approximate gradient in this "safe" range
of harmonic energies.
To better appreciate the improvements, Fig. 4 shows the dif-
ference in errors by subtracting the error metric values of the
approximate gradient method from the error metric values of
the exact gradient method when using the dissimilarity metric
istration. The final deformations were applied using FS reorien-
tation. The error bars indicate that the exact gradient method
is statistically significantly better than the approximate gradient
method over the entire range of harmonic energies and all the
error metrics (
for almost entire range of harmonic en-
ergies). We emphasize that the improvements persist even when
we evaluate a different dissimilarity measure or use a different
reorientation strategy from those used during registration.
Similarly, we find that the exact FS gradient method achieves
the lowest errors when using LOG-SSD similarity metric and
Log-Euclidean interpolation for registration, regardless of
whether FS or PPD reorientation was used to apply the final
deformation. We omit the results here in the interest of space.
2) Inverse Consistency: Fig. 5 shows the inverse consistency
errors (averaged over 20 sets of forward and backward pair-
wise registrations) with respect to the harmonic energies. Once
again, we find that all harmonic energy levels, the exact FS gra-
dient method achieves the lowest errors, regardless of whether
EUC-SSD and Euclidean interpolation or LOG-SSD and Log-
Euclidean interpolation were used. Similarly, the approximate
gradient method outperforms the fixed image gradient method.
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Page 12
Fig. 3. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD,
Euclidean interpolation and FS reorientation for registration. The final deformations are applied using PPD reorientation. We find that the exact FS gradient method
achieves the best performance.
F. Quantitative Evaluation II
We perform a second set of experiments to evaluate the algo-
rithm's ability to recover randomly generated synthetic warps.
Given a DT image, we first generate a set of random warps by
pendent and identically distributed (I.I.D.) Gaussian. The fore-
ground mask is then used to remove the velocity field from the
tially with a Gaussian filter. We compute the resulting displace-
ment field by "scaling and squaring." This displacement field is
used to warp the given DT image using Log-Euclidean interpo-
lation. We use either FS or PPD to reorient the tensors. I.I.D.
Gaussian noise is added to the warped DT image.
We pick a single DT image and generate 40 sets of random
warps. We obtain an average displacement of 9.4 mm over the
foreground voxels. The average harmonic energy is 0.15. We
then perform pairwise registration between the DT image and
the warped DT image using LOG-SSD. Once again, we con-
sider a wide range of smoothing kernel sizes. We also compute
the registration error defined to be the average difference be-
tween the ground truth random warps and the estimated defor-
under the identity transformation, the average registration error
is 9.4 mm.
the three gradients we are considering. From the plots, when the
synthetic warps were applied using FS reorientation, the exact
FS gradient recovers the ground truth warps up to 1.56 mm or
17% error with respect to the average 9.4 mm random warps.
The approximate gradient achieves 2.34 mm or 25% error. Fi-
nally, the fixed image gradient achieves 2.73 mm or 29% error.
Therefore, the exact FS gradient achieves an average of
and 43% reduction in registration er-
rors compared with the approximate gradient and fixed image
gradient, respectively.
When the synthetic warps were applied using PPD reorienta-
or 30% error with respect to the average 9.4 mm random warps.
The approximate gradient achieves 3.20 mm or 34% error. Fi-
nally, the fixed image gradient achieves 3.50 mm or 37% error.
Therefore, the exact FS gradient achieves an average of 13%
and 20% reduction in registration errors compared with the ap-
proximate gradient and fixed image gradient respectively.
Consistent with the previous experiments, using the exact
FS gradient leads to the lowest registration errors regardless
of whether FS or PPD reorientation were used to apply the
synthetic deformation fields. We note that the registration er-
rors inevitably increase when PPD were used to apply the syn-
thetic deformation field, since we use FS reorientation during
Since Gauss–Newton optimization allows the use of "big
steps" in the optimization, it might cause the approximate
gradient to be more sensitive to the reorientation. It is possible
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Page 13
YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1925
Fig. 4. Comparison of exact FS gradient and approximate gradient over an entire spectrum of harmonic energy (?-axis) using EUC-SSD, Euclidean interpolation
and FS reorientation for registration. The final deformations are applied using PPD reorientation. ? -axis shows the difference in errors obtained by subtracting
the error metrics of the approximate gradient method from the error metrics of the exact gradient method. Negative values imply that the exact gradient method
outperforms the approximate gradient method. The error bars show the statistical variability (and thus significance) of the results.
