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Is Terra Nitrogen's Robust 9% Dividend Yield Too Good to Be True?
In this era of 1% CDs, this type of ratio is highly coveted, but it doesn't mean much if it isn't sustainable. Does this high-income MLP pass muster?
Chiradeep BasuMallick
Apr 28, 2016 2:10 PM EDT
Nitrogen product manufacturer Terra Nitrogen (TNH) is a promising income-generating investment, sporting a 9%-plus dividend yield.
In fact, the company boasts several positives: a distinct advantage over peers with its zero-debt balance sheet, a variable-rate dividend policy and yields of more than 5% since 2011.
But is the yield sustainable? Let's take a closer look at whether this stock should be a part of a long-term portfolio.
Terra Nitrogen, a master limited partnership, has a prized nitrogen manufacturing facility in Verdigris, Okla., with the ability to produce annually 1.9 million tons (32% nitrogen basis) of urea ammonium nitrate solutions and 1.1 million tons of ammonia, the basic ingredient for most nitrogen fertilizers and several other industrial products.
All ammonia and UAN units at this facility are fully operational following a month-long outage.
The outages are an important element of the mix because Terra Nitrogen's dividend is a variable-rate scenario. It is required to distribute 100% of its available cash each quarter, as defined by the partnership contract, to unit holders and the general partner.
The quantity of obtainable cash is, therefore, contingent on, among other issues, Terra Nitrogen's profitability, working capital variations, capital spending and future money requirements.
And so, if the facility experiences any downtime, it isn't the greatest idea to try and pay out shareholders without significant cash flowing in.
But the stock is nevertheless a smart buy.
Terra Nitrogen has benefited from the shale gas revolution in the United States. Given this landscape and the highly cost-effective feedstock that gives the company a massive cost advantage over global suppliers, Terra Nitrogen is primed for growth.
With no debt, the stock is most certainly a very rare commodity, compared with chemical and minerals peers such as Ciner Resources ($110 million in debt), CVR Partners ($125.12 million), Incitec Pivot($1.99 billion), Israel Chemicals ($3.48 billion) and Westlake Chemical Partners ($384.1 million).
Thanks to its lack of debt, low-cost, raw-material advantage and solid management, Terra Nitrogen has delivered nearly $300 million to $500 million in annual free cash flow every year for the past half a decade.
The MLP's variable dividend rate policy has led to $300 million to $550 million worth of dividends paid every year between 2011 and 2015.
What's more, investors gained between a $10 and $17 per-share dividend during this time as earnings per share grew by close to 44% annually, with the yield remaining above 5% at all times.
The trend should continue for the foreseeable future even as Terra Nitrogen hits the sweet spot and further reinforces itself financially.
With its free cash flow most years covering dividends, there is minimal risk in owning the stock as a high-income yield generator with a cost advantage and a hold on end markets.
The verdict? At an enterprise value to earnings before interest, depreciation, taxes and amortization ratio of 6.50 times, the $2.27 billion market value company is a solid investment, standing tall against competitors.
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This article is commentary by an independent contributor. At the time of publication, the author held no positions in the stocks mentioned.
Natural GasEnergyOilInvestingDividendsAgricultureOpinionMLPs


Come Check Out Our New Venue - The Orchard!
Thank you for your interest in the Wedding Connection. With 32 years of planning and coordinating experience, we have organized events and have worked all over the Bay area, Santa Cruz County, Monterey Bay and Carmel Valley and even Hawaii,Santa Fe and Southern California.
We offer varying levels of service so whether you are in the beginning stages or at the end; we can be of help to you. Initially, you may need our help in finding a site that fits within your budget and meets your esthetic criteria. After your location is secured and your date is set, we step into the next phase of the planning.


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Loh Kean Yew's quarter-final streak ends after Indonesia Masters loss
Limp defeat by Indonesia's Chico Aura Dwi Wardoyo end his participation at the round-of-16 stage in Jakarta
Chia Han Keong
·Editor
Singapore shuttler Loh Kean Yew returns a shot at the 2023 Malaysia Open. (PHOTO: Shi Tang/Getty Images)
SINGAPORE — Loh Kean Yew's streak of reaching every tournament quarter-final this year came to an end on Thursday (26 January), when he crashed out in the round of 16 at the Indonesia Masters.
The world No.8 was out of sorts as he lost to Indonesia world No.23 Chico Aura Dwi Wardoyo 18-21, 15-21 in just 42 minutes at the Istora Gelora Bung Karno in Jakarta.
It was nowhere near his impressive win over China's Li Shifeng in his opening-round match a day earlier, and he fell behind by as many as 9-20 in the second game before making the scoreline a little more respectable.
The former world champion's lacklustre display cost him a chance to end his title drought in the BWF World Tour circuit that has stretched all the way back to when he won his monumental world title in December 2021.
It is all the more galling as Kunlavut Vitidsarn, the Thai world No.6 who had eliminated him in both his first two tournaments of the year - the Malaysia Open and the India Open - had already withdrawn from the Indonesia Masters after winning the India Open last week.
Singapore's other representative at the Indonesia Masters, women's singles shuttler Yeo Jia Min, was narrowly defeated in her opening-round match against Denmark's Line Christophersen on Wednesday, losing 20-22, 21-14, 20-22.

redpajama_set_name: RedPajamaCommonCrawl


Just 4 Steps to Get a Final Price.
Paper Mart is proud to be the only paper bag company to offer custom printed shopping bags printing where you are able to create your own custom printed shopper online without lots of art and proofs and time wasted. These printed shoppers are the same shopping bags that we sell as unprinted stock shopping bags, but here you can have that paper twisted handle bag printed with to your own satisfaction. We hope you like this new custom printed shopping bag experience and will come back with all your printed shopping bag needs.


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Lived And Simply Forgotten TV Series From The '70s
by mwildanf738 on February 5, 2017 in Shopping Guess
The image for the planet Venus is the sign also utilized in biology for the female sex. eight It's a stylized illustration of the goddess Venus's hand- mirror or an abstract symbol for the goddess: a circle with a small equilateral cross beneath. The Venus symbol additionally represented femininity , and in historical alchemy stood for copper Alchemists constructed the symbol from a circle (representing spirit ) above an equilateral cross (representing matter ).
A compelling chance is Cush Jumbo. The work of this younger Anglo-Nigerian actor – from her solo stage show Josephine and I to a number one function in The Good Spouse – has demonstrated an electrifying presence and talent to play each comedy and ache that could fascinatingly extend the Whovian succession. She additionally has useful franchise type, having starred within the 2009 season of the spin-off Torchwood. Jumbo's availability, although, would depend upon whether or not her involvement in ABC's The Good Fight, itself a spin-off from The Good Spouse, goes past a primary season.
Priscilla Barnes was Rebecca Tomkins and Debra Clinger was Amy Waddell, two lovely reporters who worked for The American Report a TV news program. Rebecca was the extra skilled reporter, a city girl who was witty, horny and complicated, while Amy, a small town girl, was fresh out of college and had quite a bit to learn. The pair traveled all over the nation in a van that was outfitted with the most recent know-how that allowed them to report stories even from the most distant of places. David Spielberg was their producer, Francis X. Casey and the information packages anchor/host was Jason Cook played by William Prince.
There are not any steps into the foyer and the Box Office is on the appropriate. There are 21 steps all the way down to the stalls, 7 steps up to the Dress Circle and 40 steps as much as the Upper Circle. All stairwells have handrails. The auditorium is open 30 mins earlier than the performance. There aren't any spaces for wheelchair customers who have to stay in their chairs. The most effective seating possibility is generally the back of the gown circle stage or Field A. Box A is accessible by way of a aspect hearth exit, which has 5 steps. Wheelchairs and scooters will likely be stored in a secure place.
Cicely Tyson was scheduled to play the position of Christie Love, however when her film The Autobiography of Miss Jane Pittman ran over schedule, they known as in Teresa. There was some hypothesis that Cicely had changed her mind concerning the role of Christie Love because of her work within the Pittman movie. She did not need to dilute the impact and importance of that film however doing Christie Love.… Read More
by mwildanf738 on February 1, 2017 in Activewear
If you are married, in what means is the lady you dated different from the woman you married? In case you are single or relationship, in what manner would you like the woman you marry to be completely different from the lady you are relationship? What ideas come to your mind? I guess you may have a few ideas, I also do.
When the boys weren't delivering supplies, they discovered their fun at a nightclub close to their base which was run by Madame Dalacort as played by Penny Santon. It was schedule towards the profitable collection The Jeffersons on CBS and the primary half of Emergency on NBC. There are more 70s reveals that match into this category, however a web page this size merely cannot listing them all. In the event you can consider one which does (and also you in all probability can), be happy to write about it in the guestbook. It took producers $30,000 to outfit one room stuffed with orchids so they weren't seen typically within the sequence. The collection first aired March 23, 1982 and lasted for a whole 6 episodes, ending on April 27. 1982. Nashville ninety nine was a alternative series for The Sonny and Cher Present that was pulled from the CBS schedule. Notable visitor stars: Denny Miller, Christopher Connelly, George Takei, Rene Auberjonoi.
Bonnie Bedelia was Anna Larsen and Scott Thomas was her husband, Christian, they have been Swiss immigrants who had been making an attempt to build a life collectively, with their youngsters, in the wilderness near Solna, Minnesota. Alongside for the journey was Rev.Lundstrom, Christian's brother, as played by Donald Moffat and his wife, Molly as played by Gwen Arner. Lou Frizzel was Murdock the proprietor of the native normal retailer.
She said she deliberate to eat the sufferer's coronary heart after she killed him, and stated she had already written a word to leave on his body explaining that she planned to kill again, in line with police. Former football player, Merlin Olsen, starred as Buddy Landau just an extraordinary man who cherished sports activities and and how ended up teaching his son, Lanny's baseball group. Lanny has no real talent for the sport and is barely concerned in it to please his dad. The show lasted for six weeks, airing 6 of the thirteen episodes before getting hit with the cancellation axe. He was like a pregnant woman, only that the factor inside him was not a baby but a youth. Trivia: Debra did a stint on Saturday morning TELEVISION as Superchic on The Kroftt Supershow from 1977. She was part of the singing group who hosted the present, Kaptain Kool and the Kongs. The woman is presently speaking with police. She was shaken up however was not severely injured.
What happens when science teacher Ted Stein (Jeffrey Kramer) discovers that he has inherited the run down Victorian Brightwater Inn. Well, he decides to promote it, that's until he discovers that the handyman, Frank, is … Read More


Image of performers doing a traditional Dragon dance in ChinaTown.
In America, we have many holidays that are celebrated from all the different cultures that make us up as a country. While many have already celebrated New Year's Day on January 1st, many people are preparing to celebrate Chinese Lunar New Year as well on the 5th of February.
Each Chinese Lunar New Year celebrates an animal that represents the new year. In this case, the pig is the animal that represents 2019. Along with this tradition, people celebrate with their families and friends by giving money in red envelopes to each other and decorate their surroundings with red as well, which symbolizes good luck and wealth.
As for entertainment, pyrotechnics are often used during celebrations as well. Firecrackers are made from strings of rolled red paper containing gunpowder that, when set off, leave shreds of scarlet paper all around. It is believed that the loud noise of the firecrackers scares away evil spirits.
Alongside firecrackers, traditional dances such as the Dragon Dance can be seen in this celebration as well. The Dragon Dance is a dance where performers dress up as a colorful dragon and dance down the streets with music.
Besides celebrating the Lunar New Year with dancing and festivities, people also celebrate by doing other activities such as cleaning and praying in order to welcome the New Year. During Chinese New Year windows are scrubbed, floors are swept, and furniture is dusted in preparation for the New Year, sweeping away the bad luck of the past year.
Another thing that people do during the Chinese Lunar New Year is going to temples and praying to gods. People pray for many things such as good health or a better upcoming year.
People during this time also have a feast with family and dine on a different array of foods, some food even has a specific meaning during the New Year. Dumplings are known to be eaten and are considered as a representation of wealth. Steamed fish is one of the many things that is eaten as well and it increases in prosperity. Nian gao is a glutinous rice cake that represents a higher income or higher position.
May people have a happy Chinese New Year and welcome the year of the pig.


