For someone raised Catholic, visiting the Non-Catholic Cemetery in a city with such an incredible wealth of churches meriting attention seems almost heretical. But I am drawn to cemeteries.
This one has a reputation as a particularly soothing one, one where cats choose to live out all nine of their lives. And, as an act of advance penance, I posted a "genuflection" to Santa Maria Maggiore first.
The Non-Catholic Cemetery is a pilgrimage must for many because here lie the remains of John Keats (1795-1821). Plagued by tuberculosis, the medically trained poet traveled from England to Italy in hope the climate would result in a cure. Some believe he was self-prescribing unsafe dosages of mercury at the time, perhaps to treat venereal disease. The combination proved lethal, and Keats died in Rome at age 25.
Welcoming him we lose with scarce extinguished breath.
The ashes remaining from the scandal-ridden life of this productive young poet joined his friend Keats amongst the young graves scarcely more than a year later.
In the summer of 1822, the Courier, a leading Tory newspaper in London, carried a brief obituary that began: "Shelley, the writer of some infidel poetry, has been drowned: now he knows whether there is a God or no." From this moment on, the dramatic death of Percy Bysshe Shelley in the Gulf of Spezia was set to become one of the most powerful of all Romantic legends. And also perhaps the most misleading.
Shelley had been sailing during stormy weather with two others aboard his small racing schooner, The Don Juan, on a return trip to Lerici from Livorno after visiting Lord Byron (1788-1824), who, too, would perish at an early age. But we have killed off enough romantic poets for one day, and Byron's bones do not reside within the shelter of these walls.
Shelley's death left a young widow, Mary Wollstonecraft Shelley (1797-1851), behind. Although their marriage was a rocky one, some claim Mary sentimentally and literally retained Shelley's heart, which sounds nightmarishly apocryphal save she is the literary birth-mother of Frankenstein.
Why did I not die? More miserable than man ever was before, why did I not sink into forgetfulness and rest?
Death snatches away many blooming children, the only hopes of their doting parents: how many brides and youthful lovers have been one day in the bloom of health and hope, and the next a prey for worms and the decay of the tomb!
Of what materials was I made, that I could thus resist so many shocks, which, like the turning of the wheel, continually renewed the torture?
But I was doomed to live….
*And the incongruous presence of a pyramid by the graveyard? Things Egyptian became fashionable in Rome after the conquest of Egypt in 30 B.C. (No, I am not getting waylaid by the story of Anthony and Cleopatra.) At 118 feet-tall, this pyramid of marble-clad brick and cement is the tomb of Gaius Cestius Epulo, a wealthy Roman who died about 15 B.C. The unlikely landmark survived all the subsequent years of development in Rome possibly because of its incorporation into the city's fortification walls.
In 1945, at the height of her career, the government commissioned (Maria) Izquierdo to do a mural for a building in Mexico City. (Diego) Rivera and (David Alfaro) Siqueiros, two of Mexico's great muralist painters, blocked her from getting the job. When she dared to denounce them in public, she received little help and a lot of strong criticism….
Izquierdo began to experience nightmares that left her sleepless. One day, she arose and drew what she remembered… a clear vision of herself, in a window of metaphysical dimension, holding her own decapitated head as her body, still walking, becomes lost in the distance of steps leading to a void. That year, 1947, she painted "Sueños y Pensamento," a premonitory painting that heralded great pain for her future. It was the last of her great works.
We journeyed to Panteon de Dolores, home to a population of a million old souls qualifying it as Mexico's largest cemetery, on All Saints Day. We encountered less than a handful of families celebrating Day of the Dead traditions graveside with their ancestors. Perhaps more ventured out on the following day, All Souls Day?
While many graves were colored with an abundance of marigolds, the majority appeared untended by those left behind on earth. Perhaps the more antiquated term of Hallowmas is a more fitting name to apply to the day in this neighborhood occupying close to 600 acres between two main sections of the sprawling Chapultepec Park. Numerous graves were adorned with a jumbled combination of ancient Day of the Dead traditions with more recently imported Halloween decor – spiders, plastic pumpkins, orange and black plastic festoons and fake spider webs.
There was an ongoing mixture of entertainment, ranging from an annoying clownish play to a talented female vocalist while we were there, staged in the plaza of the Rotunda de las Personas Illustres. At dusk, children appeared in Halloween or Catrina costumes carrying plastic pumpkin baskets for trick-or-treating.
While the dearth of ancient practices was disappointing, change happens. And I need no flowers or incense to encourage me to wander through a cemetery. So many stories shout at you from all directions.
Despite the rejection of her mural, Maria Izquierdo gained admittance to the portion of the cemetery dedicated to the illustrious of Mexico. Perhaps her fellow muralists, Rivera and Siqueiros, forgave her for her earlier criticisms of muralists including political messages in their works before they joined her there. The excerpt above is a link worth tapping to begin to learn about her life. I found myself wandering on the internet to discover more about the fascinating artist who was the first Mexican woman to mount a major solo exhibition in the United States.
But there are others. Composer Agustin Lara, who left the women swooning with his "Senora Tentacion" in 1956.
In addition to those celebrated within the inner circle of the cemetery, there are close to a million others with stories worth telling. Hopefully, the trials, tribulations and joys they experienced are preserved within their families' oral histories, repeated over and over at holiday celebrations lest the tales be lost.
And then, there are the eerily spooky graves. The angel guarding the rusty doors of a crypt, unhinged as though indicating the residents fled the confined space long ago. The coffin rusting above ground. Occupied, empty or home to a vampire planning to emerge with the rise of the next full moon?
A belated happy Hallowmas from Mexico City.


text:
The British Light Heavyweight Championship was a top  British wrestling championship found throughout the country's circuit. The title's broken history dates back to the 1920s. After going inactive in 2004, the title was re-activated in 2014.

The championship was recognised and defended on matches screened by UK national television network ITV as part of the professional wrestling slot on World of Sport as well as standalone broadcasts.  Pre-publicity for these championship match broadcasts was given in ITV's nationally published listings magazine TVTimes.

Title history
This is the combined list of different versions of the British Heavyweight Titles, each of which was probably the most significant version at the time. Each version may or may not be connected to another. However, all title changes are either actual or "official" unless indicated otherwise.

