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Welcome to our biking community website. We hope you find this site useful and informative.
I don't consider myself a typical lawyer. I wear jeans and ride a bike to work when not in Court. So do some of our lawyers. We are the biggest and best biker friendly law firm in the Country. Sadly most other lawyers, judges, jurors and others don't even understand the biking community and what it is all about. It is their loss but regardless, it leaves many of them questioning our values, integrity and morality. Understanding that ignorance and confronting it properly is an important part of legal representation. We do it best because we understand it best. If you need a lawyer, you need to call us. Consultation are free to our brothers and sisters in the biking community.
The other thing we ALL need to do is get more involved in the issues important to our lifestyles. From illegal profiling, to safety issues to helmet laws. Help support the cause of protecting what we all share, the freedom of riding a bike.
Is it Waco, TX or Whacko, TX?
Too many questions, not enough answers, even less outrage
In the aftermath of the terrible outbreak of violence that occurred at the Twin Peaks in Waco Texas, there is an increasing tide of confusion, misinformation, and obfuscation. And the number of unanswered questions is rapidly increasing.
We understand from the police statements that the initial meeting of bikers at the Twin Peaks was a conference to discuss territorial disputes between rival outlaw gangs. But, this is not true. The conference was a regularly scheduled meeting organized by the Texas Confederation of Clubs & Independents to discuss current legislation that affects biker safety and rights. This meeting has occurred regularly every couple of months for many years in communities around the state without there ever being any prior incidents of notable violence.
Knowing that, a number of questions start to arise about the police presence.
· When did the Waco Police Department (WPD) or other law enforcement (LE) first find out about the potential for two or more bike clubs meeting at the Twin Peak (TP) restaurant on the date of the incident?
· What was the basis of the information?
· Did the Waco Police Department (WPD) ever receive any information that the Twin Peak Restaurant meeting of bike clubs was a legislative assembly meeting?
· Did the WPD prior to the incident ever understand that the event was being arranged by the Texas Confederation of Clubs and Independents (TCCI)?
· Did the WPD have information prior to the event that the TCCI event was an invitation to a large and varied number of clubs from the region?
· Did the WPD have information prior to the event of the background and composition of the various member clubs to the TCCI?
· When did the Waco Police Department (WPD) or other law enforcement (LE) first find out about the potential for two or more bike clubs meeting at the Twin Peak (TP) restaurant on the date of the incident?
· What was the basis of the information?
· Did the Waco Police Department (WPD) ever receive any information that the Twin Peak Restaurant meeting of bike clubs was a legislative assembly meeting?
· Did the WPD prior to the incident ever understand that the event was being arranged by the Texas Confederation of Clubs and Independents (TCCI)?
· Did the WPD have information prior to the event that the TCCI event was an invitation to a large and varied number of clubs from the region?
· Did the WPD have information prior to the event of the background and composition of the various member clubs to the TCCI?
Read More
The leaked video from the Don Carlos parking lot capturing a small piece of the conflict in Waco may not show us exactly what transpired on May 17th, but it very well may reveal the key witness and what triggered the shooting. The man in the blue shirt must be identified. Full Story - Watch the Full Leaked Video
Sturgis Biker Rally
Click for information on the Sturgis Biker Rally
Daytona Bike Week
Click for information on Daytona's Bike Week
From our Friends at http://bikerresources.thunderpress.net
Bike Friendly
Too loud for the local crowd? Check out our extensive lisitings of biker-friendly establishment!
Looking for a used bike, new job, or new business? Shift over to our classified listings!
Want to hop on a hog and ride, just for the weekend? Need to get your bike tuned? Check out our listings!
T-Shirt Shop
Looking for a killer new T-shirt? We have the best and the darkest here in our T-shirt shop listings!
Problem With Profiles: By David "Double D" Devereaux, a motorcycle rights activist from Washington State, and a leading advocate for laws addressing the issue of motorcycle profiling in America.
1%'ers: The Confederation of Clubs of Washington State serves as the official voice of motorcycle club members present at the bill signing on April 13, 2011, and would like to address the controversy surrounding a picture taken with 1%'ers and Governor Gregoire.
LETTERS: Two critics ask questions about the controversial May 17 biker shootout at Twin Peaks: Click to read letters regarding the Waco Tragedy.
See all Perspective and Opinions
The National Council of Clubs Addresses Leesburg Bikefest Shooting The National Council of Clubs (NCOC) issued a Press Statement today addressing the recent shooting that occurred April 29th, 2017, during the annual Bikefest in Leesburg, Florida.
They brought raffle tickets to a gunfight.Four members of the Grim Guardians motorcycle club showed up to Twin Peaks restaurant in Waco, Texas, on May 17, 2015, for a meeting about state politics, they claim. Instead, rival motorcycle gangs opened fire on each other before police moved in.
Their attorneys say Baylor University should also get some of the blame.The lawsuit, filed in the U.S. District Court, Western Division in Austin, is connected to the deadly shooting at Waco's Twin Peaks restaurant two years ago.
Lawyers for three of the bikers at the Twin Peaks restaurant in Waco, Texas, filed legal documents to have the district attorney recused from their criminal cases. On Monday, a hearing on their motions put the McClennan County District Attorney on the hotseat. D.A. Abel Reyna, the police chief, and other law enforcement officials, were all placed on the stand to testify during the hearing.
In the quest to properly identify potential Law Enforcement shooters from the Texas Department of Public Safety here is a short list: Lt. Steven D. Schwartz Special Agent Christopher Dean Frost Sgt. Cory J. Ledbetter SGT. Christopher Hugh Dale
One year ago Tuesday, law enforcement officials here broke up one of the biggest and deadliest clashes of motorcycle gangs in the country. Gunfire erupted outside a Twin Peaks restaurant at a meeting of a regional coalition of motorcycle clubs after an altercation between two rival groups, the Bandidos and the Cossacks. So many bikers were arrested — nearly 200 — that officials used the Waco Convention Center for initial processing.
On May 17, 2016 at 10am, on the 1 year anniversary of the tragedy in Waco, the National Council of Clubs would like to invite you to a press conference announcing a National Protest Against Motorcycle Profiling and Discrimination. Motorcycle profiling and discrimination has reached epidemic levels in America threatening core constitutional principles and the basic civil liberties of millions of individuals.
A civil rights violation lawsuit was filed last Friday by one of those arrested at the Waco Shootout. The facts alleged in the complaint, if true, only confirms what I have believed all along.
The criminal-justice system in Waco, Texas, continues to boggle the mind. Last month, a grand jury in McClellan County held a marathon session to consider whether District Attorney Abel Reyna had presented enough evidence to justify indictments in the shootout at a May gathering of bikers where nine people were killed.
With word that Waco Police shot four of the bikers who died during the melee at outside the Twin Peaks restaurant in Waco seven months ago comes the question: Why didn't authorities come clean months ago?