John Darling adopted his sweet senior Tinker from The Humane Society of Ventura County a little over 2.5 years ago when Tinker was seven years old. Sadly she recently passed away, but not before enjoying what were most likely the best years of her life with John and his family.
Tinker's passing came suddenly leaving a terrible void in the family's heart. As part of John's healing process he single-handedly organized a program to benefit senior dogs in shelters. In a matter of a few weeks John raised the funds to cover over 20 senior adoptions and the HSVC was the lucky recipient of his kindness and generosity.
During our Open House on June 11th, John Darling set up a booth to promote the benefits and joy of adopting senior dogs. Since Saturday, 5 senior dogs have found their forever homes and we expect that number to grow as our community becomes more aware of all the rewards of a senior dog!
Senior sweetheart Penelope getting some one on one attention from John.
Arlene MacSween, Assistant Shelter Director, and Jolene Hoffman, Shelter Director, accepting a generous donation from John Darling who raised funds to cover the cost of senior adoptions.
Having adapted a senior 8yr old English Springer Spaniel plus a senior 10 yr old Siamese cat, let me tell you that our lives were over-filled with their love. They brought incomparable joy into our lives.
The Board of Directors along with Jolene and Arlene would like to say a special thank you to all of our staff, volunteers, sponsors and donors that made our open house a huge success. We had a great day thanks to all of you.


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B/R Football Ranks: the show that takes the football world and puts it in the correct order. Ranker-in-chief Sam Tighe and B/R Insider Dean Jones join host Jack Collins in ranking all that's important (and all that's not) in football and life.
B/R Football Ranks Bleacher Report
Ranking Teams We're Worried About This Season
Which clubs are in trouble as the window draws to a close? The Rank Squad take a look at some of the teams across Europe who look right now that they might have some real problems going into the second half of the season, from those desperately in need of signings, to those desperately in need of some leadership, to those desperately in need of a soul.
Hot Takes includes a discussion on Josip Ilicic, the man on fire in Europe right now; where Krzysztof Piątek could end up if he leaves the red and black half of Milan this window; and a debate on if Jurgen Klopp is right to not be in attendance for Liverpool's FA Cup replay against Shrewsbury, including a Reds view from John Gibbons of the Anfield Wrap.
Deano gives out another Melon, we talk the most exciting leagues in Europe and the wildest moments of the window in Roulette, and Sam talks the most difficult everyday tasks in this week's Nonsense Rankings.
Ranking the Best January Transfers
It's the January Transfer Window and to many that means wasted money, desperate bids, and deadline day madness - but it doesn't have to be so! Jack, Dean and Sam take a look at some of the best January transfers to go down in recent years, from those who have flipped a huge profit on sensible recruits, to the players who have gone on to be the bedrock of their teams, leading them to new heights and glories.
We start with a good old game of Who Am I? Before launching into Hot Takes, where we discuss Ole's future at Manchester United and how the club are planning to go about their business in this window and the next, Riqui Puig's introduction in Setien's first Barca game and what that means for both fans and for other La Masia graduates, and some of the chaotic storylines across Europe that you might have missed - namely Theo Hernandez and Leon Bailey.
Roulette contains a discussion of whether it would be better for Liverpool to go Invincible or smash the PL Points Record, who we'd vote for as our PFA Player of the Year and which club are the favourites for the title in the newly returned Bundesliga, before Sam talks football rules that he would add to spice up the season in his Nonsense Rankings.
Barca's New Manager, Mourinho Therapy & Things Rhys James Doesn't Get About Football
Barcelona have themselves a new manager, Quique Setien, and he's a man well liked around these parts. Sam Tighe and Jack Collins give you the lowdown on what to expect from the new man in the Nou Camp hot seat and how he'll impose his style of play on a side who many feel have not reached their potential under Erneste Valverde.
Our favourite comedian, Rhys James, joins us for his hat-trick appearance on the show, and as a Tottenham fan, we give him a therapy session on how he has coped with adjusting to life under Jose Mourinho and the absence of Mauricio Pochettino in his life.
With that done, we move onto Rhys' main ranking-things he doesn't understand about the modern game, ranging from VAR decisions to how new football chants get off the ground in the terraces. As you'd expect, there's a lot of laughs. Dean rings in from Michigan with his Melon of the Week, before Sam rounds things off by ranking his favourite numbers. It's raucous. It's rough and ready. It's Ranks.
To get involved, follow @brfootball on Twitter and Instagram, and use the hashtag #BRFootballRanks!
Top 5 Predictions for 2020
New Year, New Predictions. With a new calendar on the go, Sam Tighe, Dean Jones and Jack Collins take a look to the 12 months ahead to see what the big stories are going to be. We talk Gabriel Martinelli's rise to stardom, whether Leeds can return to the big time in England, Kylian Mbappe's goalscoring feats, Kalidou Koulibaly's future and who will be lifting the Champions League trophy in May.
In a revamped Hot Takes, we discuss James Maddison's future amid speculation Manchester United have made an offer for him, new Juventus man Dejan Kulusevski and how he'll get on in Turin next season, and Mikel Arteta's Arsenal after their win over Bielsa's Leeds in the FA Cup.
Roulette returns with a discussion of Soyuncu and Maguire, the most disrespected player in the Premier League and the worst transfers from the summer transfer window, before a certain Portuguese manager is named Melon of the Week and Sam talks New Year's Resolutions in his Nonsense Rankings.
The Biggest Transfer Stories This Window
Happy New Year Rank Squad! We're back for 2020 with a corker, a Dean Jones special ranking of the five biggest transfer storylines that you'll hear in this January Window. We discuss Christian Eriksen, Gareth Bale, Lucas Paqueta, Edinson Cavani, Wilfried Zaha and Jadon Sancho and where their respective futures may lie.
Hot Takes are renamed Haaland Takes for one week only as we look at Dortmund's new signing, working out whether he's made the right move, if the price tags add up and if his arrival makes Dortmund Bundesliga title favourites.
The Roulette Wheel returns, bringing us questions about if Mikel Arteta will improve Arsenal and who United should be looking at signing this month, before Dean names his melon of the decade and Sam ranks his best and worst Christmas presents.
To get involved, hit us up on Twitter or Instagram, and use #BRFootballRanks
Calcio Sundays: Serie A Mid-Season Update
We're at the half-way point in the Italian season, the famous Winter Break, and it couldn't be any tighter at the top-with Maurizio Sarri's Juventus and Antonio Conte's Inter locked in a two-way tussle for the Scudetto, inseparable on 42 points. We decided it was a good time to look back at the season so far, and to do so, we brought back our Calcio expert, Bleacher Report's very own Alex McGovern, to talk all things Italian football.
We discussed the ups and downs of 2019/20 so far, including the teams and players who had both impressed and disappointed us over the last few months, banged on for a bit about how much we love Cagliari, revelled in Atalanta's Champions League fairytale, and all things considered, had a very nice time. We hope you enjoy our trip to La Repubblica, and as ever, you can get involved by hitting us up on social media or using the hashtag #BRFootballRanks.
© Bleacher Report - 2020 Bleacher Report. All Rights Reserved.
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So yesterday, I went to see one of my good friends who just happens to do Jade therapy facials. If you're not familiar with that particular type of facial therapy, it's a process that detoxes your lymphatic system through your face. While I was in my therapy session, she noticed that my sinus cavities in my face were inflamed which caused inflammation in my sinuses and additional cavities all the time. Not to mention, slight difficulty breathing, unfortunately dark circles and not getting enough oxygen. I didn't even realize I had all of this going on until she noticed a picture I posted of myself on social media.
While she was doing the treatment, which felt amazing by the way, she began to explain that most of, if not, all of our health comes from our head and face. It made total sense because during the treatment, at one point, my whole body became relaxed and calm. I even got goosebumps.
Our health comes from our head. Which can also mean, our way of thinking can affect us as well.
Have you ever noticed that when you're stressed, experience anxiety or even have a headache it can affect every single part of us?
And all of those factors come from our head or mind.
I don't know how many times I've had bouts with anxiety and I either got a stomachache, headache or I just didn't feel right. Or even when I was stressed to the max, I didn't want to do anything because my whole body felt the strain and became weak.
A lot of times we can't function because our mind isn't right. Medically when we have migraines sometimes we can't even open our eyes.
Our head and mind are the most powerful organ in our body. So why not take care of them?
Yes we do therapy on our legs, feet, back, arms, etc. but what about our minds?
Getting that Jade therapy facial yesterday not only felt amazing but I was able to receive the oxygen I needed to treat the ailments I was having.
Our minds need oxygen too because sometimes our thoughts are inflamed. Inflamed with lies we believe about ourselves whether we're thinking them or believe someone who is saying them to us. Stressful situations that are getting the best of us. Anxiety that is literally taking over our bodies.
So in addition to giving our bodies physical therapy, let's give ourselves mental therapy.


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Q: Create shapefile based on raster values in QGIS/GDAL I need to find all pixels with a certain value and create a shapefile from those locations. Any idea how to get started ?

A: You may use a simple workflow (a bit annoying, but it could be easily implemented in the QGIS Modeler if you need to perform it for several values).
1) Generate a grid of points
Generate a grid of regular points centered on the pixel centroids of your raster. You may use the Regular points algorithm from Processing:

*

*The first parameter is the extent of your input raster (you may select it from the ... button on the right);

*The second parameter is the size of your input raster (you may select it from the ... button on the right):

*The third parameter is the half of the size of your input raster (you may select it as above, but remember to divide the size by two).