Original

All Star Wrestling/ WAW/ RBW

Footnotes

See also

Professional wrestling in the United Kingdom

References

External links
British Light Heavyweight Title

Light heavyweight wrestling championships
National professional wrestling championships
Professional wrestling in the United Kingdom

redpajama_set_name: RedPajamaWikipedia


Omnes Trio, the Latin translation of 'Trio for All', was written as an expression to promote equality within the percussion field. With homogeneous instrumentation and a shared set-up, the piece is characterized by hocketed rhythmic lines, trading grooves between players, and soloistic passages that create rapidly flowing textures and quasi-melodic lines. The elegiac opera gong motive serves as a sonic cushion between dark silence and intensive waves of rhythm. This piece is dedicated to the University of North Florida Class of 2021.
You will receive an email within 24 hours with a PDF download.
Dr. Andrea Venet wrote the first version of Kibo in 2005 while studying rudimental snare drumming with the famous Bob Becker. She later updated the piece after studying other great rudimental works by John S. Pratt and Joseph Tompkins. Elements of these composers can be found in Kibo, such as a variety of rudiments, a host of sticking permutations, various treatments of ruffs and flams, and a wide range of dynamics.
Nannou is a percussion adaptation of the piece originally written by Aphex Twin. The overall effect is one of a groovy techno music box. It was originally arranged for the Arizona Academy of Performing Arts front ensemble percussion section, and was premiered at the Tempe Center Of The Arts Theater Show in 2010.
This arrangement of Eric Sammut's Rotation 1 by Andrea Venet is for a percussion ensemble of 10-14 players. With an expansion of orchestration and the addition of percussion parts, this arrangement provides a fun and groovy addition to your program. Each of the marimba and vibraphone parts may be doubled. Both synthesizer parts are optional.
Double Concerto for Vibraphone, Marimba, and Orchestra, by Emmanuel Sejourne. Arranged for Vibraphone, Marimba, and Percussion Ensemble by Andrea Venet.
Clear Midnight was commissioned by and dedicated to the fabulous Escape X Percussion Duo. The work is essentially two movements in one. The first part functioning much like an introduction or prologue, and the second, a 3 part movement unto itself. The piece is intended to be a "tour de force" for these two dynamic performers showcasing their versatile skills and impassioned musical voice.
Commissioned by the Escape X Percussion Duo, jazz vibraphonist/composer Joe Locke crafted a programmatic tribute to his new home of New Jersey, and, more specifically, the Garden State Parkway. Driving along the tree-lined roads of the Parkway with a cloudless, blue sky on the horizon can be blissful, but New Jersey's fast drivers can often make the drive somewhat less than calm. Locke perfectly captured this juxtaposition by writing for marimba and vibraphone. He also incorporated Escape X's iconic style of adding extra sounds to the keyboards by writing for tuned gongs and crotales.
It is hard not to be curious about something hawked by a hundred billboards as you drive through the back of beyond. Each of the colorful dinosaurs which stand in the backyard of the gift shop seems to have been molded from a vague memory of a picture of a dinosaur the creator once saw. Many of these campy roadside attractions are eerily abandoned but still a worthy destination for the curious motorist and touring percussion duo. Abandoned Roadside Chorale was commissioned by the percussion duo Escape X.
Album of new and commissioned works for marimba duo by Escape Ten. Released in 2014 by Escape Ten Records. By Andrea Venet & Dr. Annie Stevens.


"What is the purpose I should detox?" It is an important part of a weight loss plan. It helps to rid your physique of harmful toxins and harmful small minions that hide in our bodies and we don't know it.
Some of these best detox weight loss ideas are simply based on the plan that if harmful meals goods are taken out from the diet plan and healthy new fruits and vegetables put in their place. However, not every detox diet plan you listen to about will be so simple. Consume at minimum two quarts of water daily.Consider vitamin C regularly. Make certain you breathe deeply to permit oxygen to flow into more totally in your system.
But the actuality is that most of these programs are going to inquire you both to starve your self or to fast for times on end. And there are a number of problems with this type of approach. First, fasting or close to hunger diet programs can be harmful, particularly if you are sick or severely overweight. 2nd, these types of approaches to detox merely direct to binge eating and yo-yo dieting after you arrive off them. Certain you may briefly drop 10 lbs. following two weeks of fasting, but you're a great deal much more most likely to place it on correct away if you don't view what you consume.
Acidic Blood/Ketosis: When a diet is changed from a high carbohydrate diet plan to a diet plan that does not provide sufficient carbohydrate to replenish glycogen shops, the body goes through a set of phases to enter ketosis, with side results including fainting, dizziness, reduced blood pressure, heart arrhythmia (irregular heart beat), hunger, vomiting and much more.
Reason #3 – Some pesticides as well as other toxins are proving to interfere with our hormonal stability and chemical processes. These in consist of urge for food control, the ability to shed excess weight and even each working day metabolism. In reality, some chemical toxins are confirmed to inhibit thyroid perform.
CRUMBLE: Mix with each other crushed up rolled oats, crushed walnuts, include small raisins or sliced & diced raisins and a sprinkle of cinnamon powder. This mix can be to your taste so have fun creating your personal wholesome mix! The crumble makes up the foundation and 1/3 of the dessert. Place this crumble straight into your dessert bowls / cups. I usually provide this dessert in glass cups.
So, there are numerous advantages to a detox diet, but don't take them to extremes. A change in lifestyle might be more possible for you if you begin it with a big occasion like a fast or a diet, but the proof indicates that we ought to not make a habit of it.