Watch Brent Coon on the on Antidote, hosted by Michael Parker, speak about the Twin Peaks Biker Shooting in Waco. Click Here.
Get ready to ride! The 75th Anniversary of Daytona Bike Week is quickly approaching and we are totally excited with all the great things planned for this amazing once in a lifetime landmark event in 2016!! All Riders who want to be a part of this grand milestone will be glad they made their way to the Daytona Beach area for this very special occasion. Already our businesses and community are preparing an amazing array of fun and activities that are sure to create fond memories of fun and riding over ten days in this beautiful seaside resort destination.
Read More
We have collected a number of the video stories regarding the Waco Tragedy. Click to view our library of Waco Tragedy Videos.
The Lip.TV - Antidote - Waco Biker Gang Shooting: Cops and Court Conspiracy with Brent Coon
Cop Pepper Sprays Bikers - Fort Worth Texas
Local motorcycle club members barely missed Waco shooting
Autopsy: 9 killed in Waco shooting died from gunshot wounds
170 arrested in connection with Waco biker shooting
The Latest on Waco shooting: Biker group gets proclamation
Biker arrested in Waco Twin Peaks shooting speaks
Deadly motorcycle gang shootout in Waco started with parking dispute
Inside Twin Peaks: New Details Update Timeline of Deadly Waco Biker Brawl
Purple Heart recipient among bikers killed in Waco shooting
Complete the form below to contact Brent Coon and Associates for help with any biker related issue.
Brent Coon and Associates is experienced in many areas of personal injury including Motorcycle Accidents. Founder Brent Coon, while probably best known for his work against corporate greed including the Texas City Explosion and the Gulf Coast Disaster, is an avid motorcycle rider. Brent Coon is a recipient of the Steven Sharp "Trial Lawyer of the Year Award" from the AAJ and the Clarence Darrow Award, a finalist as lawyer of the year with Trial Lawyers for Public Justice, an ALM Top 100 Plaintiff Lawyers in the Country, a founding member of The National Trial Lawyers Association, a Fortune 500 "Top 100 Go To Attorneys" along with numerous other accolades. And as an avid motorcyclist, Brent Coon understands bikes, bikers and the complexities of motorcycle injury lawsuits and litigation.
The law firm of BCA has motorcycle accident attorneys all over the country and can help you with your motorcycle accident lawsuit in various districts all over the country. Our motorcycle accident attorneys will work tirelessly to make sure every aspect of your motorcycle accident lawsuit is reviewed and help insure you gain the maximum recovery the law allows. BCA has relationships with some of the top motorcycle investigators from around the country to help assist with the complexities which may arise from a motorcycle accident litigation.
As motorcycle accidents often result in serious injury or fatalities, we understand the stress an accident puts on both the injured and loved ones. With the worries of increased medical bills, recovering from injury, returning to your occupation, BCA understands the pressures that result from motorcycle accident. Our motorcycle lawyers will work with you hand in hand and provide constant communications on your motorcycle lawsuit.
Learn more about BCA Motorcycle Legal Team.
This section is coming soon
Brent Coon & Associates was founded in 2001. Today, with offices spanning the country, it is one of the largest trial law firms in the nation and the epitome of the 21st century law practice. Brent Coon & Associates employs a team of aggressive litigators, with solid experience in individual and complex multi-party, occupational/environmental, health and personal injury cases.
Brent Coon & Associates prides itself on being a public policy law firm. "Our first and foremost responsibility is to our clients and recovering the best possible settlement or verdict on their individual cases," says BCA founder Brent Coon.
The Law Offices of Brent Coon & Associates - Copyright 2015 - All Rights Reserved.
Brent Coon - Beaumont, Texas - 215 Orleans Beaumont, Texas 77701 - Disclaimer


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subset: commoncrawl


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Vivica A. Fox Implies 50 Cent Is Gay on 'Watch What Happens' MP3
Vivica A. Fox and Kandi Burruss were the special guests on tonight's Watch What Happens. During the live episode, Fox had some interesting claims about her ...
Watch What Happens MP3
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Vivica A. Fox Calls Out 50 Cent On Gay Comments - WWHL MP3
During a round of questions Andy Cohen asks Vivica A. Fox about her role in season 2 of "Empire," her thoughts on a possible spinoff, and Vivica appears to ...
Tony Bennett, Buddy Rich Orchestra - Watch What Happens MP3
From "Tony Sings Buddy Swings.
Cate Blanchett on Watch What Happens Live October 13 2015 MP3
Best of Oprah on Watch What Happens Live - WWHL MP3
Andy Cohen had Oprah on the show. These are the best moments from the Oprahocolypse. "Watch What Happens" as Bravo Andy Cohen interviews today's ...
Michel Legrand & Phil Woods 4tet 2001 Montreal - Watch What Happens MP3
Michel Legrand & Phil Woods Quartet July 1, 2001 22. Festival International de Jazz de Montreal, Spectrum of Montreal, Montreal, Canada Michel Legrand ...
Father buys his daughter Disney stock for Christmas: Watch what happens next MP3
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Kate Hudson on the Nick Jonas Dating Rumors - WWHL MP3
A WWHL caller asks actress Kate Hudson if she is dating singer Nick Jonas and Kate wants to move on to the next question. "Watch What Happens" as Bravo ...
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I added the "write it good" scene to this because it's the original Newsies pair, and who wouldn't want to add in Jeremy Jordan into their videos!! Music by: Alan ...
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Vivica A. Fox insinuated that 50 Cent is gay on "Watch What Happens Live" and the internet went crazy!
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Posted by ESPN.com's Bill Williamson
DENVER -- Champ Bailey was asked Sunday evening if he ever thought the new-look Denver Broncos' defense could possibly be this good when he headed to training camp in July.
Bailey carefully considered the question. Then, he smiled and slyly attempted to avoid it.
"I don't know," the usually quick-to-answer Bailey said after a long delay. "I'm not one for predictions."
Yeah, Champ, we didn't think it would be this good, either. But after the first quarter of the NFL season, his crew is one of the top stories in the league.
Facing the first test in a tough 10-game stretch, Denver's defense belted the Dallas Cowboys around for the final three quarters and survived a broken play in the final moments to secure a 17-10 win. Denver is one of a quartet of 4-0 teams in the NFL. Minnesota is 3-0 and it plays Green Bay on Monday night.
Under offensive-minded coach Josh McDaniels, the Broncos are winning on the strength of a 3-4 scheme under new defensive coordinator Mike Nolan. His Broncos are physical and unrelenting. (In fact, the former San Francisco head coach could be working his way back into head-coaching consideration with this masterful job.) The Broncos have allowed an NFL-low 26 points this season. In the final three games of the 14-year Mike Shanahan head coaching era, Denver allowed 112 points.
Denver's defense was so bad the past two seasons, it was often referred to as the "Enver Broncos" because there was no "D" in the city. Denver is back and that's no joke.