This will be the result:

2) Add raster values to points
Use the Add raster values to points SAGA tool from Processing (it doesn't need more explanation).
3) Extract the point of interest
Use the Extract by attribute algorithm from Processing and select the value you need (in my example, equals to 1880):

4) Enjoy

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Jesé could get minutes during the FIFA Club World Cup
By Lucas Navarrete@LucasNavarreteM Dec 16, 2014, 10:23am CET
Share All sharing options for: Jesé could get minutes during the FIFA Club World Cup
Denis Doyle/Getty Images
Ancelotti has used him throughout meaningless games for Real Madrid like the return leg of the Copa del Rey tie against UE Cornellà and the home match against Ludogorets when Los Blancos had already clinched the first spot in their Champions League group. But it seems that Jesé Rodríguez is confident about getting some action in the FIFA Club World Cup that starts today.
We're taking it easy, we don't want scares. I think coach will give me a chance in Morocco, we need to make some rotations because a tough stretch of games is upon us. We're all going to have our chance.
It would definitely make sense from Carlo Ancelotti to leave some of his stars on the bench against Cruz Azul -- maybe Gareth Bale or Cristiano Ronaldo -- in order to test Jesé in a competitive match and see how ready he is to make a real impact in a game.
Real Madrid Vs Cruz Azul, FIFA Club World Cup 2014: All the updates
Ancelotti's trust in Iker Casillas paying off
Jesé could get minutes in the FIFA CWC
Match Preview: Real Madrid vs Cruz Azul


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Archie Lee Rundle
December 25, 1916 - September 7, 2017
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Archie Lee Rundle, 100, passed from natural causes September 7th, 2017 in Aguila, Arizona. Archie was born in December 24, 1916 in the small town of Wibaux, Montana to William Oliver Rundle and Margaret Cora Rundle. Archie was the youngest of four children. At the age of 22 he registered for World War II listed in Wells, Nevada. One of his greatest interests in life was building and flying airplanes. In 1938, he was certified as a member of the OX5 Aviation Pioneers for participating in the design, construction and operation of the powered aircraft prior to the 1940s. Another memorable accomplishment was that he knew Amelia Earhart and they flew together in Oakland, CA. After one date, Archie married Estella Blanche Clement on May 16, 1946 in Dillon, Montana. He later accepted a job and moved to California to eventually settle down in Alameda, CA. and raise a family. Archie & Estella later divorced however, he bought a house around the corner and rode his bike daily to visit her. Once Archie was unable to no longer care for himself he moved to Aguila, AZ with his daughter Margaret to help care for him.
Archie is preceded in death by his parents, siblings, Irene, Ralph, Glad, and daughter Barbara Gehring. He is survived by his daughter Mary Wright, Margaret Sawyer, sons Gaylord Rundle and Steve (Debbie) Rundle, grandchildren Barbara Rather, Kenneth (Melissa) Sawyer, Katherine Sawyer, Victoria Wright, Nicholas Rundle, Chelsie Rundle, Melissa Rundle and great-grandchildren Taylor Sawyer, Dylan McGrew, Connor Sawyer, Aaron Sawyer, Breana Sawyer and Victor Wright. He is also survived by extended family with nieces and nephews and their families. His daughter Margaret is acknowledged for all the time, care and love that she provided so that he did not have to be placed in a nursing home years ago. She was present when he passed and felt the comfort that he was at home rather than a facility. A Graveside Service will be held at Green Acres Cemetery in Scottsdale on Tuesday, September 19, 2017 at 11:30 a.m.
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Home / West Linn Paper to receive $1.5 million Diesel Emission Reduction Act grant
West Linn Paper to receive $1.5 million Diesel Emission Reduction Act grant
WEST LINN, OR, May 4, 2015 (Press Release) -The U.S. Environmental Protection Agency announced a $1.5 million Diesel Emission Reduction Act grant to the Oregon Department of Environmental Quality (DEQ) that ultimately will remove tons of air pollution from diesel-fueled activities at and near the Port of Portland. EPA Administrator Gina McCarthy visited Portland to announce the grant at an event at Northwest Container Services with Oregon DEQ Administrator Dick Pedersen. Amongst other projects, the EPA-funded grant will assist local companies in replacing 23 heavy-duty short-haul diesel trucks, traveling more than 1.3 million miles in the Portland metro area each year.
West Linn Paper Company will play a significant role in the project as the future owner of at least twelve of the new trucks. The company will take older, less efficient equipment out of use and replace it with new vehicles with improved energy efficiency and more effective filters to remove emissions. The positive impact to the environment will be significant - West Linn Paper anticipates that the project will result in a 20% improvement in fuel economy, and filters capable of removing as much as 90% of the tailpipe emissions.
"We know this investment will not only improve air quality in Portland, but will also improve people's health and quality of life," said EPA Administrator Gina McCarthy in EPA's news release announcing the grant. "Every dollar we invest in clean diesel generates between $7 and $18 in public health benefits. By promoting clean diesel technologies, we can help save lives locally and play a leadership role on climate change globally."
"West Linn Paper is pleased to be a part of this program," commented Brian Konen, West Linn Paper COO. "The contribution from these projects will be significant and felt on several different levels, from the local health and environmental improvements to the financial benefits of having more fuel efficient equipment." West Linn Paper has begun negotiating with equipment suppliers and hopes to have the first of the new fleet on the road by the end of the year.
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UPM introduces new bio-boiler plant at its plywood mill in Joensuu, Finland
UK's PHS Group announces new partnership with Viridor to divert customer waste from landfill into power generation
AF&PA applauds congressional action for continued recognition of biomass carbon neutrality
Suzano to invest R$933.4 million in three projects in Espírito Santo state including tissue converting


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Companies in Russia Cautiously Optimistic After Trump Election
By Daniel Schearf
2446332_1551229892 video player.
MOSCOW - The election of Donald Trump as the next U.S. president has raised hopes in Russia for improved relations that could help investment and trade, especially if Western sanctions imposed over Ukraine are eased. U.S. and Russian companies at Moscow government-supported industrial park Technopolis are cautiously optimistic about prospects for business.
If Western sanctions are eased, fast-growing Russian startup TEXEL says its 3-D scanning software business will get a boost in the domestic market.
"There would be more funding available for the companies and they would be able to spend more on the new technologies, and they would be able to take a higher risk with new technologies like ours, and therefore there would be an increase in sales," said TEXEL's business development director, Sergei Klimentyev.
Currency's impact
If the Russian currency, the ruble, recovers, then exports could be hurt, Klimentyev acknowledged.
"That would mean that the cost of our product would increase and it will make us less competitive, for example, with Chinese companies," he said.
Most of TEXEL's business in scanning software, used for 3-D digitization of models and printing, is in exports to Europe and Asia.
A small turnout at the Technopolis-hosted second Russian Export and Investment Fair this month did not dampen optimism from businesses located at the industrial park.
U.S. company NeoPhotonics partners with Russian state nanotechnology corporation Rusnano to manufacture opto-electronic modules and subsystems for high-speed communications networks.
Despite sanctions and a shrinking economy, they are confident business in Russia is changing for the better, and are considering expanding their manufacturing at Technopolis.
"Cooperation and business activity on [the] Russian market — it's a huge benefit for international companies," said manufacturing director Dmitry Morgun — and not just for U.S. firms, but for those in European Union nations as well.
Gradual process
Even if Western sanctions against Russia are lifted, say analysts, it would be a gradual process requiring the Kremlin to improve relations with Ukraine.
Meanwhile, Russia's economy remains dependent on exports of raw materials, making it vulnerable to low oil prices.
"That's why serious reforms are needed," said Mikhail Subbotin of the Russian Academy of Science's Institute of International Relations and World Economy. "And that's where the problem lies. The reforms should be of a double character, both political and economic."
A lack of political will and tensions with the West mean continued or even stronger sanctions are more likely.
Putin Delivers Conciliatory Annual 'State of Nation' Speech
Russian President Vladimir Putin's annual "state of the nation" address Thursday was less remarkable for what it contained than for what it lacked -- anger and blame heaped on countries supposedly looking to make an enemy of Russia. Last year's speech at the Kremlin to lawmakers and dignitaries, carried live on Russian TV, focused on demonizing Turkey after its air force shot down a Russian…
OPEC in First Joint Oil Cut with Russia Since 2001; Saudis Take 'Big Hit'
OPEC agreed on Wednesday its first oil output cuts since 2008 after Saudi Arabia accepted "a big hit" on its production and dropped its demand on arch-rival Iran to slash output.Non-OPEC Russia will also join output reductions for the first time in 15 years to help the Organization of the Petroleum Exporting Countries prop up oil prices.Brent crude jumped over 9 percent to more than $50 a barrel as Riyadh reached a compromise with Iran and after fast-growing producer Iraq…
Daniel Schearf


The cold front is forecast to reach the Interstate 95 corridor Monday morning, bringing significantly cooler weather and drier air.
The system is expected to hit the state late Sunday night and the risk of severe weather lasts until around dawn, according to the National Weather Service. The Angelina County Sheriff's Office says an 8-year-old and 3-year-old died after the tree hit the back of the auto in Lufkin, about 115 miles (185 kilometers) northeast of Houston. No injuries were reported there, but officials said several businesses and vehicles were damaged.
Some of the greatest threats include scattered tornadoes, wind gusts reaching 70 miles per hour and hail up to 1.5 inches in diameter.
Robertson County Texas Sheriff Gerald Yezak (YEZ'-ik) told The Associated Press that a suspected tornado hit Franklin on Saturday, overturning mobile homes and damaging other residences.
"Some storms may also produce large hail or torrential downpours resulting in localized flash flooding", the weather service tweeted. A tornado was reported in the area at the time. Damage was reported early Saturday evening in Vicksburg, Mississippi, as severe thunderstorms rumbled through the city.
University spokesman Sid Salter said some debris, possibly carried by the tornado, was found on campus, but no injuries were reported and no buildings were damaged. Franklin is located about 125 miles (200 kilometers) south of Dallas.
The Washington Post went on to report that an elderly man was killed "when a tree fell on his trailer in Hamilton, Mississippi".
Population centers in the zone of greatest risk, 4 out of a possible 5 on the Storm Prediction Center's scale, include Monroe and Alexandria, Louisiana. Some people had to be extricated from damaged dwellings.
Now, a study has highlighted how getting adequate nutrients from food sources is more beneficial than consuming supplements. Some studies have shown that vitamin D supplements may reduce risk of death and disease, while some have not.
House of Representatives has passed a pro-net neutrality bill largely on party lines. Some agreed not to enforce the laws pending the outcome of the Mozilla case.
As one would expect, there is a lot of hype surrounding Murray before the Arizona Cardinals make the first pick in the draft. The Washington Redskins are next on the list for Murray to visit, and they'll host him next week.
Davis acknowledged the fact that he could be playing in New Orleans when next season gets underway . In 56 games this year, the center averaged 25.9 points, 12.0 rebounds and 3.9 assists.
Sam Curran , Jos Buttler , Rory Burns , Tammy Beaumont and Virat Kohli have been named as Wisden's Five Cricketers of the Year . Curran scored 272 runs and took 11 wickets to be named man of the series in England's 4-1 win over India in September.
He also used Trump-like tactics, calling the corruption investigations a "witch hunt" and denouncing journalists covering them. As a result, it will nearly certainly be Netanyahu who is called upon by President Reuven Rivlin to form the next government.
Disney is late to the party and in its absence Netflix has built the world's leading streaming service with a near-global reach. It'll also be available on day one , while The Magic of the Animal Kingdom will come within a year of the launch date.