text:
When readers complain about a photo, they usually say the picture was "too much." In this case, they complained there was "not enough."
When The Virginian-Pilot and The Ledger-Star published a photograph of a nude man mowing his grass, some people complained about the coverage, or actually, the lack of it. It wasn't exactly naked rage, but about 60 readers said they found the photo offensive; some felt it was even pornographic.
The photo, which was four columns wide and about 6 inches deep, ran on the front page of the local news section. The subject of the photo is a member of a clothing-optional group. His decision to do his yardwork in the buff landed him in court on charges of indecent exposure.
Assistant Managing Editor Ron Speer said the decision to run the photo required no "soul-searching." He simply looked at the picture closely "to make sure nothing was revealed that shouldn't be."
After the initial round of complaints, the Norfolk newspapers decided to ask readers whether the photograph offended them. Using Infoline, the newspapers' telephone information service, 4,361 calls were logged — 65 percent of them disagreeing with the readers who said that the photo was in bad taste.
One supporter responded, "Once again, we have 60 people out of how many in your readership trying to tell me what I should read. I find that more offensive than any picture."
Another reader said, "To me it's not as bad as some of the photos the newspaper runs when there has been a tragic accident."
From another supporter: "Neither my 95-year-old mother nor I thought the picture was offensive. To tell the truth, the first thing I thought of was, 'He isn't afraid of skin cancer.'"
A "pro-photo" caller who said she was the 83-year-old mother of two sons said she found the picture of the man "perfectly beautiful." She added, "God help us, if we had to look at most men in the buff, my sons included."
But these comments were in the 1500 calls objecting to the picture:
"I have a 9-year-old daughter who now knows what a naked man looks like thanks to you . . . I will no longer buy your paper until you raise your standards."
"The editor who decided to print this needs counseling."
"Nudity is not expected in a newspaper. Nudity is abnormal. If it were normal, the editor who put this picture in the paper would be sitting there with no clothes on.
Speer (who does wear clothes to work) said he's been amazed at the amount of reaction. He said if he had the decision to make again, he "perhaps wouldn't play the picture so big."
In case you're wondering, the indecent exposure charges against the naked rider were dismissed. The story accompanying the photograph said the judge decided "it's OK to expose one's private parts in the private parts of one's yard."

redpajama_set_name: RedPajamaCommonCrawl


text:
UPDATE 1-Belarus accuses Russia of election meddling, seeks talks with Putin
Andrei Makhovsky
* Belarus holds presidential election on Aug. 9
* President Lukashenko is seeking to extend his long rule
* Lukashenko has cracked down on opponents (Adds quotes, details, background)
MINSK, June 25 (Reuters) - Belarusian President Alexander Lukashenko accused Russian and Polish forces on Thursday of meddling in his country's presidential election campaign and trying to discredit him.
The former collective farm boss, who has ruled Belarus for 26 years and allowed little dissent, said he would discuss the alleged meddling with Russian President Vladimir Putin. He did not say when.
Lukashenko, 65, faces the strongest challenge in years to his hold on power in the Aug. 9 election as frustration mounts over his handling of the coronavirus pandemic and grievances grow over the economy and human rights.
In comments made while appointing new ambassadors, Lukashenko said external forces stood behind his opponents and were disseminating information discrediting him and his family members.
"It's clear that the puppeteers are behind them. They are on the one side and the other side. They live in Poland and are supplied from Russia," he said. "We will talk about this with President Putin in the near future at a meeting."
Belarus sovereign bonds fell as much as 2.5 cents after Lukashenko's comments.
RUSSIAN DENIAL
Kremlin spokesman Dmitry Peskov denied Lukashenko's allegations.
"Russia has never, is not and does not intend to interfere in election processes, particularly in the ones of our Belarusian ally," he told reporters.
Relations between Belarus and traditional ally Russia have been strained in recent months as Moscow reduced subsidies that have propped up Lukashenko. But a crackdown by Lukashenko on opponents is likely to hinder his efforts to mend fences with the West.
A top security official under Lukashenko has accused Viktor Babariko, widely seen as Lukashenko's most powerful electoral opponent, of being controlled by Russian "puppeteers".
Babariko was head of Belgazprombank, the local unit of Russia's Gazprombank, before running for president. Lukashenko said the bank's money was being used to finance Babariko's campaign and the authorities have opened a criminal case against the lender.
Babariko's campaign team called the allegations against him "an absurdity". He was arrested last week, sparking a new round of protests. Human rights activists say 360 people were arrested in a subsequent crackdown in recent days. (Writing by Matthias Williams, Editing by Timothy Heritage)


Liberty: Teresa Lewis didn't pull the trigger. Why is she on death row?
Teresa Lewis didn't pull the trigger. Why is she on death row?
The Commonwealth of Virginia already has a serious relationship with its death penalty. In the past three decades, only Texas has executed more inmates. But on Sept. 23, the Old Dominion will enter new territory when it executes a female inmate for the first time in nearly a century.
Her name is Teresa Lewis, she is the only woman on death row at the Fluvanna Correctional Center for Women, and her appeals have all but expired. If she is executed, she will become another glaring example of the unfairness of our death penalty system.