The Broncos are stopping the run. That was a huge problem in recent years. Dallas had 74 yards rushing. Denver is rushing the passer with vengeance. Linebacker Elvis Dumervil, who converted from end this year, has eight sacks in four games. The Broncos have claimed 10 turnovers.
"I think [the change] is that we are so prepared and so confident," Bailey said. "We're ready for anything."
That was evident in the final minute of the game.
After Dallas jumped out to a 10-0 lead in the first quarter, Denver's defense shut down the Cowboys. Dallas quarterback Tony Romo was pressured in the final three quarters and harassed into making one errant throw after another. Dallas couldn't establish any ground-game rhythm.
Then there was the broken play. On fourth down and three from its own 27 with 1:16 to go and Denver leading 17-10, Romo shot out of pressure and hit Sam Hurd about 20 yards down field. Hurd bolted 53 yards down to the Denver 20.
Did the old, porous Denver defense resurface at the worst possible time?
"It was just one play," Dumervil said. "We had to strap it back on and get after it. There was still 20 yards to go."
Dallas couldn't complete the journey.
In the final nine seconds of the game, from the Denver 2, Romo went after Bailey. Yes, Bailey. The Broncos cornerback beautifully defended two straight passes intended for Hurd in the end zone to win the game.
Bailey is rarely challenged. But Dallas went after him all game. Bailey made a difficult interception at the Denver 3 in the third quarter.
"That's Champ Bailey, one of the best cornerbacks of all time," Denver linebacker D.J. Williams said. "I don't know why they would throw at him."
Asked if he was shocked that the ball came his way on the final two plays, all Bailey said was, "I'm glad they did."
So are the Denver faithful.
But Denver's defense is more that just 10 scrubs and a Hall of Fame cornerback.
This is a cohesive group that is playing well on all three layers of the unit. The defensive front three, one of the most inexperienced groups in the league, is setting the tone in the run game. Linebackers Andra Davis, Williams, Mario Haggan and Dumervil are creating chaos. The secondary has been spectacular. New safety Brian Dawkins plays like he is 30, not nine days away from celebrating his 36th birthday.
"Everyone here holds each other accountable," said right cornerback André Goodman."On that last drive, no one gave up. We just got stronger. That's been the way we've done it all season."
Offensively, Denver has been very timely. Case in point: Brandon Marshall 's jaw-dropping 51-yard catch and run from Kyle Orton to give Denver the lead with 1:46 remaining. Orton has not thrown an interception all season and the Broncos' offense is doing just enough to help its defensive mates.
In the past, it was the Denver defense that couldn't help out the offense. But a lot has changed in Denver.
The "D" has come back to the city.


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The following is a continuation of Forum:BO2 Zombies map names.
Let me explain why Green Run, Great Leap Forward, Alcatraz Island, and Resolution 1295 are the actual maps.
What does everyone see when loading a multiplayer game? They first see the name of the map they are playing on, then the location of the map below it (not really important but its there), then the game mode they are playing. same for zombies games. For example, load a game of grief on Borough.
Resolution 1295
Primary Plates
Now load a Buried game.
Resolution 1295
Primary Plates
That would also explain why Buried, MOTD, and Die Rise are also gamemodes. If TranZit is a gamemode, then the rest (Die Rise, Mob of the Dead, Buried, and most likely Origins too) are also gamemodes.
That being said, if Green Run is a map, then Great Leap Forward, Alcatraz Island, and Resolution 1295 are also maps. SKITTLESINMYASS (talk) 01:52, August 13, 2013 (UTC)
DiscussionEditWith that logic we'd have to rename Nuketown Zombies to "Nevada, U.S.A.".
literally, thats true. The actual map is Nevada. but people can still call it Nuketown, and same goes with the other maps. but on the wiki, isnt supposed to state the facts? meaning what im saying is actually true SKITTLESINMYASS (talk) 02:06, August 13, 2013 (UTC)
Rather than going by what is technically true, I prefer how the place with most action (as I said in the last forum) and content is called the map, i.e. Buried, because the easter eggs/buildables/new characters/other notable content are there instead of the other subsidiary places such as Borough in Turned. Not to mention that the maps are advertised and officially (by that I mean Treyarch) spoken of as Buried and the like. Also, I discontinued the last forum because it was completely inactive, not because I wanted to ignore your points; if this becomes such a ghost forum that goes nowhere I'll close this too. -- laagone (talk) 10:10, August 14, 2013 (UTC)
This isnt gonna become a ghost forum. im actually following up on this
And in the previews, ive noticed that Treyarch has never called them maps. They say "a brand new zombies experience" . i think that the template should say the maps are Green Run, Great Leap Forward, Alcatraz Prison, Resolution 1295, and whatever Origins is going to be under. You guys already have Green Run under there so the others should be there too.
There might not be pages for Alcatraz and Resolution 1295, but how about I make them? ill do the work.
and i know, i forgot to sign againWWZ Veteran (talk) 11:57, August 14, 2013 (UTC)
I meant ghost forum as in uninterested by others than a couple of users. And I think it's sufficient that for example in promotional posters there are the regular multiplayer maps, and the Zombies one. Don't change the template, at least yet. -- laagone (talk) 12:58, August 14, 2013 (UTC)
When you say "at least yet", you mean it may be changed eventually?
And I see what you mean by ghost forum, but what can i do?
Also, if i can ask, what do YOU think? you read my other comment about the loading screens, no? whats your opinion on this?WWZ Veteran (talk) 17:03, August 14, 2013 (UTC)
The template might get changed if it comes to that, but for now let it be like it is. It's a little bit difficult to attract other people's attention to things they might not care or even know about, but making consensus out of two people's opinions when both of them differ is impossible. As to the loading screens I already stated what I think of them as the first thing I said on this forum. I think the maps are Green Run (also, I'm not exactly saying TranZit is a gamemode, it's the "campaign" of BO2 Zombies that includes Green Run, Die Rise and Buried), Die Rise, Mob of the Dead, Buried and Origins given everyone (including the officials) barring a few like you speaks of them as the maps, despite what loading screens tell me. -- laagone (talk) 17:40, August 14, 2013 (UTC)
ok, i see what youre saying about getting other people, yes they might not care. i dont know yet on what to do about that.
and i dont think TranZit is the overall gamemode including everything. people have said that from the start because that was the only one available when the game came out. but then when Revolution was released, people (not everyone unfortunately) saw that they were wrong, including ME. TranZit is called that because it features a bus (Transit with a Z). The only way its related to the other maps is through story, like you said, Die Rise and Buried.
and what the loading screens tell you, that proves what im saying is true, but that doesnt mean people have to call them Great Leap Forward, Alcatraz Island, etc. The public is free to call them whatever suits them best, but the way i see it, the wiki should state the facts like this, meaning that the names of the maps should be changed because its the truth.
and one last thing a little off topic: im not, like, angry or pissed off about this and im not trying to argue with any of you guys. im debating. if it ends up being official and stays the same, i dont really care because i have a military career to pursue. but i just thought i would help out a little, and i thought this was right, and i still think its right.WWZ Veteran (talk) 23:56, August 14, 2013 (UTC)
- Closed - No consensus as well as going off-topic.