text:
Science research pays off
MIT economists herald public investment in science as an economic cure
Peter Dizikesarchive page
Cover of book reviewed
The Human Genome Project helped scientists make the first complete map of human genes. But the $3 billion, 15-year project was also a remarkably successful investment: in 2012, human genome sequencing accounted for an estimated 280,000 jobs, $19 billion in personal income, $3.9 billion in federal taxes, and $2.1 billion in state and local taxes, all for just $2 per year per US resident.
"It's an incredible rate of return," says Sloan economist Simon Johnson, PhD '89.
It's hardly the only science-based payoff out there. Every $10 million in public science funding granted to the National Institutes of Health produces 2.7 patents and an additional $30 million in value for the private-sector firms that own them; meanwhile, each dollar in publicly funded military R&D leads to another $2.50 to $5.90 in private-sector investment.
Once upon a time, the US steadily increased public investment in science. In 1938, the country allocated under one-tenth of 1% of its national income to science research. By 1944, the war had boosted that to 0.5%, and by 1964, 2% of GDP was invested in science research. But today, that figure has dropped to 0.7% of GDP, and overall growth rates have lagged compared with the first postwar decades.
In Jump-Starting America (PublicAffairs, 2019), Johnson and his MIT colleague Jonathan Gruber '87 make the case that reviving public support of science and technology would be an excellent way to generate economic growth—and more high-quality jobs throughout the economy.
Recent books from the MIT community
The Architecture of Banking in Renaissance Italy: Constructing the Spaces of Money
By Lauren Jacobi, associate professor of architecture Cambridge University Press, 2019, $99.99
Strange Frequencies: The Extraordinary Story of the Technological Quest for the Supernatural
By Peter Bebergal, MIT technology licensing officer TarcherPerigee, 2018, $28
Ugliness and Judgment: On Architecture in the Public Eye
By Timothy Hyde, associate professor of architecture Princeton University Press, 2019, $35
Active Defense: China's Military Strategy since 1949
By M. Taylor Fravel, professor of political science and member of the Security Studies Program Princeton University Press, 2019, $35
Gyorgy Kepes: Undreaming the Bauhaus
By John R. Blakinger MIT Press, 2019, $55
Alabamians in Blue: Freedmen, Unionists, and the Civil War in the Cotton State
By Christopher M. Rein '92 Louisiana State University Press, 2019, $47.50
Send book news to MIT News MITNews@technologyreview.com or 1 Main Street, 13th Floor Cambridge, MA 02142
The payoffs from tech investment could also be geographically widespread, the authors contend. Lately, growth has occurred in "superstar" areas (think Silicon Valley, San Francisco, New York, Los Angeles, Boston, Seattle, and Washington, DC), leading to increased congestion and sky-high housing prices.
"The prosperity has been concentrated in some places where it's become incredibly expensive to live and work," Johnson says. But the authors believe many US cities have the right ingredients for tech growth: universities, a capable workforce, and residual industrial know-how. Places like Orlando, a center for computer modeling, are quietly there already. The authors list another 102 metro areas they think are ripe for investment and growth, starting with Pittsburgh, Rochester, and three Ohio cities: Cincinnati, Columbus, and Cleveland.
Johnson and Gruber note that private--sector R&D is vital as well, but not sufficient for a large-scale economic boost given the constraints of investor and shareholder demands. Entrepreneurs and venture capitalists readily acknowledge that "their incentives are to make money relatively quickly," driving investor money disproportionately into software, Johnson says.
The authors also recommend an annual innovation dividend for taxpayers—like the one Alaskans receive on energy revenues.
"We wanted to put some ideas out there and really start a debate," Johnson says.
by Peter Dizikes


text:
\section{Introduction}
Single Image Super-Resolution (SISR), which aims to restore a visually pleasing High-Resolution (HR) image from its Low-Resolution (LR) version, is still a challenging task within computer vision research community \cite{NTIRE2017, NTIRE2018}. Since multiple solutions exist for the mapping from LR to HR space, SISR is highly ill-posed. To regularize the solution of SISR, various priors of natural images have been exploited, especially the current leading learning-based methods \cite{SCN2015, SRCNN2016, REDNet2016, VDSR2016, DRCN2016, DRRN2017, MemNet2017, EDSR2017, CARN2018, DDBPN2018, MSRN2018, RDN2018} are proposed to directly learn the non-linear LR-HR mapping.

By modeling the sparse prior in natural images, the Sparse Coding (SC) based methods for SR \cite{ScSR2008, JianChaoYang2010, IFC2014} with strong theoretical support are widely used owing to their excellent performance. Considering the complexity in images, these methods divide the image into overlapping patches and aim to jointly train two over-complete dictionaries for LR/HR patches. There are usually three steps in these methods' framework. First, overlapping patches are extracted from input image. Then to reconstruct the HR patch, the sparse representation of LR patch can be applied to the HR dictionary with \textit{the assumption that LR/HR patch pair shares similar sparse representation}. The final HR image is produced by aggregating the recovered HR patches.

\begin{figure}[t]
	\centering
	\includegraphics[width=\linewidth, keepaspectratio]{psnr_vs_names.pdf}
	\caption{\label{fig:psnr:names} PSNRs of recent state-of-the-arts for scale factor $\times 4$ on Set5 \cite{Set5:2012} and Urban100 \cite{Urban100:2015}. \textcolor{red}{Red} names represent our proposed models.}
	\vspace*{-.5cm}
\end{figure}

Recently, with the development of Deep Learning (DL), many researchers attempt to combine the advantages of DL and SC for image SR.
Dong \etal \cite{SRCNN2016} firstly proposed the seminal CNN model for SR termed as SRCNN, which exploits a shallow convolutional neural network to learn a nonlinear LR-HR mapping in an end-to-end manner and dramatically overshadows conventional methods \cite{JianChaoYang2010, Aplus2014}.
However, sparse prior is ignored to a large extent in SRCNN for it adopts a generic architecture without considering the domain expertise. To address this issue, Wang \etal \cite{SCN2015} implemented a Sparse Coding based Network (SCN) for image SR, by combining the merits of sparse coding and deep learning,  which fully exploits the approximation of sparse coding learned from the LISTA \cite{LISTA2010} based sub-network.

\begin{figure}[t]
	\centering
	\includegraphics[width=\linewidth, keepaspectratio]{figure2.png}
	\caption{\label{fig:figure2} Visual comparisons between our model CRNet-B and other state-of-the-art methods on ``img004'' from Urban100 \cite{Urban100:2015} for scale factor $\times 4$}
	\vspace*{-.5cm}
\end{figure}

It's worth to note that most of SC based methods utilize the sparse prior locally \cite{papyan2017working}, \ie, coping with overlapping image patches.  Thus the consistency of pixels in overlapped patches has been ignored \cite{CSC-SR2015, papyan2017working}.
To address this issue, CSC is proposed to serve sparse prior as a global prior \cite{Zeiler:2010kx,papyan2017working, papyan2018theoretical} and it furnishes a way to fill the local-global gap by working directly on the entire image by convolution operation. Consequently, CSC has attained much attention from researchers \cite{Zeiler:2010kx, Bristow:2013fw,Heide:2015hq,CSC-SR2015,LCSC2018,GarciaCardona:2018id}. However, very few studies focus on  the validation of CSC for image SR \cite{CSC-SR2015}, resulting in no work been reported that CSC based image SR can achieve state-of-the-art performance. Can CSC based image SR  show highly competitive results with recent state-of-the-art methods \cite{SRCNN2016, REDNet2016, VDSR2016, DRCN2016, DRRN2017, MemNet2017, SRDenseNet2017, EDSR2017, MSRN2018, RDN2018}? To answer this question, the following issues need to be considered:

\noindent\textbf{Framework Issue}. Compared with SC based image SR methods \cite{ScSR2008, JianChaoYang2010}, the lack of a unified framework has hindered progress towards improving the performance of CSC based image SR.

\noindent\textbf{Optimization Issue}. The previous CSC based image SR method \cite{CSC-SR2015} contains several steps and they are optimized independently. Hundreds of iterations are required to solve the CSC problem in each step.

\noindent\textbf{Memory Issue}. To solve the CSC problem, ADMM \cite{ADMM2011} is commonly employed \cite{Bristow:2013fw, Wohlberg:2014iy, Heide:2015hq, wohlberg2016boundary, GarciaCardona:2018id}, where the whole training set needs to be loaded in memory. As a consequence, it is not applicable to improve the performance  by enlarging the training set.

\noindent\textbf{Multi-Scale Issue}. Training a single model for multiple scales is difficult for the previous CSC based image SR method~\cite{CSC-SR2015}.

Based on these considerations, in this paper, we attempt to answer the aforementioned question. Specifically, we exploit the advantages of CSC and the powerful learning ability of deep learning to address image SR problem. Moreover, massive theoretical foundations for CSC \cite{papyan2017working, papyan2018theoretical, garcia2018convolutional} make our proposed architectures interpretable and also enable to theoretically analyze our SR performance. In the rest of this paper, we first introduce CISTA, which can be naturally implemented using CNN architectures for solving the  CSC problem. Then we develop a framework for CSC based image SR, which can address the \textbf{Framework Issue}. Subsequently, CRNet-A (CSC and Residual learning based Network) and CRNet-B inspired by this framework are proposed for image SR. They are classified as pre- and post-upsampling models \cite{Wang2019DeepLF} respectively, as the former takes Interpolated LR (ILR) images as input while the latter processes LR images directly. By adopting CNN architectures, \textbf{Optimization Issue} and \textbf{Memory Issue} would be mitigated to some extent.  For \textbf{Multi-Scale Issue}, with the help of the recently introduced scale augmentation \cite{VDSR2016, DRCN2016} or scale-specific multi-path learning \cite{EDSR2017, Wang2019DeepLF} strategies, both of our models are capable of handling multi-scale SR problem effectively, and achieve favorable performance against state-of-the-arts, as shown in Fig.~\ref{fig:psnr:names}.