Royal Infirmary of Edinburgh (RIE), 51 Little France Crescent, Edinburgh EH16 4SA.
Royal Hospital for Sick Children (RHSC), 9 Sciennes Road, Edinburgh EH9 1LF.
St John's Hospital, Howden Road, West Howden, Livingston, West Lothian EH54 6PP.
Roodlands General Hospital, Hospital Road, Haddington, East Lothian EH41 3PF.
Herdmanflat Hospital, Aberlady Road, Haddington, East Lothian EH41 3BU.
Western General Hospital, Crewe Road South, Edinburgh EH4 2XU.
Princess Alexandra Eye Pavilion, Chalmers Street, Edinburgh EH3 9HA.
Chalmers Hospital, 2A Chalmers Street, Edinburgh EH3 9HQ.
Astley Ainslie Hospital, 133 Grange Loan, Edinburgh EH9 2HL.
Lauriston Building, Lauriston Place, Edinburgh EH3 9HA.
The NHS Minor Ailments Service is provided by local pharmacists. You can use this service if you are registered with a GP Surgery in Scotland, and if you do not live in a Nursing or Residential Care Home.
Please note that this information has been taken from the NHS leaflet ''The NHS Minor Ailments Service at your local Pharmacy''. Please ask for further information at your local Pharmacy, or click here.
All women aged between 50 and 70 are currently offered breast screening by mammography every 3 years in Scotland. Women under the age of 50 are not offered screening unless they have previously had breast cancer, have a close relative who developed breast cancer early (usually when aged under 45), or are known to carry certain genes that predispose them to early-onset breast cancer. The screening programme is centrally coordinated with invitations issued on a 3 yearly basis. The contact details for the Lothian Breast Screening Programme are Ardmillan House, 42 Ardmillan Terrace, Edinburgh EH11 2JL, and their telephone number is 0131 537 7400.
If you have a positive home pregnancy test, the first step is to self-refer to the Midwife by calling 0131 536 2009 (Monday to Friday from 9 a.m. to 5 p.m.) for an appointment. Or, if you are unsure about continuing with the pregnancy, you should book an appointment with your GP on 01875 610248 as soon as possible for advice and support.
At your first Midwife appointment at the surgery (usually on Monday afternoons) at around 8-10 weeks into your pregnancy, you will be asked detailed questions about your past medical history, family history and medications, and have blood tests taken. Following this, you will receive an appointment for your first scan that is usually done when you are around 12 weeks pregnant. This booking scan confirms the dates of pregnancy, and you will also have a later detailed scan at around 20 weeks that looks at your baby's heart, kidneys, skull and spine.
You should be taking a supplement of Folic Acid (400 micrograms for most women) for the first 12 weeks of your pregnancy, as this has been shown to reduce the risk of a malformation in the baby's spinal cord called Spina Bifida. You may need a higher dose if you are Diabetic, on medication for Epilepsy, or have a previous or family history of problems related to Spina Bifida - please ask your Midwife or GP if you are unsure. You should eat a healthy, balanced diet, and should avoid eating liver and pate, raw or partially cooked eggs, raw or uncooked meats, raw shellfish, unpasteurised milk and some types of cheese. It is best not to drink alcohol at all as it will reach and may harm your baby. Smoking harms you and your baby. If you need help to stop smoking during pregnancy, please phone 0800 848484.
Our Midwifery colleagues who work in teams in the community deliver the majority of antenatal care. Our local Midwife team visits the surgery every Monday afternoon, and can be contacted on 0131 536 5804 outwith these hours. For further information about keeping you and your baby well during pregnancy, please click here.
In East Lothian, you can self-refer to Physiotherapy by phoning NHS24 MATs (Musculo-skeletal Advice and Triage service) Monday to Friday 2pm - 8pm on 0845 604 0001. This service is for Adults only with Musculo-skeletal problems. It offers triage and early access to self-management advice. This advice line is able to refer directly to Physiotherapy if required. Please note that you can also be referred by your GP for Physiotherapy if you have consulted them about the problem. For further information on Physiotherapy in East Lothian or to self-refer, please click here.
You can obtain details of NHS Dentists currently registering NHS patients by phoning 0131 537 8444. If you are not registered with a Dentist, or are unable to contact your Dentist, an emergency service only is available for patients 16 years and over at Chalmers Dental Centre, 3 Chalmers Street, Edinburgh (telephone number: 0131 536 4800).
The NHS in Scotland recommends that everyone has a free eye check every 2 years, or more frequently if advised. It is important for adults to have eye examinations on a regular basis to check for problems that may require specialist care, such as glaucoma, macular degeneration and cataract. Please note that you are no longer able to self-refer to the Emergency Eye Clinic at the Princess Alexandra Eye Pavilion in Edinburgh.
Unfortunately, we do not offer a NHS Podiatry clinic based within the Practice, although one is located nearby at Tranent Health Centre in Tranent. You can refer yourself by picking up a form from Reception at the surgery or clicking here. You should then either return the completed form to us or post it to Tranent Health Centre directly. Please note that you cannot self-refer for Domiciliary Podiatry as you require a GP referral for this service.
Podiatrists offer a range of services dedicated to foot care, including Orthotics. Please note NHS Podiatry is unable to offer routine nail care and cutting unless there are relevant medical circumstances. For further information about Podiatry, please click here.
Why use NHS Lothian Stopping Smoking Clinic?
For further information about East Lothian Stopping Smoking Services, please click here.
CHANGES is a Community Health Project in Musselburgh that promotes the positive well-being of people. They are able to help you if you are experiencing a range of mental health issues, such as stress, anxiety and depression. You can self-refer for counselling or psychotherapy to CHANGES by phoning 0131 653 3977 or by emailing info@changeschp.org.uk.
Please click here for further information and advice. If you wish to make an appointment at this strictly confidential clinic, please telephone 0131 536 1070.
Cardiac and Pulmonary Rehabilitation exercise classes take place every Wednesday from 11.00 a.m. to 12.00 noon in The Coalgate, The Wynd, Ormiston. This class is open to anyone, but particularly those who have at some time experienced heart or breathing problems. It is facilitated by a fully qualified Cardiac Rehabilitation Fitness Instructor. A programme is devised to meet the requirements of each individual and refreshments are provided. Please ask at Reception for further details.