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An investigation by the BBC's Money Programme exposed a London agency, Amelia King, which advertised a property without the vendor's permission or knowledge - and at more than his asking price.
• Weak dollar sparks online Christmas shopping bonanza
The practice is a criminal offence, in breach of the Estate Agents Act. A successful prosecution could result in a fine of up to £25,000.
To avoid the commission charged by estate agents, Simon Holder offered his Greenwich home for sale privately through Propertybroker.com.
He provided a description of the house, Propertybroker took photos of it, and these appeared on thewebsite.
But Mr Holderwas shocked to learn the property was also being marketed by Amelia King on estate agents' website Rightmove - even though he had never spoken to the company.
His description was used word for word with the pictures from the Propertybroker website. After the Money Programme found two similar cases, he called the agent to ask where it had found his property's details.
Mr Holder, who recorded the call, was told: "I got it from Propertybroker and I was just advertising your property and finding the customers."
The agent agreed that it was not instructed to advertise the house.
Mr Holder also asked why his home was listed at £378,000 when his asking price was £370,000. The agent said this was because it was trying "to find customers free of charge to the owners".
However, the extra £8,000 was just over two per cent of his asking price, an agent's normal commission.
Worse, as his house is now under offer, his buyer would be upset to find the house still on the market.
Amelia King manager Anton Mara was not available when contacted to ask why his company was marketing the property and others without the owners' permission, in breach of the Estate Agents Act. He did not return telephone calls.
Miles Shipside, of Rightmove, said: "If agents are not complying with the Act then it will be our policy to suspend them from membership. That appears to be the case in this instance and that's what we've done."
Kay and Keith Davis, who run Propertybroker.com and have saved customers as much as £82,000 in agents' fees, said: "This is not the first case we know. This offence is widespread and becoming more so as the market gets increasingly competitive. It must be nipped in the bud.
"We are reporting Amelia King and Rightmove to Trading Standards and will strongly urge them to prosecute."
Legislation coming into effect in spring 2008 should stop the practice by requiring all agents to belong to the Ombudsman's redress scheme, which Amelia King does not do.
However, the programme concludes that with the number of private sales websites rising - helping to save sellers thousands - and the public's lack of faith in estate agents, the fight between the two will only intensify.
- More about:
- Bonanza
- Greenwich
- Housing Prices
- Internet
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Escorts in The World's Greatest City
The heart off London is rich with British history and culture and has many sights and attractions that are not to be missed. Call up and book one of the stunning and beautiful London escorts to show you around and provide you with an unforgettable experience.
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Phillies unravel Red Sox to take interleague openerMay 19, 2012 - 4:00 AM
Ty Wigginton and John Mayberry added an RBI apiece for the Phillies, who extended their season-high win streak to six games.
Cole Hamels (6-1) worked seven solid innings, allowing six hits and three runs while striking out nine.
"It was nice to get some runs right off the bat," said Hamels, whose club staked him to a 4-0 lead when he went out for the second inning. "I was able to go out and keep the momentum and wear down the opponents."
Jonathan Papelbon continued his great start with his 12th save in as many chances.
Mike Aviles, Cody Ross and Adrian Gonzalez went deep for the Red Sox, who have dropped two of three.
Daniel Bard (3-5) departed after five frames, charged with five runs and three hits while issuing five walks.
Gonzalez hit the facing of the lower deck in right off Chad Qualls with one down in the eighth to bring Boston within 5-4, and after Ross fanned swinging, Antonio Bastardo retired David Ortiz on a grounder to first.
Galvis provided an insurance run on a line homer with one out in the home half, then Papelbon worked around a Kelly Shoppach infield hit to start the ninth to close out the win.
"It wasn't much different -- just a little more juice in the ballpark," said Papelbon about facing the team for whom he helped win two World Series. "But it was just another ballgame and another big save."
After Hamels worked around Aviles' leadoff double in the first, the Phils picked up four in the home half on a two-run single from Ruiz, a line-drive sac fly from Wigginton and an RBI double by Mayberry.
"I'm still in reliever mode and I have to learn how to be more aggressive," Bard said of his disastrous opening frame. "This one is on me."
Boston responded with a run each in the third and fourth frames on an Aviles homer and Daniel Nava sac fly, respectively.
Pence made it 5-2 with a homer to left-center with one out in the fifth, but Ross equalized in the sixth to pull the Sox back within two.
Game Notes
Boston still leads the overall series 27-20, though Philadelphia has won four of the last five meetings...Phils manager Charlie Manuel did not appear, as he was serving his one-game suspension for an argument with umpire Bob Davidson earlier in the week for which Davidson was also suspended...The Phils last won six games in a row from September 5-10...Red Sox manager Bobby Valentine was ejected in the top of the eighth inning for arguing a call at first base.
May 18 9:57 PM
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rootsRUWTbot Added 38 roots
Red Sox 4, Phillies 6 Top 9, 1 OutMay 18 9:52 PM
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Red Sox 4, Phillies 6 End 8May 18 9:45 PM
- Freddy Galvis one-out, solo Home Run (2) to left. Antonio Bastardo due up.
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- Adrian Gonzalez one-out, solo Home Run (3) to right. Cody Ross due up.
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- Carlos Ruiz singled to center scoring Jimmy Rollins and Shane Victorino with one out. On the play, Hunter Pence advances to third. Runners on first and third with Ty Wigginton due up.
Red Sox 0, Phillies 2 Bot 1, 1 OutMay 18 7:26 PM
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Related News
- Ventura: Relievers, media pose threats to Bradley's streak May 23
- LEADING OFF: Kershaw tests streak vs Reds, Colon faces Nats May 23
- Baseball Capsules May 22
- Phillies-Tigers Preview May 22
- Ortiz gets 4 hits, Bradley's streak at 27, Bosox top Indians May 22
- Eickhoff shuts down Braves over 7 innings, Phillies win 5-0 May 22
- Eickhoff shuts down Braves over 7 innings, Phillies win 5-0 May 22
- Boston's Bradley extends hitting streak to 27 games May 22
- LEADING OFF: Bradley's 26-game hit streak, Braun recovering May 22
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University of Arizona, College of Optical Sciences, 1630 East University Boulevard, Tucson, Arizona 85721
We theoretically investigate the effect of noise on frequency-domain heterodyne and/or homodyne measurements of intensity-modulated beams propagating through diffusive media, such as a photon density wave. We assumed that the attenuated amplitude and delayed phase are estimated by taking the Fourier transform of the noisy, modulated output data. We show that the estimated amplitude and phase are biased when the number of output photons is small. We also show that the use of image intensifiers for photon amplification in heterodyne or homodyne measurements increases the amount of biases. Especially, it turns out that the biased estimation is independent of AC-dependent noise in sinusoidal heterodyne or homodyne outputs. Finally, the developed theory indicates that the previously known variance model of modulation amplitude and phase is not valid in low light situations. Monte-Carlo simulations with varied numbers of input photons verify our theoretical trends of the bias.