Fig. 5. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis). ? -axis shows the
inverse-consistency errors averaged over 20 sets of forward and backward pairwise registrations. We find that the exact FS gradient method achieves the lowest
that other optimization methods, such as the conjugate gradient,
might improve the results of using the approximate gradient,
by allowing for "smaller steps" and reorient after each "small
step." Possible future work would involve comparing the
exact gradient and approximate gradient under an optimization
framework that takes small steps in the optimization procedure.
However, from optimization theory and from our experience,
Gauss–Newton method requires much fewer iterations to con-
verge than conjugate gradient. Furthermore, conjugate gradient
requires a line search, resulting in many function evaluations.
Function evaluations are quite expensive in our case, because of
the need to reorient and perform "scaling and squaring" of the
velocity field. On the other hand, we find that in practice, line
search is not necessary with Gauss–Newton optimization.
We should also emphasize that ignoring the gradient of the
reorientation term can lead to registration errors that cannot
be recovered regardless of any gradient-based optimization
scheme. For example, consider the registration of a 2-D diffu-
sion tensor image consisting of only horizontal tensors and a
2-D diffusion tensor image consisting of tensors orientated at
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Page 14
Fig. 6. Comparison of exact FS gradient, approximate gradient and fixed image gradient over an entire spectrum of harmonic energy (?-axis) using synthetic
images with ground truth deformations. ? -axis shows the registration errors (in millimeters) averaged over 40 pairwise registrations. We find that the exact FS
gradient method achieves the lowest errors regardless of whether FS or PPD reorientation were used to generate the warps.
a 10 angle. In this case,
approximate gradient and fixed image gradient updates are both
zeros. In contrast, the exact gradient update is nonzero due to
the reorientation.
In our experiments, we show that using the exact gradient re-
However, the inverse consistency is not perfect. Incorporating
an inverse consistency constraint as suggested by the recent ex-
tension of the diffeomorphic Demons algorithm [41] should not
be difficult.
An interesting observation from the synthetic warp experi-
ment in Section V-F is that the best registration occurs when the
is close to the average harmonic energy of the synthetic warps.
In practice, no ground truth deformation is available, making
selection of the optimal kernel size difficult. Furthermore, we
believe the amount of deformation required is dependent on the
application of interest; the appropriate kernel size is likely to
vary with applications.
require different optimal warp smoothness. While using a dif-
ferent kernel at each spatial location is possible, this would re-
duce the efficiency of the demons algorithm. More importantly,
tial smoothing) is justified. To get around such a situation, one
could instead shift the burden to step 1 of the demons algo-
rithm. In this paper, thevariability matrix
tity matrix. Allowing for a nonconstant diagonal matrix
effectively result in spatially varying warp smoothness, since
smaller values of the th diagonal entry place greater emphasis
on matching the
th tensor of the fixed image to the moving
image. Estimating
and an optimal registration regularization
tradeoff is an active area of research [4], [19], [33], [40], [44],
[45] that we do not deal with in this paper.
of deformation or dissimilarity metric from the ones we employ
in this paper. Mutual Information (MI) has been proposed as a
criterion to register diffusion images [16], [28], [39]. Because
MI can handle nonlinear change in intensities across images, it
can potentially handle diffusion image registration without any
reorientation. In fact, [39] suggests that MI without reorienta-
and are both zeros, so the
is set to be theiden-
tion results in better registration than MI with iterative reorien-
tation with either FS or PPD. Future work could involve testing
their observation when reorientation is properly taken into ac-
In this work, we derive the exact differential of the FS
reorientation. We propose a fast diffeomorphic DT image
registration algorithm DT-REFinD using the exact FS differ-
ential. We show that the use of the exact gradient achieves
better tensor alignment than the approximate gradient which
ignores reorientation, over an entire spectrum of harmonic
energies. The improvements persist even if we use an error
metric different from the objective function we optimize and
if we use PPD reorientation for applying the final deforma-
tion. We also show that the exact gradient method recovered
randomly generated warps significantly better than the approx-
imate gradient method—1.56 mm versus 2.34 mm error on
average. DT-REFinD has been incorporated into the freely
available MedINRIA software, which can be downloaded at
In [20], the differential of the matrix
derived, where
of labeled points and
rigid body motion.
componentoftherigidmotion usingtheleast-squaresestimate
. Finding the differential
andtherefore allows the error analysis of the estimate
the measurements andare noisy.
by ignoring second order terms [20]. Defining
, we have
andandare matrices. In
contains their measured positions after
andcan be used to estimate therotation
in terms of
, we get
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Page 15
YEO et al.: DIFFUSION TENSOR REGISTRATION WITH EXACT FINITE-STRAIN DIFFERENTIAL1927
From (38),
the form
is a skew symmetric matrix, and therefore takes
We define
Then, the major result of [20] can be expressed as follows:
product operator.