Fuzzrocious Pedals, LLC's products are handmade in Mount Laurel, NJ by a husband and wife team with help from their children. You are purchasing a pedal made for you by us. We are working as hard as we can for you. By purchasing from us, you accept these terms and conditions in their entirety.
Our guitar pedals are made on a build to order basis, so wait times may vary from month to month, depending upon the amount of pending orders in the queue. Fuzzrocious Pedals, LLC estimates orders handpainted by Shannon Ratajski with will ship in sixteen (16) weeks or less from date of payment. Screenprinted orders with will ship in three (3) weeks or less from date of payment. This wide estimate is in place to ensure we drill, paint, install hardware, build, test, and ship your pedal properly – you are having a product built just for you.
Fuzzrocious Pedals, LLC assumes responsibility for all domestic shipping fees. International customers assume responsibility for all international shipping fees via USPS Priority Mail International.
Fuzzrocious Pedals, LLC is not responsible for VAT/Customs fees outside of the US.
Fuzzrocious Pedals, LLC accepts returns of mint condition (unused, unmarked, working order) pedals within seven (7) days of delivery of product to customer, minus a 25% restocking fee. Customer assumes responsibility for return shipping. Fuzzrocious Pedals, LLC also reserves the right to refuse returns based on condition of pedal. We recommend packaging all returns similarly to how you received it to ensure you can receive a refund and that the pedal is not damaged in the mail.
After seven (7) days of customer's receipt of pedal, Fuzzrocious Pedals, LLC will not accept returns.
All clothing or other apparel sales are final.
Fuzzrocious Pedals, LLC will cancel and refund an order, minus 10% for processing fees and restocking, if the request for cancelation by the customer is made before Fuzzrocious Pedals, LLC notifies customer via email that the pedal has shipped. Due to the labor involved, custom orders do not qualify for cancellation.
Fuzzrocious Pedals, LLC Pedals' pedals are guaranteed to be free from manufacturer and design defects for a minimum of one (1) year from the date of purchase. If an issue arises after one (1) year from date of purchase, the customer assumes a $25/hour repair fee and the cost of shipping both to Fuzzrocious Pedals, LLC for repair, and after repair, for return from Fuzzrocious Pedals, LLC to the customer.
If a customer owns a preowned Fuzzrocious pedal and there is fault with the pedal from reasonable usage or defect of hardware and is in a domestic US address, the customer assumes shipping to/from Fuzzrocious Pedals, LLC. International customers pay shipping to the US; Fuzzrocious Pedals, LLC will assume return shipping costs. Fuzzrocious Pedals, LLC is not responsible for VAT/Customs fees outside of the US.
Instruction manuals for each pedal are available on our website, and will not be provided in a printed form by Fuzzrocious Pedals, LLC. It is strongly recommended that the customer review the manual in its entirety in order to not damage the pedal.
If there is an issue with the pedal from reasonable usage within 30 days of receipt of the pedal to a domestic US address, Fuzzrocious Pedals, LLC will pay for shipping to and from the customer at no cost to the customer. International customers pay shipping to the US; Fuzzrocious Pedals, LLC will assume return shipping costs. Fuzzrocious Pedals, LLC is not responsible for VAT/Customs fees outside of the US.
By purchasing a pedal from Fuzzrocious Pedals, LLC, you agree to hold harmless, release, defend, and indemnify Fuzzrocious Pedals, LLC, as well as all agents and representatives of Fuzzrocious Pedals, LLC, against any claims arising from injury or damage to personal property as a result of using our products. It is your responsibility as the customer to read all manuals included with our pedals, and to operate the pedals in accordance with the directions given in the manuals. All issues arising from the sale of any of our products are governed by New Jersey State Law. By purchasing from us, you agree to submit to binding impartial arbitration in the Burlington, New Jersey area to settle any disputes.


text:
\section{Introduction}
	\label{sec:intro}
	Surface normal is an informative 3D visual feature used in various computer vision and robotics applications, such as object recognition and scene understanding \cite{PAMI2012, klasing2009comparison,fan2019pothole}. To date, there has not been extensive research on surface normal estimation, as it is merely considered to be an auxiliary functionality for other computer vision and robotics applications \cite{3F2N}. As such applications are typically required to perform robustly and in real time, surface normal estimators (SNEs) must be sufficiently accurate and computationally efficient \cite{3F2N}.

	\begin{figure}[!t]
		\centering
		\includegraphics[width=0.48\textwidth]{fig/fig_front.pdf}
		\vspace{+0.5em}
		\caption{Comparison between 3F2N (recently published SoTA SNE) \cite{3F2N} and our proposed SDA-SNE. A significantly improved region is marked with a dashed blue circle.}
		\label{fig.front}
	\end{figure}

	The state-of-the-art (SoTA) SNEs \cite{klasing2009realtime,jordan2014quantitative,klasing2009comparison,3F2N,CP2TV,raposo2020accurate,daniel2015optimal,barath2019optimal} generally select a set of 3D points and compute surface normals via planar fitting, geometric transformation, or statistical analysis. However, such approaches are infeasible to estimate surface normals near/on spatial discontinuities, \eg, edges and ridges (see Fig. \ref{fig.front}), as they generally introduce adjacent 3D points on different surfaces unintentionally \cite{JV2018}. Bormann \emph{et al.} \cite{IROS2015} proposed the first edge-aware SNE, capable of adaptively selecting reasonable adjacent 3D points on the same surface. It performs significantly better than prior arts near edges. Nonetheless, their method focuses only on edges and requires a manually-set discontinuity awareness threshold. This results in low adaptability to different scenarios and datasets. Hence, there is a strong motivation to develop an SNE capable of tackling all types of spatial discontinuities to achieve greater robustness and environmental adaptability.

	The major contributions of this work are summarized as follows:\\
	(1) \textbf{Spatial discontinuity-aware surface normal estimator (SDA-SNE)}, a highly accurate SNE, significantly outperforming all other SoTA SNEs, especially near/on spatial discontinuities. \\
	(2) \textbf{Multi-directional dynamic programming (DP)} for depth gradient refinement. It iteratively introduces inliers (co-planar pixels) by minimizing a smoothness energy.\\
	(3) \textbf{Path discontinuity (PD) norm}, a semi-norm evaluating the discontinuity of a given path. It reflects the depth gradient estimation error generated by the Taylor expansion.\\
	(4) \textbf{Recursive polynomial interpolation (RPI)}, an ultrafast polynomial interpolation algorithm. Compared to Lagrange and Newton polynomial interpolation, it iteratively refines the first-order derivative of depth with a more efficient polynomial interpolation strategy.

	\section{Related Work}
	\label{sec:RelatedWork}

	The existing SoTA SNEs can be categorized into four classes: (1) optimization-based \cite{klasing2009comparison}, (2) averaging-based \cite{klasing2009realtime}, (3) affine-correspondence-based \cite{barath2019optimal}, and (4) depth-to-normal \cite{wang2021sne}.

	The optimization-based SNEs, \eg, PlaneSVD \cite{klasing2009realtime}, PlanePCA \cite{jordan2014quantitative}, VectorSVD \cite{klasing2009realtime}, and QuadSVD \cite{2001Cloud,2005On}, compute surface normals by fitting local planar or curved surfaces to an observed 3D point cloud, using either singular value decomposition (SVD) or principal component analysis (PCA). The averaging-based SNEs, \eg, AreaWeighted \cite{klasing2009comparison} and AngleWeighted \cite{klasing2009realtime}, estimate surface normals by computing the weighted average of the normal vectors of the triangles formed by each given 3D point and its neighbors. However, these two categories of SNEs are highly computationally intensive and unsuitable for online robotics and computer vision applications \cite{3F2N}. The affine-correspondence-based SNEs exploit the relationship between affinities and surface normals \cite{daniel2015optimal, raposo2020accurate, barath2019optimal}. Nevertheless, such SNEs are developed specifically for stereo or multi-view cases and cannot generalize to other cases where monocular depth images are used.