Medical imaging in the regime of visual wavelengths, such as diffuse optical tomography (DOT) or fluorescence lifetime imaging (FLI), has emerged as a non-invasive, safe, and cost-effective method to investigate physiologic and/or biological quantities in tissues.1,2 Frequency- or time-domain measurements are widely used data-acquisition methods to improve the system performance in diffusive imaging. These methods measure the frequency-response of diffusive media in addition to the steady-state response (DC). The additional information measured from frequency- or time-domain measurements generally alleviates disadvantages inherent in DC-based diffusive imaging.1,3 For example, the crosstalk between scattering (μs) and absorption (μa) coefficients of reconstructed functional images of tissues in DOT can be dramatically reduced by employing frequency-domain measurement.4,5 Also, the problem of ill-posedness and non-unique solutions typically happens in reconstruction algorithms of tomographic diffusive imaging, which can be effectively managed using frequency- or time-domain measurements.1,6
In addition to acquiring various types of data, such as a frequency-response of tissue, collecting large datasets also enhances system performance in both optical tomography and topography imaging.2 Researchers have considered collecting large datasets using imaging systems with detector arrays, such as charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS) detectors, in diffusive imaging as opposed to conventional fiber-based systems that collect data at relatively a few measurement points.7 It is demonstrated that DC data measured by a CCD can relieve the ill-posedness and improve the resolution of reconstructed images in DOT.8 As another example, a CCD-based DOT system operating at up to 1 GHz frequency of intensity modulation was built for investigating small tissue volume.9 It is also reported that the reconstructed images from fluorescent optical tomography was improved by large DC datasets.10 Recently, the Hotelling observer performance for frequency-domain phased array systems with a detector array is investigated showing that stable and high detectability is achieved at very low modulation frequencies.11
For frequency-domain systems, the intensity of an incident beam is modulated at, typically, hundreds of MHz. The modulation amplitude and phase of an output modulation beam from a tissue are attenuated and delayed due to scattering and absorption characteristics of the tissue. Since the typical modulation frequency in diffusive imaging is usually higher than the achievable sampling rates of current CCD or CMOS detectors, a gain-modulated image intensifier is used to make the modulation amplitude and phase of the output measurable. There are two methods of the gain-modulation measurement: the heterodyne and homodyne methods.3 The heterodyne method makes the output modulate at a beat frequency, that is the difference between source and gain-modulation frequencies. The beat frequency is selected from the purpose of diffusive imaging considering the maximum frame rate of the CCD or CMOS detectors. The homodyne method is conceptually the same as the heterodyne in that a beat frequency output is detected, but the output beam intensity is varied along the beat frequency trajectory by changing the relative phase between gain and source modulations. For this case, the output images are stable in time, so it is easier to average multiple images to reduce measurement noise. The measurement speed of the homodyne method is, however, limited due to instrumentation required to change the modulation phase, which indicates that homodyne measurements might not be appropriate for very fast dynamic diffusive imaging, where heterodyne measurements operating at up to a few kHz beat frequency have the potential to monitor the dynamic variations of biological quantities.12,13
Time-domain systems are conceptually similar to frequency-domain systems. The impulse response function is attenuated and distorted by scattering and absorption in a tissue, which contains the frequency-response information of attenuated amplitude and delayed phase. The time-domain approach has the benefit that one can estimate the frequency-response of tissue for a wide range of frequencies from a single measurement. Furthermore, for very thick tissues, where it is hard in a frequency-domain measurement to extract meaningful data due to very small number of output photons, an impulse response function can be measured by the time correlated single photon counting (TCSPC) method.5,14 However, the instrumentation for time-domain systems is usually more complicated and costly than that of frequency-domain systems. More discussion for the comparison between frequency- and time-domain measurements can be found in Refs. 1 and 3.
For both frequency- and time-domain measurements, the estimated DC, amplitude, and phase are random variables due to measurement noise. The randomness of these quantities detrimentally affects tomographic reconstructions employing these quantities as well as biological parameters estimated from these quantities. Therefore, it is important to investigate the statistical characteristics of the frequency-response quantities with an appropriate noise model. V. Toronov et al.15 theoretically derived variances of modulation amplitude and phase in a frequency-domain DOT by Fourier transforming heterodyne outputs with quantum shot noise. These analytic expressions of amplitude and phase variances have been used to investigate optimized modulation frequencies for small volume tissues and to evaluate a CCD-based phased array system in DOT.11,16 In another study, the influence of system parameters on amplitude and phase signal-to-noise ratios (SNRs) was analyzed.17 The statistical characteristics of the frequency-response quantities also play an important role in FLI, where the fluorescent lifetime of fluorophores, an indicator of biological characteristics of tissues, is estimated. Instead of directly measuring the lifetime, which is typically a few or tens of nanoseconds, the frequency-response of fluorophores is measured using intensify-modulated beams. In FLI, the effect of Poisson noise might be significant because the number of fluorescent output photons is occasionally very small (∼several hundreds).18,19 E. Gratton et al.20 simulated the disagreement of modulation amplitudes and phases between frequency- and time-domain measurements in accordance with output photon counts in FLI. J. Philip and K. Carlsson19 theoretically studied the effect of photon noise on the fluorescent lifetime SNR estimated from various methods, such as lock-in amplifiers, frequency-domain measurements, and demodulation. A. Esposito et al.21 investigated photon economy and acquisition speed in various FLI detection techniques using Monte-Carlo simulations, where they observed that when the number of detected photons is small, the algorithm for estimating the florescent lifetime converges to incorrect values. Recently, Y. Lin and A. Gmitro22 theoretically found out that fluorescent lifetimes estimated by a lock-in amplifier method are biased in the situation of the low number of detected fluorescent photons because of Poisson photon noise.
The discrete-Fourier transform is widely used to estimate the attenuated amplitude and delayed phase of the measured heterodyne or homodyne or impulse time response outputs, since it is simple and fast.7,9,14,19–21 In this paper, we derive expressions for means and variances of modulation amplitude and phase that are estimated by Fourier transforming noisy sinusoidal measurements from heterodyne or homodyne methods. Other methods, such as data fitting or maximum-likelihood estimation can be used to extract the frequency-response information, but they generally require more computation time than Fourier analysis.23 When collecting large datasets in diffusive imaging, the computational advantage of Fourier methods over other fitting methods becomes enormous, because the number of detector pixels is very large. Although the kind of methods to estimate modulation amplitudes and phases from measured noisy data might affect the statistics of these quantities, the result of the developed theory in this paper provides insights about the effect of noise on a frequency-domain measurement.