Recall that we are interested in
. By setting
, we obtain
denotes the th column ofanddenotes the cross
, where
and. Therefore
, by multiplying (41) by, we obtain
By setting,
expression for
and, we finally arrive at the
where we have used the fact that
For completeness, we now derive the expressions for
, whereare the neighboring voxels
of voxel
are the neighbors of voxel
the displacement field in the
. Recall that
in the
we denote
. Using the chain rule, we have
The second and third equalities come from evaluating
, which are mostly zeros. No-
tice that
voxels. Similarly, we have
are evaluated at two different
The authors would like to thank D. Ducreux, M.D., Ph.D.,
Bicêtre Hospital, Paris, for the DTI data and the reviewers for
their manyhelpful suggestions. B. T.T. Yeo would like to thank
C.-F. Westin, G. Kindlmann, and M. Sabuncu for useful discus-
sions and feedback.
[1] A. Alexander, K. Hassan, G. Kindlmann, D. Parker, and J. Tsuruda,
"A geometric analysis of diffusion tensor measurements of the human
brain," Magn. Reson. Med., vol. 44, no. 2, pp. 283–291, Aug., 2000.
[2] D.AlexanderandJ.Gee,"Elasticmatchingofdiffusiontensorimages,"
Comput. Vis. Image Understand., vol. 77, pp. 233–250, 2000.
[3] D. Alexander, C. Pierpaoli, and J. Gee, "Spatial transformations of dif-
fusion tensor magnetic resonance images," IEEE Trans. Med. Imag.,
vol. 20, no. 11, pp. 1131–1139, Nov. 2001.
[4] S. Allassonniere, Y. Amit, and A. Trouvé, "Toward a coherent statis-
tical framework for dense deformable template estimation," J. R. Stat.
Soc. B, vol. 69, no. 1, pp. 3–29, 2007.
[5] V. Arsigny, O. Commowick, X. Pennec, and N. Ayache, "A Log-Eu-
clidean framework for statistics on diffeomorphisms," in Proc. Int.
Conf. Medical Image Computing and Computer Assisted Intervention
(MICCAI).New York: Springer, 2006, vol. 4190, Lecture Notes
Computer Science, pp. 924–931.
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text:
"Science and Cooking" was the topic of a HarvardX lecture offered at the new Harvard Ed Portal in Allston.
Step by step, a growing Harvard women's student group is helping to change the male-dominated culture of computer science by creating fresh realities.
Researchers from the Harvard School of Engineering and Applied Sciences and from universities in Chile, Costa Rica, and Brazil have been studying the secret power of the velvet worm.
Sculptor Kim Bernard, known for her spinning, swaying, bouncing, moving creations, is artist-in-residence in the Physics Department.
Harvard and Brazilian students spent 10 days visiting sustainability-related sites around São Paulo as part of a field course sponsored by Harvard's School of Engineering and Applied Sciences, the David Rockefeller Center for Latin American Studies, and the University of São Paulo.
For doctoral student Sarah Rugheimer, the study of atmosphere holds deep promise in the search for extrasolar life.
The Global Network of Internet and Society Research Centers and the Berkman Center for Internet & Society at Harvard University have released a report on "multistakeholder governance groups" to better inform the discussion over Internet governance models and mechanisms.
In Harvard's high-tech cleanroom, applied physicists produce vivid optical effects — on paper.
At the annual CS50 Fair, students of history, literature, music, and more create tools to share knowledge across fields.
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Journalist Walter Isaacson and College students talk about the achievements and challenges for women in the field of computer science, including pioneer Grace Hopper.
Nobel winner Steven Weinberg brought his thoughts on a "theory of everything" to the Physics Department's Lee Historical Lecture.