	Recently, enormous progress has been made in end-to-end depth-to-normal translation \cite{3F2N, fan2020sne, CP2TV}. Such SNEs have demonstrated superior performance in terms of both speed and accuracy. Fan \emph{et al.} \cite{3F2N} proposed a fast and accurate SNE referred to as 3F2N, which can infer surface normal information directly from depth or disparity images, with two gradient filters and one central tendency measurement filter (a mean or median filtering operation), as follows:
	\begin{equation}
		\begin{split}
			&{n_x= {\partial d}/{\partial u}, \ \ \ \ \ \ \ \ \ \ \  n_y= {\partial d}/{\partial v}},\\
			&{{n}_z}=-\Phi\Bigg\{ \frac{({x_i}-x) n_x + ({y_i}-y) n_y }{{z_i}-z} \Bigg\},\ i = 1,\dots,m,
			\label{eq.3F2N}
		\end{split}
	\end{equation}
	where $\mathbf {n} = [n_x,n_y,n_z]^\top$ is the surface normal of a given 3D point $\mathbf{p}^C= [x,y,z]^\top$ in the camera coordinate system, $\mathbf{p}^C$ is projected to a 2D pixel $ \mathbf {p}  = [u, v]^\top $ via $z[{\mathbf {p}}^\top,1]^\top=\mathbf{K}\mathbf{p}^C$ ($\mathbf{K}$ is the camera intrinsic matrix), $\mathbf{p}_i^C=[x_i,y_i,z_i]^\top$ is one of the $m$ adjacent pixels of $\mathbf{p}^C$, $d$ is disparity (or inverse depth), and  $\Phi\{\cdot\}$ represents the central tendency measurement filtering operation for $n_z$ estimation. Nakagawa \emph{et al.} \cite{CP2TV} presented an SNE based on cross-product of two tangent vectors (CP2TV): $\mathbf n(u,v)=\bm t_u\times \bm t_v$, where
	\begin{equation}
		\begin{split}
			\bm t_u & = \begin{bmatrix}\begin{displaystyle}\frac{\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u}\end{displaystyle}\end{bmatrix}^\top \\
			\bm t_v & = \begin{bmatrix}\begin{displaystyle}\frac{\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}\end{displaystyle}\end{bmatrix}^\top,
		\end{split}
		\label{eq.CP2TV.1}
	\end{equation}
	\begin{equation}
		\begin{split}
			\frac{\partial x}{\partial u} & = \frac{z}{f_u}+\frac{u-c_u}{f_u}\frac{\partial z}{\partial u}\\
			\frac{\partial y}{\partial u} & = \frac{v-c_v}{f_v}\frac{\partial z}{\partial u}\\
			\frac{\partial x}{\partial v} & = \frac{u-c_u}{f_u}\frac{\partial z}{\partial v}\\
			\frac{\partial y}{\partial v} & = \frac{z}{f_v}+\frac{v-c_v}{f_v}\frac{\partial z}{\partial v},
		\end{split}
		\label{eq.CP2TV.2}
	\end{equation}
	$\mathbf{c}=[c_u,c_v]^\top$ is the principal point (in pixels), and $f_u$ and $f_v$ are the camera focal lengths (in pixels) in the $u$ and $v$ directions, respectively. Since the depth or disparity gradient estimation has a direct impact on the surface normal quality, this paper focuses thoroughly on the strategy to improve the accuracy of depth gradient $\nabla \hat z = [ \hat z_{u},\hat z_{v} ]^{\top}$\footnote{In this paper, the variables with and without hat symbols denote the estimated and theoretical values, respectively.}. The proposed strategy can also be applied to improve disparity gradient estimation, as it is inversely proportional to depth.

	\section{Methodology}
	\label{sec:Methodology}
	As discussed in Sec. \ref{sec:RelatedWork}, surface normal estimation was formulated as a depth-to-normal translation problem in SoTA approaches, which have demonstrated superior performances over other methods in terms of both speed and accuracy \cite{3F2N}. The accuracy of such approaches is subject to the depth gradient quality. Since depth gradients typically jump near/on discontinuities, \eg, edges and ridges, the estimated surface normals near/on such discontinuities are always significantly different from their ground truth. Therefore, we propose a multi-directional DP strategy to translate a depth or disparity image into a surface normal map in a coarse-to-fine manner. It first initializes $\nabla \hat z$ using finite difference interpolation and then optimizes $\nabla \hat z$ using our proposed multi-directional DP. Accurate surface normals can then be obtained by plugging the optimum depth gradients into either the central tendency measurement filtering operation used in 3F2N (see (\ref{eq.3F2N})) or the cross-product operation used in CP2TV (see (\ref{eq.CP2TV.1})-(\ref{eq.CP2TV.2})).

	\begin{figure*}[!t]
		\centering
		\includegraphics[width=1\textwidth]{fig/fig_dp.pdf}
		\caption{
			(a) An illustration of the energy transfer process, where the inliers (desired pixels) are gradually introduced in each iteration; (b) The relationship among smoothness energy, state variable, depth gradient, and surface normal.}
		\label{fig.dp}
	\end{figure*}