The theoretical derivation of the mean and variance of a modulation amplitude and phase is described in Sec. 2. We first develop the theory by considering that sinusoidal outputs are contaminated by temporally uncorrelated noise. Then the derived noise model of heterodyne outputs is applied to the developed results. We theoretically observed that estimated modulation amplitude and phase are severely biased from the actual values, when the number of output photons is small. Furthermore, the amplitude and phase SNRs show large discrepancies from the previously known SNR model of V. Toronov's et al.15 Sec. 3 shows computational results based on the developed theory. Monte-Carlo simulations are also presented, which verify the characteristics of biases. Extending the developed theory is briefly discussed in Sec. 4 along with conclusions.
By Fourier transforming noisy heterodyne/homodyne measurements, we can estimate the modulation amplitude and phase. We denote the true amplitude and phase as A and φ, and estimates of them as A^ and φ^, respectively. To describe the estimation process, we first describe the Fourier transform of the noisy data evaluated at the beat frequency Ω via Display Formula
Assuming temporally uncorrelated noise for i(t) in Eq. (1), the mean and covariance for i(t) can be written as Display Formula
For the statistics of A^ and φ^, the statistics of X^ considering Eqs. (3) and (4) should be investigated first. The mean of X^ is derived as Display Formula
It is reasonable to consider X^ in Eq. (1) is complex multivariate Gaussian from the central limit theorem, because all points of the noisy signal i(t) are weighted and summed to form X^. Considering Eqs. (6) and (11), the random variable transforming25 from (X^R,X^I) of the multivariate Gaussian to (A^,φ^) generates probability density functions (PDF) for A^ and φ^ as Display Formula
The mean and variance of A^ can be analytically calculated from Eq. (12), which are Display Formula
With the similar mathematical procedure for calculating moments of X^R(Ω0) and X^I(Ω0), the mean and variance for estimated DC D^ can be derived from that Ω=0 is substituted to Eqs. (5), (7), and (8), instead of considering Ω=Ωo. The condition of Ω=0 generates 〈X^R(0)〉=qβmTaDC, 〈X^I(0)〉=0, σR2=q2βc2TσDC2, and σI2=0, which denotes the PDF of D^ is univariate Gaussian. Notice that the variance of D^, q2βc2TσDC2 is the same as σA2 in Eq. (16) only if A¯=A. We will observe in simulation that A¯ is generally greater than A, unless the number of measured photons is very large, so it can be stated that the amplitude variance is generally smaller than the DC variance in frequency-domain measurements. Since D^ is univariate Gaussian, it is not difficult to show that D¯ is unbiased.
Equations (12)013014015–(16) are generally applied for estimating the amplitude and phase of any sinusoidal signal having the noise property of Eq. (3). Recently, the covariance of a heterodyne and/or homodyne output was theoretically derived from the random amplification of a temporal point process.27 The model of covariance implies that the variance of heterodyne outputs can be expressed as the form of σp2(t) in Eq. (4) and both σDC2 and σAC2 are affected by a Fano factorF24 of image intensifiers, which is experimentally verified. A Fano factor is defined by the variance divided by the mean of amplified (secondary) photons per input (primary) photon in image intensifiers. If the gain of an image intensifier in the heterodyne measurement is assumed to be high, and the spatial variation of the amount of primary photons incident on the image intensifier is not abruptly changed, the covariance model specifies aAC and σDC2 in Eq. (14) as27Display Formula
Figures 1(a) and 1(b) show some examples of prA^(A^) and prφ^(φ^) for arbitrarily chosen values of γ, A, and φ, respectively, to investigate characteristics of these PDFs. The values of βm and βc in Eq. (3) are also arbitrarily chosen as 0.5 and 0.6, respectively, which will be maintained for all simulations in this paper. Having βm, βc<1 indicates the conversion loss from photons to photoelectrons without an electronic gain in the CCD or CMOS detectors. As expected from the definition of PDF, all PDFs plotted in Fig. 1 are non-negative functions. Although all PDFs appear Gaussian shaped, they are actually not, as mathematically verified from Eqs. (12) and (13). Figure 1 indicates that the asymmetry in both prA^(A^) and prφ^(φ^) increases as γ decreases, which results in biased means. From characteristics of prA^(A^) shown in Fig. 1(a), it is expected that A¯ is higher than A for the smaller value of γ. Oppositely, the behavior of prφ^(φ^) in Fig. 1(b) shows that φ¯ is smaller than φ because the value of prφ^(φ^) is increased around −π as γ decreases. From other simulation results, we confirmed that when φ is negative, prφ^(φ^) around π starts increasing as γ decreases. Therefore, it can be stated that φ¯ is always under-estimated regardless of its actual mean. Similarly, we observed that A¯ is always over-estimated. It can be also conjectured from Fig. 1 that the amount of bias for both A^ and φ^ is dependent on their actual mean values. Furthermore, it can be expected that smaller γ increases the bias of A¯ and φ¯ as well as the standard deviations of A^ and φ^.
Some examples of PDFs of the modulation amplitude and phase are plotted with arbitrary parameters. Notice that these PDFs deviate from the symmetry as γ is decreased.
Since it is definite that γ is critical for statistics of A^ and φ^, as observed in Fig. 1, it is worth to briefly discuss the effect of heterodyne measurement parameters on γ in Eqs. (14) and (19). It is clear from Fig. 1 that higher γ is generally better to correctly estimate A^ and φ^, which is achieved by the larger amount of primary photon, Tcp in Eq. (19). If modulating primary photons are assumed to be directly detected without heterodyne and/or homodyne processes, it is reasonable to consider that the main noise for the primary is Poisson. For Poisson photon noise, σDC2=cp and aAC=cpmp in Eq. (14), so γ is Display Formula
We assumed an image intensifier's gain is large enough to validate Eqs. (17) and (18), so NF=4 is selected for simulations. Figure 2 shows percent differences between actual and estimated means for A^ and φ^, respectively, with parameters of mg=1=1.3, and ∑g=1∞mg2=1.5×m12 for γ in Eq. (19). A value of 1.3 for mg=1 is considered from the experimental result27 and ∑g=1∞mg2=1.5×m12 indicates the summation of mg≠12 is assumed to be 50% of m12. Considering the typical quantum efficiency of a photocathode in image intensifiers, c0 in γ is set to 0.3. The larger value among estimated and actual means is used as a denominator to set the percent difference within ±100%. Since T is usually much longer than the modulation period of the primary, T·cp in x-axes of Fig. 2 can be considered as the average primary photons during T. For frequency-domain diffusive imaging, mp is largely varied across a phantom exit surface. Thus, A¯ and φ¯ in Fig. 2 are simulated for three different mp values. As shown in Fig. 2, both A¯ and φ¯ are severely biased when the number of primary photons from a phantom is small, where A¯ and φ¯ show over- and under-estimations, respectively, as expected from the results in Fig. 1. It is verified from other additional simulations that φ¯ always approaches zero as the primary photon number decreases. Furthermore, Fig. 2(b) indicates the amount of bias for φ¯ is strongly dependent on φ, which is consistent with the behavior of prφ^(φ^) in Fig. 1(b). It should be noticed that the typical range of c0k¯nI is from hundreds up to ten thousands, so the number of secondary photons for the primaries in Fig. 2 is actually very large. Figure 2 implies that for the diffusive imaging, where the number of output photons from a phantom is small, tomographic functional images or biological quantities estimated from A^ and φ^ may be highly inaccurate.