Author Walter Isaacson's new book is "The Innovators: How a Group of Hackers, Geniuses and Geeks Created the Digital Revolution." Here is an excerpt about computing pioneer Grace Hopper from his book.
On Friday, leaders in the field of navigation converged on Radcliffe's annual science symposium to discuss findings in everything from brain science to animal navigation to the psychology of how a lost person behaves — which can give rescuers important cues about where to look.
Steve Ballmer was joined by President Drew Faust and School of Engineering and Applied Sciences (SEAS) Dean Cherry Murray at an iLab event to formally announce that the University will increase its computer science faculty by 50 percent over the next few years, to 36 from 24.
This month, the Harvard Physics Department and swissnex Boston, a cultural and technological exchange effort by the Swiss consulate, are sponsoring a photo exhibit that focuses on the people of CERN — laughing, napping, and thinking — and the sometimes ordinary-looking places where they unearth the extraordinary.
In a study reported in Nature Biotechnology, a team of Harvard scientists and engineers has developed a new surface coating for medical devices using materials already approved by the Food and Drug Administration (FDA). The researchers noted that the coating repelled blood from more than 20 medically relevant substrates (glass, plastic, and metal) and also suppressed biofilm formation.
A team of scientists from Harvard University and MIT has developed a theoretical model of a material that could one day anchor the development of highly efficient solar panels.
Best-selling author Walter Isaacson '74 talks about the history of the computer and the Internet.
THATCamp forum allows practitioners of digital humanities to define their concerns, devise solutions for them.


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Congratulations to the San Francisco Giants! Did you watch game 7 of the World Series? Do you prefer it when a sporting event goes to all of the games, or when there's a sweep at the beginning?
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What Olympic events are you most interested in, or not interested in?
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I'm a terrible bowler. When we were first married my husband and I bowled on a league, but that didn't last very long because neither of us could handle the amount of cigarette smoke. Thankfully, smoking isn't allowed indoors anymore, however I'm still a lousy bowler.
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Seattle Sonics Blog
Seattle SuperSonics and NBA news, including the latest information on the proposed new Seattle arena and the quest to bring the Sonics back to Seattle.
Retrospective: The last time Seattle had an NBA team …
Visit seattlepi.com for more Seattle news. Contact sports editor Nick Eaton at firstname.lastname@example.com or @njeaton. [Read More]
Kevin Durant: Happy for Seattle, sad for Sacramento
That Kevin Durant is one good basketball player. But maybe his future lies in politics. He has told an Oklahoma City newspaper that he's happy that Seattle, where he, of course, began his NBA career, might be getting a team. And he's sad that Sacramento might be losing its team to Seattle. The reason he […] [Read More]
Possible hiccup revealed in Kings sale to Chris Hansen
The several minority owners of the Sacramento Kings have been denied their "first right of refusal" to buy a controlling stake in the NBA franchise, according to The Sacramento Bee, representing a possible hiccup in the planned sale to Seattle arena investor Chris Hansen. David Flemmer, a court-appointed bankruptcy trustee who controls 7 percent of […] [Read More]
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Seattle mayor says city will be ready for the Sonics to return
If arena investor Chris Hansen's deal to buy the Sacramento Kings is approved by the NBA, and the league also OKs the team's relocation to Seattle, the new SuperSonics will be playing for two to three years at KeyArena while a new venue is constructed. But first, KeyArena needs to be updated so it can […] [Read More]
If Seattle steals the Kings, are we all huge hypocrites?
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Could Larry Bird run Seattle's new NBA franchise? Adrian Wojnarowski at Yahoo Sports quotes sources as saying that Bird could be targeted as Seattle's general manager. There are, of course, several steps before that. The sale of the Sacramento Kings to Chris Hansen and other investors must go through. The NBA must approve moving the […] [Read More]


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33-year-old Billy F Lush lives in Houston, TX, but has also spent time in Lubbock, TX. Companies in which Billy has a work history include Without A Trace, Arc, Law & Order: Special Victims Unit, Er, Mercenary, Hack, Norman, Trauma, Clubhouse, Six Feet Under, Cold Case, Straw Dogs, One Night, Beautiful Dreamer, Stateside, Huff, Generation Kill, Television Actors, Law & Order: Criminal Intent, Csi: Crime Scene Investigation, Terminator: The Sarah Connor Chronicles, The Black Donnellys and Flights.