The main contributions of this paper include:
\begin{itemize}[noitemsep,topsep=0pt]
	\item We introduce CISTA, which can be naturally implemented using CNN architectures for solving the CSC problem.
	\item A novel framework for CSC based image SR is developed. Two models, CRNet-A and CRNet-B, inspired by this framework are proposed for image SR.
	\item Experimental results demonstrate our proposed models outperform the previous CSC based image SR method \cite{CSC-SR2015} by a large margin and show superior performance against recent state-of-the-arts, \eg, EDSR/MDSR \cite{EDSR2017}, RDN \cite{RDN2018}, as depicted in Fig.~\ref{fig:figure2}.
	\item The differences between our proposed models and several SR models with recursive learning strategy, \eg, DRRN \cite{DRRN2017}, SCN \cite{SCN2015}, DRCN \cite{DRCN2016}, are discussed.
\end{itemize}

\section{Related Work}

\subsection{Sparse Coding for Image Super-Resolution}\label{sub:sparse:coding}

Sparse coding has been widely used in a variety of applications \cite{Zhang:2015kb}. As for SISR, Yang \etal \cite{ScSR2008} proposed a representative Sparse coding based Super-Resolution (ScSR) method. In the  training stage, ScSR attempts to learn the LR/HR overcomplete dictionary pair $\bm{D}_l$/$\bm{D}_h$ jointly by given a group of LR/HR training patch pairs $\bm{x}_l$/$\bm{x}_h$. In the test stage, the HR patch $\bm{x}_h$ is reconstructed from its LR version $\bm{x}_l$ by assuming they share the same sparse code. Specifically, the optimal sparse code is obainted by minimizing the following sparsity-inducing $\ell_1$-norm regularized objective function
\begin{equation}\label{equ:ScSR}
\bm{z}^\ast=\arg \min_{\bm{z}}\left\Vert\bm{x}_l-\bm{D}_l \bm{z}\right\Vert_{2}^{2}+\lambda\Vert\bm{z}\Vert_{1},
\end{equation}
and then the HR patch is obtained by $\bm{x}_h=\bm{D}_h \bm{z}^\ast$. Finally, the HR image can be estimated by aggregating all the reconstructed HR patches. Inspired by ScSR, many SC based SR methods have been proposed by using various constraints on sparse code and dictionary \cite{yang2012coupled, wang2012semi}.

\begin{figure}
	\centering
	\includegraphics[width=\linewidth, keepaspectratio] {CRNet-flowchart.pdf}
	\vspace*{-.6cm}
	\caption{\label{fig:flowchar} Our framework for CSC based image SR. }
	\vspace*{-.4cm}
\end{figure}

\subsection{Convolutional Sparse Coding for Image Super-Resolution}

Traditional SC based SR algorithms usually process images in a patch based manner to reduce the burden of modeling and computation, resulting in the inconsistency problem \cite{papyan2017working}. As a special case of SC, CSC is inherently suitable for this issue \cite{Zeiler:2010kx}. CSC is proposed to avoid the inconsistency problem by representing the whole image directly. Specifically, an image $\bm{y}\in\mathbb{R}^{n_r\times n_c}$ can be represented as the summation of $m$ feature maps $\bm{z}_i\in\mathbb{R}^{n_r\times n_c}$ convolved with the corresponding filters $\bm{f}_i\in\mathbb{R}^{s\times s}$: $\bm{y} = \sum _ { i = 1 } ^ { m } \bm { f } _ { i } \otimes \bm { z } _ { i }$, where $ \otimes $ is the convolution operation.

Gu \etal \cite{CSC-SR2015} proposed the CSC-SR method and revealed the potential of CSC for image SR. In \cite{CSC-SR2015}, CSC-SR requires to solve the following CSC based optimization problem in both the training and testing phase:

\begin{equation}\label{equ:csc}
\min _ {\bm{f},\bm{z} } \frac {1}{2} \left\| \bm { y } - \sum _ { i = 1 }  ^ { m } \bm {f} _ {i } \otimes \bm { z } _ { i } \right\| _ { 2 } ^ { 2 } + \lambda \sum _ { i = 1 } ^ { m } \left\| \bm { z } _ { i } \right\| _ { 1 }.
\end{equation}
\cite{CSC-SR2015} solves this problem by alternatively optimizing the $\bm{z}$ and $\bm{f}$ subproblems \cite{Wohlberg:2014iy}. The $\bm{z}$ subproblem is a standard CSC problem. Hundreds of iterations are required to solve the CSC problem and the  aforementioned \textbf{Optimization Issue} and \textbf{Memory Issue} cannot be completely avoided.
Inspired by the success of deep learning based sparse coding \cite{LISTA2010}, we exploit the natural connection between CSC and CNN to solve the CSC problem efficiently.

\section{CISTA for solving CSC problem}

\begin{figure}

	\centering
	\renewcommand{\thesubfigure}{(a)}
	\vspace*{-.2cm}
	\subfloat[$K = 1$]{\label{sub:demo:k1}\includegraphics[%
		width=0.466\linewidth, keepaspectratio]{residual_vs_noresidual_k1.pdf}}\hspace*{.03\linewidth}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[$K = 5$]{\label{sub:ndemo:k5}\includegraphics[%
		width=0.46\linewidth, keepaspectratio]{residual_vs_noresidual_k5.pdf}}
	\vspace*{-.2cm}
	\caption{\label{fig:figure3} Performance curve for residual/non-residual networks with different recursions. The tests are conducted on Set5 for scale factor $\times 3$.}
	\vspace*{-.3cm}
\end{figure}

\begin{figure*}
	\centering
	\includegraphics[width=\textwidth, keepaspectratio] {CRNet-A.pdf}
	\caption{\label{fig:model} The architecture of the pre-upsampling model CRNet-A. The proposed CISTA block with $K$ recursions is surrounded by the dashed box and its unfolded version is shown in the bottom. $\bm{S}$ is shared across every recursion.}

	\vspace*{-.3cm}
\end{figure*}

CSC can be considered as a special case of conventional SC, due to the fact that convolution operation can be replaced with matrix multiplication, so the objective function of CSC can be formulated as:
\begin{equation}\label{equ:rewritten}
\min_{\bm{z}}\left\Vert\bm{y} - \sum_{i=1}^{m}\bm{F}_i\bm{z}_i\right\Vert^2_2+ \lambda\sum_{i=1}^{m}\left\Vert\bm{z}_i\right\Vert_1.
\end{equation}
$\bm{y}, \bm{z}_i$ are in vectorized form and $\bm{F}_i$ is a sparse convolution matrix with the following attributes:
\begin{equation}\label{equ:relation}
\begin{aligned}
\bm{F}_i\bm{z}_i &\equiv \bm{f}_i \otimes \bm{z}_i \\
\bm{F}_i^T\bm{z}_i &\equiv \texttt{flipud}(\texttt{fliplr}(\bm{f}_i)) \otimes \bm{z}_i \\
&\equiv \texttt{flip}(\bm{f}_i) \otimes \bm{z}_i
\end{aligned}
\end{equation}
where $\texttt{fliplr}(\cdot)$ and $\texttt{flipud}(\cdot)$ are following the notations of Zeiler \etal \cite{Zeiler:2010kx}, representing that array is flipped in left/right or up/down direction.

Iterative Soft Thresholding Algorithm (ISTA) \cite{ISTA2004} can be utilized to solve \eqref{equ:rewritten}, at the $k^{th}$ iteration:
\begin{equation}\label{equ:temp}
\bm{z}_{k+1} = h_{\theta}\left(
\bm{z}_k + \frac{1}{L}\bm{F}^T\left(
\bm{y} - \bm{F}\bm{z}_k\right)
\right)
\end{equation}
where $L$ is the Lipschitz constant, $\bm{F} = [\bm{F}_1, \bm{F}_2, \ldots, \bm{F}_m]$ and $\bm{F}\bm{z} = \sum_{i=1}^{m}\bm{F}_i\bm{z}_i$. Using the relation in \eqref{equ:relation} to replace the matrix multiplication with convolution operator, we can reformulate \eqref{equ:temp} as:
\begin{equation}\label{equ:almost}
\begin{aligned}
\bm{z}_{k+1}
&= h_{\theta}\left(
\bm{I}\bm{z}_k + \frac{1}{L}\texttt{flip}(\bm{f})\otimes\left(
\bm{y} - \bm{f}\otimes\bm{z}_k\right)
\right)
\end{aligned}
\end{equation}
where $\bm{I}$ is the identity matrix, $\bm{f} = [\bm{f}_1, \bm{f}_2, \ldots, \bm{f}_m]$ and $\texttt{flip}(\bm{f}) = [\texttt{flip}(\bm{f}_1), \texttt{flip}(\bm{f}_2), \ldots, \texttt{flip}(\bm{f}_m)]$. Note that identity matrix $\bm{I}$ is also a sparse convolution matrix, so according to \eqref{equ:relation},  there existing a filter $\bm{n}$ satisfies:
\begin{equation}
\bm{I}\bm{z} = \bm{n}\otimes\bm{z},
\end{equation}
so \eqref{equ:almost} becomes:
\begin{equation}\label{equ:final}
\bm{z}_{k+1} = h_{\theta}\left(
\bm{W}\otimes\bm{y} + \bm{S} \otimes \bm{z}_k
\right),
\end{equation}
where $\bm{W} = \frac{1}{L}\texttt{flip}(\bm{f})$ and $\bm{S} = \bm{n} - \frac{1}{L}\texttt{flip}(\bm{f})\otimes\bm{f}$. Even though \eqref{equ:rewritten} is for a single image with one channel, the extension to multiple channels (for both image and filters) and multiple images is mathematically straightforward. Thus for $\bm{y}\in\mathbb{R}^{b\times c\times n_r\times n_c}$  representing $b$ images of size $n_r\times n_c$ with $c$ channels, \eqref{equ:final} is still true with $\bm{W}\in\mathbb{R}^{m\times c\times s\times s}$ and $\bm{S}\in\mathbb{R}^{m\times m\times s\times s}$.

As for $h_{\bm{\theta}}$, \cite{papyan2017convolutional} reveals two important facts: (1) the expressiveness of the sparsity inspired model is not affected even by restricting the coefficients to be nonnegative; (2) the $ReLU$ \cite{relu2010} activation function and the soft nonnegative thresholding operator are equal, that is:
\begin{equation}\label{equ:relu}
h_{\bm{\theta}}^{+}(\bm{\alpha}) = \max(\bm{\alpha} - \bm{\theta}, 0) = ReLU(\bm{\alpha} - \bm{\theta}).
\end{equation}
We set $\bm{\theta} = \bm{0}$ for simplicity. So the final form of \eqref{equ:final} is:
\begin{equation}\label{equ:cista}
\bm{z}_{k+1} = ReLU\left(
\bm{W}\otimes\bm{y} + \bm{S} \otimes \bm{z}_k
\right).
\end{equation}
One can see that \eqref{equ:cista} is a convolutional form of \eqref{equ:temp}, so we name it as CISTA. It provides the solution of \eqref{equ:rewritten} with theoretical guarantees \cite{ISTA2004}.  Furthermore, this convolutional form can be implemented employing CNN architectures. So $\bm{W}$ and $\bm{S}$ in \eqref{equ:cista} would be trainable.

\section{Proposed Method}

\begin{figure*}
	\centering
	\includegraphics[width=\textwidth, keepaspectratio] {CRNet-B.pdf}
	\caption{\label{fig:model2} The architecture of the post-upsampling model CRNet-B.}
	\vspace*{-.3cm}
\end{figure*}

In this section, our framework for CSC based image SR is first introduced. And then we implement it using CNN techniques. Since most of image SR methods can be attributed to two frameworks with different upsampling strategies, \ie, pre-upsampling and post-upsampling, we propose two models, CRNet-A for pre-upsampling and CRNet-B for post-upsampling.

\subsection{The framework for CSC based Image SR}

Analogy to sparse coding based SR,  we develop a framework for CSC based image SR. As shown in Fig.~\ref{fig:flowchar}, LR feature maps are extracted from the input LR image using the learned LR filters. Then convolutional sparse codes $\bm{z}$ of LR feature maps are obtained using CISTA with LR dictionary $\bm{W}_l$ and shared parameter $\bm{S}$, as indicated in \eqref{equ:cista}.
\textbf{Under the assumption that HR feature maps share the same convolutional sparse codes with LR feature maps}, HR feature maps can be recovered by $\bm{W}_h \otimes \bm{z}$. Finally, the HR image is reconstructed by utilizing the learned HR filters.