	\subsection{Multi-Directional Dynamic Programming}
	\subsubsection{Smoothness Energy and State Variable}
	As a general rule, only inliers (co-planar pixels) should be introduced when estimating the depth gradient of an observed pixel ${\mathbf {p}}$. Such pixels are determined by minimizing the (path) smoothness energy $\bm E=[E_u,E_v]$ of ${\mathbf {p}}$ using DP, where $E_u$ and $E_v$ represent the horizontal and vertical components of $\bm E$, respectively. To minimize the computational complexity, we recursively introduce adjacent co-planar pixels. In each iteration, two desired pixels ${\mathbf {p}}_u'$ and ${\mathbf {p}}_v'$ (corresponding to the minimum $E_u$ and $E_v$, respectively) are used to extend the DP path and  update the depth gradient of ${\mathbf {p}}$. We define a state variable $\bm S=[\bm s_u,\bm s_v]$, where $\bm s_u={\mathbf {p}}_u'-{\mathbf {p}}$ and $\bm s_v={\mathbf {p}}_v'-{\mathbf {p}}$ denote the horizontal and vertical components of $\bm S$, respectively\footnote{$\bm s_u,\bm s_v \in \big\{[i,j]^{\top}\mid i,j\in \{-1,0,1\}\big\}$ because only adjacent pixels are considered.}. In the $k$-th iteration, the desired pixels are determined by minimizing $\bm E^{\left(k\right)}$ as follows:
	\begin{equation}
		\bm E^{\left(k\right)}=\min\limits_{\bm S}\left\{\bm{\mathcal T}\left(\bm E^{\left(k-1\right)},\bm S \right)\right\}.
		\label{def.C1}
	\end{equation}
	The state variable $\bm S^{\left(k\right)}$ is transferred to
	\begin{equation} \bm S^{\left(k\right)}=\mathop{\arg\min}\limits_{\bm S}\left\{\bm{\mathcal T}\left(\bm E^{\left(k-1\right)},\bm S \right)\right\},
		\label{def.C2}
	\end{equation}
	where $\bm {\mathcal T}$ denotes the smoothness energy transfer function. Therefore, we can update the depth gradient with the desired pixels using:
	\begin{equation}
		\nabla \hat z^{\left(k\right)}=\bm{\mathcal R}\left(\nabla \hat z^{\left(k-1\right)},\bm S^{\left(k\right)}\right),
		\label{def.R}
	\end{equation}
	where $\bm {\mathcal R}$ denotes the depth gradient update function.

As shown in Fig. \ref{fig.dp}, after the initialization of depth gradient $\nabla \hat{z}$ and DP variables $\bm E$ and $\bm S$ (detailed in Sec. \ref{sec:Ini}), inliers are iteratively introduced using the smoothness energy transfer function $\bm{\mathcal T}$ (detailed in Sec. \ref{sec:C}) and the depth gradients are iteratively updated using the depth gradient update function $\bm{\mathcal R}$ (detailed in Sec. \ref{sec:R}). The optimum depth gradients can be obtained when the smoothness energy converges to the minimum. The pseudo-code of multi-directional DP is given in Algorithm \ref{alg:DP}.

		\begin{figure}[!t]

		\renewcommand{\algorithmicrequire}{\textbf{Input:}}
		\renewcommand{\algorithmicensure}{\textbf{Output:}}
		\begin{algorithm}[H]
			\caption{Multi-Directional DP}
			\label{alg:DP}
			\begin{algorithmic}[1]
				\REQUIRE Depth map $z$, Initial depth gradient $[\hat z_{u}^{\left(0\right)},\hat z_{v}^{\left(0\right)}]$, Initial smoothness energy components $E^{\left(0\right)}_u$ and $E^{\left(0\right)}_v$, Initial state variable components $\bm s_u^{\left(0\right)}$ and $\bm s_v^{\left(0\right)}$.
				\ENSURE Optimum depth gradient $[\hat z_{u}, \hat z_{v}]$
				\STATE {$[\hat z_{u},\hat z_{v}] \leftarrow [\hat z_{u}^{\left(0\right)},\hat z_{v}^{\left(0\right)}]$, \\
					$[E_{u},E_{v}] \leftarrow [E^{\left(0\right)}_u, E^{\left(0\right)}_v]$,\\
					$[\bm s_u,\bm s_v] \leftarrow [\bm s_u^{\left(0\right)},\bm s_v^{\left(0\right)}]$}

				\REPEAT

				\STATE {Initialize energy set $\Omega_u \leftarrow \varnothing,\Omega_v \leftarrow \varnothing$}
				\FOR {\textbf{each} ${\mathbf {p}}'$}
				\STATE {$\Omega_u[{\mathbf {p}}] \!\leftarrow \! \Omega_u[{\mathbf {p}}] \cup \mathcal T(E_u,E_v,{\mathbf {p}},{\mathbf {p}}',\bm s_u$})
				\STATE {$\Omega_v[{\mathbf {p}}] \leftarrow \Omega_v[{\mathbf {p}}] \cup \mathcal T(E_u,E_v,{\mathbf {p}},{\mathbf {p}}',\bm s_v$})\\
				\ENDFOR
				\STATE {$\bm s_u[{\mathbf {p}}] \leftarrow \mathop{\arg\min}\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\left\{\Omega_u[{\mathbf {p}}][{\mathbf {p}}']\right\} - {\mathbf {p}}$}
				\STATE {$\bm s_v[{\mathbf {p}}] \leftarrow \mathop{\arg\min}\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\left\{\Omega_v[{\mathbf {p}}][{\mathbf {p}}']\right\} - {\mathbf {p}}$}
				\STATE {$E_u[{\mathbf {p}}] \leftarrow \Omega_u[{\mathbf {p}}][{\mathbf {p}}+\bm s_u] $}
				\STATE {$E_v[{\mathbf {p}}] \leftarrow \Omega_v[{\mathbf {p}}][{\mathbf {p}}+\bm s_v] $}
				\STATE {$[\hat z_{u},\hat z_{v}] \leftarrow [\mathcal R(\hat z_{u},\hat z_{v}, \bm s_u),\mathcal R(\hat z_{u},\hat z_{v}, \bm s_v)]$}
				\UNTIL{all ($\bm s_u[:]=\bm 0$) \textbf{and} all ($\bm s_v[:]=\bm 0$)}
				\RETURN{$[\hat z_{u}, \hat z_{v}]$}
			\end{algorithmic}
		\end{algorithm}
 	\end{figure}