Percent differences between actual and estimated means for (a) A^ and (b) φ^ are plotted as a function of primary photon number of the output beam from a diffusive medium. The terms mp and φ indicate an actual modulation depth and phase of the output beam, respectively.
Figure 3 shows percent differences between SNR calculated from the derived theory in this paper and SNR (SNRt) from the previously known model of Eq. (2) for the same parameters as Fig. 2. For SNRt, A¯ and φ¯ are assumed to be correct, A and φ, respectively. Like Fig. 2, the larger one between SNR and SNRt is used as a denominator for calculating these percent differences. There is a good agreement between SNR and SNRt when the primary photon number from a phantom is relatively large. However, the agreement starts deviating as primary photon number is decreased, where SNRA and SNRφ are larger and smaller than SNRAt and SNRφt, respectively. For SNRφ, especially, the primary photon number starting the discrepancy and the amount of the percent difference are magnificently increased, compared with the bias of φ¯ in Fig. 2. Figure 3 shows that it is not appropriate to estimate system performance based on the previously known noise model for frequency-domain diffusive imaging systems with the small output photon numbers, such as the system observing dynamic variation of biological quantities or FLI.
Percent differences between SNR calculated from the theory and SNR from the previously known model for (a) A^ and (b) φ^ are plotted as a function of primary photon number of the output beam from a diffusive medium. mp and φ indicate an actual modulation depth and phase of the output beam, respectively.
We examine the estimated means and their characteristics using brute-force and Monte-Carlo simulations. For brute-force simulations, we generate sinusoidal signals of one period with known D and A, where white Gaussian noise is added. Fourier transforming the noise-contaminated signals determines D^ and A^ that are random. Averaging these random variables by repeating brute-force simulations can estimate D¯ and A¯. Figure 4(a) shows one example of the noisy sinusoidal signal that mimics a heterodyne or homodyne output. The standard deviation of the noisy data is 10 and the solid line indicates the mean sinusoidal signal. D¯ and A¯ estimated from the ensemble of the noisy signals are shown in Fig. 4(b) as increasing the standard deviation from 0 to 30. As theoretically investigated, D¯ is unbiased for the entire range of the noise level. However, A¯ shows the bias estimation, the amount of which is increased as the noise level increases—over-estimated. In order to investigate biased estimations in the situation similar to diffusive imaging, Monte-Carlo simulations are conducted using tMCimg30 for a 62×62×30 mm3 homogeneous phantom of μs=5 mm−1 and μa=0.005 mm−1. The tMCimg simulator is widely used in diffusive imaging Monte-Carlo simulations, where the Henyey-Greenstein phase function is assumed for scattering. The anisotropy parameter for the Henyey-Greenstein phase function is set to 0.8 in this simulation. The source of a 2 mm diameter injects photons to the phantom, of which number is varied from 0.5×108 to 4×108, as shown in Fig. 4. The degree of noise on output photons are decreased as the total photon number increases, because the increased photon number is achieved by summing independently simulated outputs. For example, the output from 4×108 of total photon number is the summation of eight outputs of 0.5×108 source photons. The 40×40 mm2 detector array with 1×1 mm2 pixels on the exit side of the phantom collects simulated output photons. Different from brute-force simulations in Figs. 4(a) and (b), temporal functions of output photons on each detector pixel are Fourier transformed to calculate D^, A^, and φ^. Since it is obvious that D¯ is unbiased from the developed theory and brute-force simulation, the term m¯p is investigated instead of A¯ to minimize the numerical error of the Monte-Carlo simulations, such as photon number mismatching. Figures 4(c) and 4(d) show the averaged modulation depth (m¯p=A¯/D¯) and φ¯ from the entire detector pixels at fixed modulation frequencies (fm). Averaging from the entire pixels avoids a heavy computational load because enormous number of simulations is required to calculate means for one detector pixel. Although actual means mp and φ are different for different detector pixels, PDFs of m^p and φ^ are the same (with different means), as shown in Eqs. (12) and (13), respectively. Therefore, biased estimations would be clearly observed for m¯p and φ¯ averaged from all detector pixels if the theoretical results derived in Sec. 2 are valid. For the comparison, all curves calculated at a fixed fm are scaled for both m¯p and φ¯ to make the values of the photon number of 0.5×108 are the same. Figures 4(c) and (d) show that m¯p and φ¯ are decreased and increased, respectively, as the number of simulated photons increases, which indicates that m¯p and φ¯ are over- and under-estimated. Furthermore, it is observed that the degree of bias is worse for higher fm, where the number of output photons contributing to the calculation for m¯p and φ¯ is smaller. It is well known that m¯p and φ¯ are decreased and increased, respectively, as fm increases, which shows the validity of the Monte-Carlo simulations. Although other systematic factors in heterodyne or homodyne measurements, such as a photon amplifier, signal and noise amplification, and photon losses in these brute-force and Monte-Carlo simulations, Fig. 4 clearly shows that characteristics of estimated quantities from simulations are coincident with what expected from the developed theory.
Estimated mean DC and AC, D¯ and A¯ are calculated from sinusoidal signals with white Gaussian noise of increased standard deviations. It is clear that D¯ and A¯ are unbiased and over-estimated, respectively, in (b). In (c) and (d), Monte-Carlo simulated means are shown as a function of simulated photons, where the relative amount of noise is decreased as the number of incident photons is increased. Estimated means of (c) mp and (d) φ^ from simulations are over- and under-estimated, respectively.
As shown from the theory and simulation, increasing the output photon number from a phantom can definitely remedy the bias estimation and make the simplified noise model of Eq. (2) acceptable. This is definitely achieved by increasing the total measurement time T which indicates that quite a number of heterodyne and/or homodyne output periods are acquired to estimate A^ and φ^. However, increasing measurement time is not allowed or difficult in some situations, where, for example, the imaging purpose is to observe the dynamic variation of biological quantities, such as hemodynamic changes.31 Alternatively, increasing the DC primary photon rate cp decreases the amount of bias estimations with a fixed T. Because the photoelectron capacity of CCD or CMOS detector pixels is usually limited, however, the gain of image intensifier should be decreased to avoid the saturation of detector outputs for this increased cp. The lower gain indicates the noise factor NF of the image intensifier should be increased. Resulting from this, γ in Eq. (19) might not be increased enough to diminish the bias even for the increased amount of primary photons.