In this work, we implement this framework  using CNN techniques. However, when combining CSC with CNN, the characteristics of CNN itself must be considered. With more recursions used in CISTA, the network becomes deeper and tends to be bothered by the gradient vanishing/exploding problems. Residual learning \cite{VDSR2016, DRCN2016, DRRN2017} is such a useful tool that not only mitigates these difficulties, but helps network converge faster. In Fig.~\ref{fig:figure3}, residual/non-residual networks with different recursions are compared experimentally and the residual network converges much faster and achieves better performance. Based on these observations, both of our proposed models adopt residual learning.

\subsection{CRNet-A Model for Pre-upsampling}

As shown in Fig.~\ref{fig:model}, CRNet-A takes the ILR image $\bm{I}_y$ with $c$ channels as input, and predicts the output HR image as $\bm{I}_x$. Two convolution layers, $\bm{F}_0\in\mathbb{R}^{n_0\times c\times s\times s}$ consisting of $n_0$ filters of spatial size $c\times s\times s$ and $\bm{F}_1\in\mathbb{R}^{n_0\times n_0\times s\times s}$ containing $n_0$ filters of spatial size $n_0\times s\times s$ are utilized for hierarchical features extraction from ILR image:
\begin{equation}\label{equ:extract}
\bm{y} = ReLU\Big(\bm{F}_1\otimes ReLU(\bm{F}_0\otimes\bm{I}_y)\Big).
\end{equation}

The ILR features are then fed into a CISTA block to learn the convolutional sparse codes. As stated in \eqref{equ:cista}, two convolutional layers $\bm{W}_l\in\mathbb{R}^{m_0\times n_0\times s\times s}$ and $\bm{S}\in\mathbb{R}^{m_0\times m_0\times s\times s}$ are needed:
\begin{equation}
\bm{z}_{k + 1} = ReLU(\bm{W}_l\otimes\bm{y} + \bm{S}\otimes\bm{z}_k),
\end{equation}
where $\bm{z}_0$ is initialized to $ReLU(\bm{W}_l\otimes\bm{y})$. The  convolutional sparse codes $\bm{z}$ are learned after $K$ recursions with $\bm{S}$ shared across every recursion. When the  convolutional sparse codes $\bm{z}$ are obtained, it is then passed through a convolution layer $\bm{W}_h\in\mathbb{R}^{n_0\times m_0\times s\times s}$ to recover the HR feature maps. The last convolution layer $\bm{H}\in\mathbb{R}^{c\times n_0\times s\times s}$ is used as HR filters:
\begin{equation}\label{equ:plus}
\bm{R} = \bm{H}\otimes ReLU(\bm{W}_h \otimes \bm{z} ).
\end{equation}
Note that we pad zeros before all convolution operations to keep all the feature maps to have the same size, which is a common strategy used in a variety of methods \cite{VDSR2016, DRCN2016, DRRN2017}. So the residual image $\bm{R}$ has the same size as the input ILR image $\bm{I}_y$, and the final HR image $\bm{I}_x$ would be reconstructed by:
\begin{equation}\label{equ:add}
\bm{I}_x = \bm{I}_y + \bm{R}.
\end{equation}
Given $N$ ILR-HR image patch pairs  $\{ \bm{I}_y^{(i)}, \tilde{\bm{I}}_x^{(i)} \}_{i=1}^{N}$ as a training set, our goal is to minimize the following objective function:
\begin{equation}\label{equ:loss}
\mathcal{L}(\bm{\Theta}) = \frac{1}{2N}\sum_{i = 1}^{N}\left\Vert \bm{I}_x^{(i)} - \tilde{\bm{I}}_x^{(i)}\right\Vert^2_2
\end{equation}
where $\bm{\Theta}$ denotes the learnable parameters. The network is optimized using the mini-batch Stochastic Gradient Descent (SGD) with backpropagation~\cite{lecun1998gradient}.

\subsection{CRNet-B Model for Post-upsampling}

We extend CRNet-A to its post-upsampling version to further mine its potential. Notice that most post-upsampling models \cite{SRResNet2017, SRDenseNet2017, MSRN2018, RDN2018} need to train and store many scale-dependent models for various scales without fully using the inter-scale correlation, so we adopt the scale-specific multi-path learning strategy \cite{Wang2019DeepLF} presented in MDSR \cite{EDSR2017} with minor modifications to address this issue. The complete model is shown in Fig.~\ref{fig:model2}. The main branch is our CRNet-A module. The pre-processing modules are used for reducing the variance from input images of different scales and only one residual unit with $3\times 3$ kernels is used in each of the pre-processing module. At the end of CRNet-B, upsampling modules are used for multi-scale reconstruction.

{
	\begin{table}
		\datasize

		\centering
		\begin{tabular}{|C{.88cm}|C{.6cm}|C{.9cm}|C{.9cm}|C{1.cm}|C{1.3cm}|}
			\hline
			\makecell{Dataset\\[-.13cm]} & \makecell{Scale\\[-.13cm]} & \makecell{Bicubic\\[-.13cm]} &
			\makecell{CSC-SR \\\cite{CSC-SR2015}} &  \makecell{CRNet-B \\ (ours)} & \makecell{Our \\ Improvement}\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Set5} & $\times 2$ & $33.66$ & $36.62$ & $\textcolor{red}{38.13}$ & $\textbf{1.51}$\\
			& $\times 3$ & $30.39$ & $32.65$ & $\textcolor{red}{34.75}$ & $\textbf{2.10}$ \\
			& $\times 4$ & $28.42$ & $30.36$ & $\textcolor{red}{32.57}$ & $\textbf{2.21}$ \\\hline
		\end{tabular}
		\vspace*{-.1cm}
		\caption{\label{tab:compare:cscsr}Average PSNRs of CSC-SR \cite{CSC-SR2015} and CRNet-B for scale factor $\times2$, $\times3$ and $\times4$ on Set5. The performance gain of our model over CSC-SR is shown in the last column.}
		\vspace*{-.4cm}
	\end{table}
}

\section{Experimental Results}

{
	\begin{table*}
		\datasize

		\centering
		\begin{tabular}{|C{1.cm}|C{.5cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|}
			\hline
			\makecell{Dataset\\[-.13cm]} & \makecell{Scale\\[-.13cm]} & \makecell{Bicubic\\[-.13cm]} & \makecell{SRCNN \\ \cite{SRCNN2016}} & \makecell{RED30 \\\cite{REDNet2016}} & \makecell{VDSR \\\cite{VDSR2016}} & \makecell{DRCN \\ \cite{DRCN2016}} & \makecell{DRRN \\ \cite{DRRN2017}} & \makecell{MemNet \\\cite{MemNet2017}} & \makecell{CRNet-A \\ (ours)}\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Set5} & $\times 2$ & $33.66$/$0.9299$ & $36.66$/$0.9542$ & $37.66$/$\textcolor{blue}{0.9599}$ & $37.53$/$0.9587$ & $37.63$/$0.9588$ & $37.74$/$0.9591$ & $\textcolor{blue}{37.78}$/$0.9597$ & $\textcolor{red}{37.79}$/$\textcolor{red}{0.9600}$\\
			& $\times 3$ & $30.39$/$0.8682$ & $32.75$/$0.9090$ & $33.82$/$0.9230$ & $33.66$/$0.9213$ & $33.82$/$0.9226$ & $34.03$/$0.9244$ & $\textcolor{blue}{34.09}$/$\textcolor{blue}{0.9248}$ & $\textcolor{red}{34.11}$/$\textcolor{red}{0.9254}$\\
			& $\times 4$ & $28.42$/$0.8104$ & $30.48$/$0.8628$ & $31.51$/$0.8869$ & $31.35$/$0.8838$ & $31.53$/$0.8854$ & $31.68$/$0.8888$ & $\textcolor{blue}{31.74}$/$\textcolor{blue}{0.8893}$ & $\textcolor{red}{31.82}$/$\textcolor{red}{0.8907}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Set14} & $\times 2$ & $30.24$/$0.8688$ & $32.45$/$0.9067$ & $32.94$/$\textcolor{blue}{0.9144}$ & $33.03$/$0.9124$ & $33.04$/$0.9118$ & $33.23$/$0.9136$ & $\textcolor{blue}{33.28}$/$0.9142$ & $\textcolor{red}{33.33}$/$\textcolor{red}{0.9152}$\\
			& $\times 3$ & $27.55$/$0.7742$ & $29.30$/$0.8215$ & $29.61$/$0.8341$ & $29.77$/$0.8314$ & $29.76$/$0.8311$ & $29.96$/$0.8349$ & $\textcolor{red}{30.00}$/$\textcolor{blue}{0.8350}$ & $\textcolor{blue}{29.99}$/$\textcolor{red}{0.8359}$\\
			& $\times 4$ & $26.00$/$0.7027$ & $27.50$/$0.7513$ & $27.86$/$0.7718$ & $28.01$/$0.7674$ & $28.02$/$0.7670$ & $28.21$/$0.7720$ & $\textcolor{blue}{28.26}$/$\textcolor{blue}{0.7723}$ & $\textcolor{red}{28.29}$/$\textcolor{red}{0.7741}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{B100} & $\times 2$ & $29.56$/$0.8431$ & $31.36$/$0.8879$ & $31.99$/$0.8974$ & $31.90$/$0.8960$ & $31.85$/$0.8942$ & $32.05$/$0.8973$ & $\textcolor{blue}{32.08}$/$\textcolor{blue}{0.8978}$ & $\textcolor{red}{32.09}$/$\textcolor{red}{0.8985}$\\
			& $\times 3$ & $27.21$/$0.7385$ & $28.41$/$0.7863$ & $28.93$/$0.7994$ & $28.82$/$0.7976$ & $28.80$/$0.7963$ & $28.95$/$\textcolor{blue}{0.8004}$ & $\textcolor{blue}{28.96}$/$0.8001$ & $\textcolor{red}{28.99}$/$\textcolor{red}{0.8021}$\\
			& $\times 4$ & $25.96$/$0.6675$ & $26.90$/$0.7101$ & $27.40$/$\textcolor{blue}{0.7290}$ & $27.29$/$0.7251$ & $27.23$/$0.7233$ & $27.38$/$0.7284$ & $\textcolor{blue}{27.40}$/$0.7281$ & $\textcolor{red}{27.44}$/$\textcolor{red}{0.7302}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Urban100} & $\times 2$ & $26.88$/$0.8403$ & $29.50$/$0.8946$ & $30.85$/$0.9148$ & $30.76$/$0.9140$ & $30.75$/$0.9133$ & $31.23$/$0.9188$ & $\textcolor{blue}{31.31}$/$\textcolor{blue}{0.9195}$ & $\textcolor{red}{31.36}$/$\textcolor{red}{0.9207}$\\
			& $\times 3$ & $24.46$/$0.7349$ & $26.24$/$0.7989$ & $27.25$/$0.8283$ & $27.14$/$0.8279$ & $27.15$/$0.8276$ & $27.53$/$\textcolor{blue}{0.8378}$ & $\textcolor{blue}{27.56}$/$0.8376$ & $\textcolor{red}{27.64}$/$\textcolor{red}{0.8403}$\\
			& $\times 4$ & $23.14$/$0.6577$ & $24.52$/$0.7221$ & $25.28$/$0.7555$ & $25.18$/$0.7524$ & $25.14$/$0.7510$ & $25.44$/$\textcolor{blue}{0.7638}$ & $\textcolor{blue}{25.50}$/$0.7630$ & $\textcolor{red}{25.59}$/$\textcolor{red}{0.7680}$
			\\\hline
		\end{tabular}
		\vspace*{-.1cm}
		\caption{\label{tab:psnr1}Average PSNR/SSIMs of \textbf{Pre-upsampling} models for scale factor $\times2$, $\times3$ and $\times4$ on datasets Set5, Set14, BSD100 and Urban100. \textcolor{red}{Red} color indicates the best performance and \textcolor{blue}{blue} color indicates the second best performance.}
	\end{table*}
}

{

	\begin{table*}
		\datasize

		\centering
		\vspace*{-.3cm}
		\begin{tabular}{|C{1.cm}|C{.5cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|C{1.4cm}|}
			\hline
			\makecell{Dataset\\[-.13cm]} & \makecell{Scale\\[-.13cm]} & \makecell{SRDenseNet \\ \cite{SRDenseNet2017}} & \makecell{MSRN \\ \cite{MSRN2018}} & \makecell{D-DBPN \\ \cite{DDBPN2018}}& \makecell{EDSR \\ \cite{EDSR2017}} & \makecell{MDSR \\ \cite{EDSR2017}} & \makecell{RDN \\ \cite{RDN2018}}& \makecell{CRNet-B \\ (ours)} & \makecell{CRNet-B+ \\ (ours)}  \\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Set5} & $\times 2$ & -/- & $38.08$/$0.9605$ & $38.09$/$0.9600$ & $38.11$/$0.9601$ & $38.11$/$0.9602$ & $\textcolor{blue}{38.24}$/$\textcolor{blue}{0.9614}$ & $38.13$/$0.9610$ & $\textcolor{red}{38.25}$/$\textcolor{red}{0.9614}$\\
			& $\times 3$ & -/- & $34.38$/$0.9262$ & -/- & $34.65$/$0.9282$ & $34.66$/$0.9280$ & $34.71$/$0.9296$ & $\textcolor{blue}{34.75}$/$\textcolor{blue}{0.9296}$ & $\textcolor{red}{34.83}$/$\textcolor{red}{0.9303}$\\
			& $\times 4$ & $32.02$/$0.8934$ & $32.07$/$0.8903$ & $32.47$/$0.8980$ & $32.46$/$0.8968$ & $32.50$/$0.8973$ & $32.47$/$0.8990$ & $\textcolor{blue}{32.57}$/$\textcolor{blue}{0.8991}$ & $\textcolor{red}{32.71}$/$\textcolor{red}{0.9008}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Set14} & $\times 2$ & -/- & $33.74$/$0.9170$ & $33.85$/$0.9190$ & $33.92$/$0.9195$ & $33.85$/$0.9198$ & $34.01$/$0.9212$ & $\textcolor{blue}{34.09}$/$\textcolor{blue}{0.9219}$ & $\textcolor{red}{34.15}$/$\textcolor{red}{0.9227}$\\
			& $\times 3$ & -/- & $30.34$/$0.8395$ & -/- & $30.52$/$0.8462$ & $30.44$/$0.8452$ & $30.57$/$\textcolor{blue}{0.8468}$ & $\textcolor{blue}{30.58}$/$0.8465$ & $\textcolor{red}{30.67}$/$\textcolor{red}{0.8481}$\\
			& $\times 4$ & $28.50$/$0.7782$ & $28.60$/$0.7751$ & $\textcolor{blue}{28.82}$/$0.7860$ & $28.80$/$\textcolor{blue}{0.7876}$ & $28.72$/$0.7857$ & $28.81$/$0.7871$ & $28.79$/$0.7867$ & $\textcolor{red}{28.93}$/$\textcolor{red}{0.7894}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{B100} & $\times 2$ & -/- & $32.23$/$0.9013$ & $32.27$/$0.9000$ & $32.32$/$0.9013$ & $32.29$/$0.9007$ & $\textcolor{blue}{32.34}$/$\textcolor{blue}{0.9017}$ & $32.32$/$0.9014$ & $\textcolor{red}{32.38}$/$\textcolor{red}{0.9020}$\\
			& $\times 3$ & -/- & $29.08$/$0.8041$ & -/- & $29.25$/$0.8093$ & $29.25$/$0.8091$ & $\textcolor{blue}{29.26}$/$\textcolor{blue}{0.8093}$ & $\textcolor{blue}{29.26}$/$0.8091$ & $\textcolor{red}{29.32}$/$\textcolor{red}{0.8103}$\\
			& $\times 4$ & $27.53$/$0.7337$ & $27.52$/$0.7273$ & $27.72$/$0.7400$ & $27.71$/$\textcolor{blue}{0.7420}$ & $27.72$/$0.7418$ & $27.72$/$0.7419$ & $\textcolor{blue}{27.73}$/$0.7414$ & $\textcolor{red}{27.80}$/$\textcolor{red}{0.7430}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Urban100} & $\times 2$ & -/- & $32.22$/$0.9326$ & $32.55$/$0.9324$ & $\textcolor{blue}{32.93}$/$0.9351$ & $32.84$/$0.9347$ & $32.89$/$0.9353$ & $\textcolor{blue}{32.93}$/$\textcolor{blue}{0.9355}$ & $\textcolor{red}{33.14}$/$\textcolor{red}{0.9370}$\\
			& $\times 3$ & -/- & $28.08$/$0.8554$ & -/- & $28.80$/$0.8653$ & $28.79$/$0.8655$ & $28.80$/$0.8653$ & $\textcolor{blue}{28.87}$/$\textcolor{blue}{0.8667}$ & $\textcolor{red}{29.09}$/$\textcolor{red}{0.8697}$\\
			& $\times 4$ & $26.05$/$0.7819$ & $26.04$/$0.7896$ & $26.38$/$0.7946$ & $26.64$/$0.8033$ & $26.67$/$0.8041$ & $26.61$/$0.8028$ & $\textcolor{blue}{26.69}$/$\textcolor{blue}{0.8045}$ & $\textcolor{red}{26.90}$/$\textcolor{red}{0.8089}$\\\hline\hline\vspace*{.03cm}
			\multirow{3}{*}{Manga109} & $\times 2$ & -/- & $38.82$/$\textcolor{red}{0.9868}$ & $38.89$/$0.9775$ & $39.10$/$0.9773$ & $38.96$/$0.9769$ & $\textcolor{blue}{39.18}$/$0.9780$ & $39.07$/$0.9778$ & $\textcolor{red}{39.28}$/$\textcolor{blue}{0.9784}$\\
			& $\times 3$ & -/- & $33.44$/$0.9427$ & -/- & $34.17$/$0.9476$ & $34.17$/$0.9473$ & $34.13$/$\textcolor{blue}{0.9484}$ & $\textcolor{blue}{34.17}$/$0.9481$ & $\textcolor{red}{34.52}$/$\textcolor{red}{0.9498}$\\
			& $\times 4$ & -/- & $30.17$/$0.9034$ & $30.91$/$0.9137$ & $31.02$/$0.9148$ & $31.11$/$0.9148$ & $31.00$/$0.9151$ & $\textcolor{blue}{31.16}$/$\textcolor{blue}{0.9154}$ & $\textcolor{red}{31.52}$/$\textcolor{red}{0.9187}$
			\\\hline
		\end{tabular}
		\vspace*{-.1cm}
		\caption{\label{tab:psnr2}Average PSNR/SSIMs of \textbf{Post-upsampling} models for scale factor $\times2$, $\times3$ and $\times4$ on datasets Set5, Set14, BSD100, Urban100 and Manga109. \textcolor{red}{Red} color indicates the best performance and \textcolor{blue}{blue} color indicates the second best performance.}
		\vspace*{-.3cm}
	\end{table*}
}

\begin{figure*}
	\renewcommand{\thesubfigure}{}
	\vspace*{-.2cm}
	\centering
	\includegraphics[width=\linewidth, keepaspectratio]{figure7.png}

	%

	%

	\caption{\label{fig:visual}SR results of ``img016'' and ``img059'' from \textbf{Urban100} with scale factor $\times 4$. \textcolor{red}{Red} indicates the best performance.}
	\vspace*{-.33cm}

	%

\end{figure*}

\subsection{Datasets and metrics}

\textbf{Training Set} By following \cite{VDSR2016, DRRN2017}, the training set of CRNet-A consists of \textit{291} images, where $91$ of these images are from Yang \etal~\cite{JianChaoYang2010} with the addition of $200$ images from Berkeley Segmentation Dataset \cite{BSD100:2001}. For CRNet-B, $800$ training images of \textit{DIV2K} \cite{NTIRE2017} are used for training.

\textbf{Testing Set} During testing, \textit{Set5} \cite{Set5:2012}, \textit{Set14} \cite{Set14:2010}, \textit{B100} \cite{BSD100:2001} and \textit{Urban100} \cite{Urban100:2015} are employed. As recent post-upsampling methods \cite{EDSR2017, MSRN2018, DDBPN2018, RDN2018} also evaluate their performance on \textit{Manga109} \cite{Manga109:2017}, so does CRNet-B.

\textbf{Metrics} Both PSNR and SSIM \cite{SSIM2004} on Y channel (\ie, luminance) of transformed YCbCr space are calculated for evaluation.

\subsection{Implementation details}
\label{subsec:details}

\textbf{CRNet-A} Data augmentation and
scale augmentation \cite{VDSR2016, DRCN2016, DRRN2017} are used for training a single model for all different scales ($\times 2$, $\times 3$ and $\times 4$). Every convolution layer in CRNet-A contains $128$ filters ($n_0 = 128$) of size $3\times 3$ while $\bm{W}_l$ and $\bm{S}$ have $256$ filters ($m_0 = 256$). The network is optimized using SGD. The learning rate is initially set to $0.1$ and then decreased by a factor of $10$ every $10$ epochs. L2 loss is used for CRNet-A, and we train a total of $35$ epochs.

\textbf{CRNet-B} Every weight layer in CRNet-B has $64$ filters ($n_0 = 64$) with the size of $3\times 3$ except $\bm{W}_l$ and $\bm{S}$ have $1,024$ filters ($m_0 = 1,024$). CRNet-B is updated using Adam \cite{Adam2014}. The initial learning rate is $10^{-4}$ and halved every $200$ epochs. We train CRNet-B for $800$ epochs. Unlike CRNet-A, CRNet-B is trained using L1 loss for better convergence speed.

\textbf{Recursion} We choose $K=25$ in both of our models. We implement our models using the PyTorch \cite{pytorch2017} framework with NVIDIA Titan Xp. It takes approximately $4.5$ days to train CRNet-A, and $15$ days to train CRNet-B.

\subsection{Comparison with CSC-SR}

We first compare our proposed models with the existing CSC based image SR method,  \ie, CSC-SR \cite{CSC-SR2015}. Since CSC-SR utilizes LR images as input image, it can be considered as a post-upsampling method, thus CRNet-B is used for comparison. Tab.~\ref{tab:compare:cscsr} presents that our CRNet-B clearly outperforms CSC-SR by a large margin.