Percent differences for A¯ and φ¯ with increased NFs are shown in Figs. 5(a) and 5(b), respectively, where NF ranges 4 to 10 and 4 to 16. It was experimentally shown that these NF ranges are easily achieved in common image intensifiers of decreased gain.29 For comparison, results of NF=4 in Fig. 2 are repeatedly presented in Fig. 5. We assume NF linearly increases for decreasing the MCP gain k¯nI that makes the number of secondary photons constant for the increased primary photons in Fig. 5. This implies σ2_DC of Eq. (18) is increased by the increased NF. Other simulation parameters in Fig. 5 are the same as Fig. 2. As shown in Fig. 5, the degree of bias is worse than the results in Fig. 2, which is caused by increased NF. Interestingly, the detrimental effect of increased NF is much more significant for A¯ than φ¯. It is observed in Fig. 5(b) that the effect of increased NF is almost ignorable for cases of φ smaller than 3π/4. Higher NF indicates smaller SNR of secondary photons for given SNR of primary photons, which physically means the secondary photons are more random. Figure 5 indicates that estimating modulation amplitude is much more sensitive to the randomness of output photons than phase.
Percent differences between actual and estimated means for (a) amplitudes and (b) phases are plotted. For the increasing primary photon number, NF is linearly increased with ranges of 4∼10 and 4∼16.
We observed that modulation amplitude and phase A^ and φ^ are biased when they are estimated by Fourier transforming the noisy heterodyne outputs. Furthermore, the previously known SNR model for A^ and φ^ is not valid when the number of output photons from a phantom is small. It is known that a very high modulation frequency is required to achieve high SNRs for φ^ in diffusive imaging for small volume phantoms9,16 because a high modulation frequency increases φ in Eq. (4). However, the theoretical result in this paper shows that high φ (i.e., the large amount of phase delay of the modulated incident beam to a phantom) deteriorates the reliability of the estimated phase φ^. Furthermore, the modulation depth mp in Eqs. (17) and (19) is dramatically decreased as the modulation frequency increases, which further degrades the estimation for φ^. Although recently developed noise models for heterodyne and homodyne measurements indicate that small mp reduces σAC2 in Eq. (4),27 the decreased σAC2 does not improve the estimation for φ^ because γ in Eq. (14) depends on only aAC and σDC2. Therefore, it can be importantly noticed that increasing SNR for φ^ does not always reduce the degree of bias estimation.
In frequency-domain diffusive imaging with a detector array, increasing detecting points usually accompanies decreasing the detector pixel size, which reduces the measured primary photons per pixel. Furthermore, for detector pixels far away from the incident source, the modulation depth mp as well as the DC photon rate cp are decreased because of increased scattering and absorbing in the diffusive medium. Therefore, the bias in A^ and φ^ is substantially increased for high-dimensional frequency-domain diffusive imaging. Additionally, the typical loss of photons in heterodyne and/or homodyne frequency-domain measurements makes the photon number necessary for correctly estimating A^ and φ^ increased. For example, in some types of systems, where the phosphor of an image intensifier is imaged onto a CCD by an imaging system, the amount of photon loss due to the imaging system is usually significant. This loss of photons is equivalent to decreasing primary photons.
Although the theory in this paper is derived from Fourier analysis of a noisy sinusoidal signal, it can be expanded for time-domain measurements. In fact, it is reported elsewhere that φ^ estimated by Fourier transforming simulated impulse outputs from TCSPC in FLI are under-estimated as compared to the exact values, where the amount of under-estimation becomes larger for a higher modulation frequency.20 Similarly, the Monte-Carlo simulation results in Fig. 4 are calculated by Fourier transforming impulse response functions of output photons at each detector pixel. As long as A^ and φ^ are estimated by Fourier transforming a temporal impulse function that is contaminated by temporally uncorrelated noise, the theoretical analysis and characteristics for the statistics of A^ and φ^ is very similar with what is observed in this paper. For example, if the impulse output ap(i)(t) is measured without amplification, σDC2 in Eq. (4) can be considered as ∫−T/2T/2ap(i)(t)/T based on that the variance of shot noise induced by a time-variant signal is described by time-averaging the signal.32.
The theoretical procedure in this paper is based on a heterodyne measurement, but the similar results can be derived for a homodyne measurement. The lock-in amplifier method is widely used for estimating fluorescent lifetimes in FLI, where two (or more) co-sinusoidal signals are multiplied to measured modulated fluorescent outputs. The relative phases between modulation outputs and multiplied signals are set to be different like in a homodyne method. Therefore, the concept of lock-in amplifier is very similar with that of heterodyne measurement except the photon amplification process in a homodyne method, which is optional in lock-in amplifiers. We expect that this similarity is the reason that fluorescent lifetimes estimated from a lock-in amplifier show bias, when the detected fluorescent photons are small.19,22 Although the theoretical approach22 to show the biased fluorescent lifetime is different from what is presented in this paper, we consider the origin of the phenomenon is conceptually similar with what we have observed in this paper.
The effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging was investigated. It was theoretically derived that the Fourier analysis for a noisy sinusoidal signal induces the estimated amplitude and phase to be biased from actual values when the detected photon number is small. By combining the developed theory with the known noise model of heterodyne and/or homodyne outputs, the degree of bias was measured as a function of the number of diffused photons exiting the phantom. Furthermore, physical insight including the effect of heterodyne and/or homodyne measurement parameters on the bias was elucidated. Especially, we showed that the amplitude and phase biases are deteriorated by the increased noise factor of image intensifiers, which is common for heterodyne and homodyne measurements with low-gain image intensifiers. It is interesting that the noise depending on output signal variation (i.e., AC variance on noisy sinusoidal signals) does not affect the estimation of modulation amplitude and phase. Finally, we pointed out that the previously known noise model for the modulation amplitude and phase is not valid for the situation of small output photons. Therefore, considering the known noise model should be careful because, for many frequency-domain diffusive imaging situations, output photon counts are low. In the near future, experimentally measuring the bias for the estimated amplitude and phase will be investigated. Also, it is worth assessing the performance of frequency-domain diffusive imaging systems in photon-limited situations based on statistical properties of the estimated amplitude and phase developed in this paper.
Dongyel Kang and Matthew A. Kupinski"Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging", J. Biomed. Opt. 17(1), 016010 (Feb 10, 2012). ; http://dx.doi.org/10.1117/1.JBO.17.1.016010
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IML, at the 13th Aesthetic & Anti-Aging Medicine Congress
IML, represented by Dr. Irene Cruz, attended the 13th edition of the Aesthetic & Anti-Aging Medicine World Congress (AMWC), held on 26 to 28 March in Monaco.
Dr. Cruz holds a Masters in Aesthetic Medicine and is member of the Instituto Medico Laser medical staff.
The congress programme integrated multiple scientific and practical sessions dealing with issues such as new approaches and techniques in rejuvenation, practical protocols and information of new technologies.