\subsection{Comparison with State of the Arts}
\label{subsec:start-of-the-arts}

We now compare the proposed models with other state-of-the-arts in recent years. We compare CRNet-A with pre-upsampling models (\ie, SRCNN \cite{SRCNN2016}, RED30 \cite{REDNet2016}, VDSR \cite{VDSR2016}, DRCN \cite{DRCN2016}, DRRN \cite{DRRN2017}, MemNet \cite{MemNet2017}) while CRNet-B with post-upsampling architectures (\ie, SRDenseNet \cite{SRDenseNet2017}, MSRN \cite{MSRN2018}, D-DBPN \cite{DDBPN2018}, EDSR/MDSR \cite{EDSR2017}, RDN \cite{RDN2018}). Similar to \cite{EDSR2017,RDN2018}, self-ensemble strategy \cite{EDSR2017} is also adopted to further improve the performance of CRNet-B, and we denote the self-ensembled version as CRNet-B+.

Tab.~\ref{tab:psnr1} and Tab.~\ref{tab:psnr2} show the quantitative comparisons on the benchmark testing sets. Both of our models achieve superior performance against the state-of-the-arts, which indicates the effectiveness of our models. Qualitative results are provided in Fig.~\ref{fig:visual}. Our methods tend to produce shaper edges and more correct textures, while other images may be blurred or distorted. More visual comparisons are available in the supplementary material.

Fig.~\ref{fig:psnr:params} shows the performance versus the number of parameters, our CRNet-B and CRNet-B+ achieve better results with fewer parameters than EDSR \cite{EDSR2017} and RDN \cite{RDN2018}. It's worth noting that EDSR/MDSR and RDN are far deeper than CRNet-B (\eg, $169$ \vs $36$), but CRNet-B is quite wider ($\bm{W}_l$ and $\bm{S}$ have $1,024$ filters). As reported in \cite{EDSR2017}, when increasing the number of filters to a certain level, \eg, $256$, the training procedure of EDSR (for $\times 2$) without residual scaling \cite{Incepv4-2016, EDSR2017}  is numerically unstable, as shown in Fig.~\ref{sub:EDSR:loss}. However, CRNet-B is relieved from  the residual scaling trick. The training loss of CRNet-B is depicted in Fig.~\ref{sub:RLCSCB:loss}, it converges fast at the begining, then keeps decreasing and finally fluctuates at a certain range.

\subsection{Parameter Study}

The key parameters in both of our models are the number of filters ($n_0, m_0$) and recursions $K$.

\textbf{Number of Filters} We set $n_0 = 128, m_0 = 256, K=25$ for CRNet-A as stated in Section~\ref{subsec:details}. In Fig.~\ref{sub:RLCSCA:filters}, CRNet-A with different number of filters are tested (DRCN \cite{DRCN2016} is used for reference). We find that even $n_0$ is decreased from $128$ to $64$, the performance is not affected greatly. On the other hand, if we decrease $m_0$ from $256$ to $128$, the performance would suffer an obvious drop, but still better than DRCN \cite{DRCN2016}. Based on these observations, we set the parameters of CRNet-B by making $m_0$ larger and $n_0$ smaller for the trade off between model size and performance. Specifically, we use $n_0 = 64, m_0 = 1024, K=25$ for CRNet-B. As shown in Fig.~\ref{sub:RLCSCB:params}, the performance of CRNet-B can be significantly boosted with larger $m_0$ (MDSR \cite{EDSR2017} and MSRN \cite{MSRN2018} are used for reference). Even with small $m_0$, \ie, $256$, CRNet-B still outperforms MSRN \cite{MSRN2018} with fewer parameters ($2.0$M \vs $6.1$M).

\textbf{Number of Recursions} We also have trained and tested CRNet-A with $15$, $20$, $25$, $48$ recursions, so the depth of the these models are $20$, $25$, $30$, $53$ respectively. The results are presented in Fig.~\ref{sub:RLCSCA:K}.  It's clear that CRNet-A with $20$ layers still outperforms DRCN with the same depth and increasing $K$ can promote the final performance. The results of using different recursions in CRNet-B are shown in Fig.~\ref{sub:RLCSCB:K}, which demonstrate that more recursions facilitate the performance improved.

\begin{figure}
	\centering
	\renewcommand{\thesubfigure}{(a)}
	\vspace*{-.2cm}
	\subfloat[Pre-upsampling models]{\label{sub:bicubic:params}\includegraphics[%
		width=0.466\linewidth, keepaspectratio]{psnr_bicubic_params.pdf}}\hspace*{.03\linewidth}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[Post-upsampling models]{\label{sub:nobicubic:params}\includegraphics[%
		width=0.46\linewidth, keepaspectratio]{psnr_nobicubic_params.pdf}}
	\vspace*{-.2cm}
	\caption{\label{fig:psnr:params} PSNR of recent state-of-the-arts versus the number of parameters for scale factor $\times 4$ on Set5. The number of layers are marked in the parentheses.}

	\centering
	\renewcommand{\thesubfigure}{(a)}
	\vspace*{-.2cm}
	\subfloat[EDSR for $\times 2$]{\label{sub:EDSR:loss}\includegraphics[%
		width=0.461\linewidth, keepaspectratio]{EDSR_without_res_scale.pdf}}\hspace*{.03\linewidth}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[CRNet-B for all scales]{\label{sub:RLCSCB:loss}\includegraphics[%
		width=0.47\linewidth, keepaspectratio]{RLCSCB_loss.pdf}}
	\vspace*{-.2cm}
	\caption{\label{fig:RLCSCB:lines} Training loss of  EDSR ($\times 2$) without residual scaling and CRNet-B (for all scales).}

	%
	%

	\centering
	\renewcommand{\thesubfigure}{(a)}
	\vspace*{-.2cm}
	\subfloat[CRNet-A]{\label{sub:RLCSCA:filters}\includegraphics[%
		width=0.46\linewidth, keepaspectratio]{RLCSCA_psnr_channels.pdf}}\hspace*{.03\linewidth}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[CRNet-B]{\label{sub:RLCSCB:params}\includegraphics[%
		width=0.471\linewidth, keepaspectratio]{RLCSCB_psnr_params.pdf}}
	\vspace*{-.2cm}
	\caption{\label{fig:num:filters} PSNR of proposed models versus different number of filters on Set5 with scale factor $\times 3$.}

	\centering
	\renewcommand{\thesubfigure}{(a)}
	\vspace*{-.2cm}
	\subfloat[CRNet-A]{\label{sub:RLCSCA:K}\includegraphics[%
		width=0.46\linewidth, keepaspectratio]{RLCSCA_psnr_K.pdf}}\hspace*{.03\linewidth}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[CRNet-B]{\label{sub:RLCSCB:K}\includegraphics[%
		width=0.46\linewidth, keepaspectratio]{RLCSCB_psnr_K.pdf}}
	\vspace*{-.2cm}
	\caption{\label{fig:num:recursions} PSNR of proposed models versus different number of recursions on Set5 with scale factor $\times 3$.}
	\vspace*{-.5cm}
\end{figure}

\section{Discussions}
\begin{figure}
	\centering

	\renewcommand{\thesubfigure}{(a)}
	\subfloat[DRRN]{\includegraphics[width=.24\linewidth, keepaspectratio]{DRRN.pdf}}
	\renewcommand{\thesubfigure}{(b)}
	\subfloat[SCN]{\includegraphics[width=.25\linewidth, keepaspectratio]{SCN.pdf}}
	\renewcommand{\thesubfigure}{(c)}
	\subfloat[DRCN]{\label{subfig:drcn}\includegraphics[width=.233\linewidth, keepaspectratio]{DRCN.pdf}}
	\renewcommand{\thesubfigure}{(d)}
	\subfloat[CRNet-A]{\includegraphics[width=.23\linewidth, keepaspectratio]{RL-CSC.pdf}}

	\caption{\label{fig:comparisons} Simplified network structures of (a) DRRN \cite{DRRN2017}, (b) SCN \cite{SCN2015}, (c) DRCN \cite{DRCN2016}, (d) our model CRNet-A.}
	\vspace*{-.3cm}
\end{figure}

We discuss the differences between our proposed models and several recent CNN models for SR \textit{with recursive learning strategy}, \ie, DRRN \cite{DRRN2017}, SCN \cite{SCN2015} and DRCN \cite{DRCN2016}. Due to the fact that CRNet-B is an extension of CRNet-A, \ie, the main part of CRNet-B has the same structure as CRNet-A, so we use CRNet-A here for comparison. The simplified structures of these models are shown in Fig.~\ref{fig:comparisons}, where the digits on the left of the recursion line represent the number of recursions.

\noindent\textbf{Difference to DRRN}. The main part of DRRN \cite{DRRN2017} is the recursive block structure, where several residual units with BN layers are stacked. On the other hand, guided by \eqref{equ:cista}, CRNet-A contains no BN layers. Coinciding with EDSR/MDSR \cite{EDSR2017}, by normalizing features, BN layers get rid of range flexibility from networks. Furthermore, BN consumes much amount of GPU memory and increases computational complexity. Experimental results on benchmark datasets under common-used assessments demonstrate the superiority of CRNet-A.

\noindent\textbf{Difference to SCN}. There are two main differences between CRNet-A and SCN \cite{SCN2015}: CISTA block and residual learning. Specifically, CRNet-A takes consistency constraint into consideration with the help of CISTA block, while SCN uses linear layers and ignores the information from the consistency prior. On the other hand, CRNet-A adopts residual learning, which is a powerful tool for training deeper networks. CRNet-A ($30$ layers) is much deeper than SCN ($5$ layers). As indicated in \cite{VDSR2016}, a deeper network has larger receptive fileds, so more contextual information in an image would be utilized to infer high-frequency details. In Fig.~\ref{sub:RLCSCA:K}, we show that more recursions, \eg,  $48$, can be used to achieve better performance.

\noindent\textbf{Difference to DRCN}. CRNet-A differs with DRCN \cite{DRCN2016} in two aspects: recursive block and training techniques. In the recursive block, both local residual learning \cite{DRRN2017} and  pre-activation \cite{he2016identity, DRRN2017} are utilized in CRNet-A, which are demonstrated to be effective in \cite{DRRN2017}.
As for training techniques, DRCN is not easy to train, so recursive-supervision is introduced to facilitate the network to converge. Moreover, an ensemble strategy (in Fig.~\ref{subfig:drcn}, the final output is the weighted average of all intermediate predictions) is used to further improve the performance. CRNet-A is relieved from these techniques and can be easily trained with more recursions.

\section{Conclusions}

In this work, we propose two effective CSC based image SR models, \ie, CRNet-A and CRNet-B, for pre-/post-upsampling SR, respectively. By combining the merits of CSC and CNN, we achieve superior performance against recent state-of-the-arts. Furthermore, our framework and CISTA block are expected to be applicable in various CSC based tasks,  though in this paper we focus on CSC based image SR.

\newpage

{\small
\bibliographystyle{ieee}

redpajama_set_name: RedPajamaArXiv


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