\subsubsection{Initialization of Depth Gradient, Smoothness Energy, and State Variable}
	\label{sec:Ini}
	In the initialization stage, the coarse depth gradient $\nabla \hat z^{\left( 0 \right)}$ of a given pixel is obtained by computing the finite difference (FD) between adjacent pixels. The FDs of depth are divided into $\Delta_f z = [\Delta_f z_u,\Delta_f z_v]^{\top}$ and $\Delta_b z = [\Delta_b z_u, \Delta_b z_v]^{\top}$, where the subscripts $u$ and $v$ represent the horizontal and vertical directions, respectively, and $\Delta_f$ and $\Delta_b$ represent forward and backward FDs, respectively. In order to achieve the discontinuity awareness ability, we set different weights to these FD operators by comparing the local smoothness measured by $\hat z_{uu}$ and $\hat z_{vv}$, the second-order partial derivatives of adjacent pixels, as follows:
	\begin{equation}
		\begin{split}
			\bm \eta=\begin{bmatrix}\eta_u,\eta_v\end{bmatrix}=\mathop{\arg\min}\limits_{[i,j]}\left\{|\hat z_{uu}(u+i,v)|+|\hat z_{vv}(u,v+j)|\right\}\\
		\end{split},
		\label{eq.ini.xy}
	\end{equation}
	where $i,j\in \{-1,0,1\}$. We can then use the smoothest pixels to linearly interpolate $\Delta_f z$ and $\Delta_b z$:
	\begin{equation}
		\begin{split}
			\nabla \hat z^{\left( 0 \right)}=\frac{1}{2}(\bm 1+\bm \eta)^{\top} \circ \Delta_f z + \frac{1}{2}(\bm 1-\bm \eta)^{\top} \circ \Delta_b z
		\end{split},
		\label{eq.ini.G}
	\end{equation}
	where $\circ$ denotes the Hadamard product and $\bm 1=[1,1]^{\top}$.
	Furthermore, we initialize the smoothness energy $\bm E^{\left(0\right)}$ as $\begin{bmatrix}\min\{|\hat z_{uu}(u+i,v)|\}, \min\{ |\hat z_{vv}(u,v+j)|\}\end{bmatrix}$ and the state variable $\bm S^{\left(0\right)}$ as $\mathrm{diag}(\eta_u,\eta_v)$.

	\subsection{Smoothness Energy Transfer Function}
	\label{sec:C}
This section discusses the formulation of the smoothness energy transfer function $\bm{\mathcal T}$. Energy transfer can be divided into two categories: 1) collinear transfer ($\bm s_u \times \mathbf e_1 = \bm 0$ or $\bm s_v \times \mathbf e_2 = \bm 0$, where $\mathbf e_1=[1,0]^\top$ and $\mathbf e_2=[0,1]^\top$ are the unit orthogonal base of 2D Euclidean space); 2) non-collinear transfer ($\bm s_u \times \mathbf e_1 \neq \bm 0$ or $\bm s_v \times \mathbf e_2 \neq \bm 0$). As a general rule, the collinear pixels should be introduced to adapt to the convex surface if they are smooth enough; otherwise, non-collinear pixels should be introduced to deal with discontinuities.

	\subsubsection{Depth Gradient Calculation}
	Based on the gradient theorem \cite{williamson2004multivariable}, the relationship among $z$, $z_u$, and $z_v$ can be written as follows:
	\begin{equation}
		z({\mathbf {p}}')-z(\mathbf {p})=\int_{\mathcal L}z_u\,du+z_v\,dv,
		\label{eq.PathIntegral.0}
	\end{equation}
	where $\mathbf {p} = [u_0,v_0]^{\top}$ is the given pixel, ${\mathbf {p}}' = [u_1,v_1]^{\top}$ is its adjacent pixel to be introduced (satisfying $|u_0-u_1| \le 1, |v_0-v_1| \le 1$), and $\mathcal L$ is the path from $\mathbf {p}$ to ${\mathbf {p}}'$. We can therefore obtain (i) the collinear transfer as follows:
	\begin{equation}
		\begin{split}
			& z({\mathbf {p}}')-z(\mathbf {p})=\int_{u_0}^{u_1} z_u\,du, 	\mbox{ or } \\
			& z({\mathbf {p}}')-z(\mathbf {p})=\int_{v_0}^{v_1} z_v\,dv,
		\end{split}
		\label{eq.PathIntegral.1}
	\end{equation}
	and (ii) the non-collinear transfer as follows:
	\begin{equation}
		z({\mathbf {p}}')-z(\mathbf {p})=\int_{\mathcal L_1}z_u\,du+z_v\,dv=\int_{\mathcal L_2}z_u\,du+z_v\,dv,
		\label{eq.PathIntegral.2}
	\end{equation}
	where $\mathcal L_1$ and $\mathcal L_2$ are two different paths from $\mathbf {p}$ to ${\mathbf {p}}'$.

	In practice, however, the discrete $\hat z_{u}$ and $\hat z_{v}$ can result in errors in (\ref{eq.PathIntegral.1}) and (\ref{eq.PathIntegral.2}). The errors of $\hat z_{u}$ and $\hat z_{v}$ can be estimated using the path integral of $z_{uu}$ and $z_{vv}$ based on the Taylor expansion (more details are provided in the supplement). Hence, in order to measure these errors in a more convenient way, we propose the PD norm, which denotes the path integral (sum) of a series of $|z_{uu}|$ and $|z_{vv}|$.

	\subsubsection{Path Discontinuity Norm}
	Let $z(u,v)$ be a function in $\mathbb{R}^2$ defined on an open set $\Omega$ and $\mathcal L$ be a specific path contained in $\Omega$. The discontinuity extent of the path can be measured using our proposed PD norm as follows:
	\begin{equation}
		||z||_{\text{PD}}= \int_{\mathcal L} |z_{uu}\,du|+|z_{vv}\,dv|.
		\label{def.PDN}
	\end{equation}
	After computing the PD norm, we can judge whether an adjacent pixel ${\mathbf {p}}'$ should be introduced. Therefore, we define the smoothness energy component as the PD norm with respect to a given path. In each iteration, we introduce two adjacent pixels which respectively minimize the smoothness energy components as follows:
	\begin{equation}
		\vspace{-3pt}
		\begin{split}
			& \bm s_u,\bm s_v=\mathop{\arg\min}\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\left\{||z||_{\text{PD}}\right\}-\mathbf {p},\\
			\mathcal L :
			& \begin{cases}
				\mathbf {p} \to {\mathbf {p}}'
				&,\mbox{if }\bm s_u \times \mathbf e_1 = \bm 0\,(\mbox{or }\bm s_v \times \mathbf e_2 = \bm 0) \\
				\mathbf {p} \to {\mathbf {p}}' \to \mathbf {p} &,\mbox{if }\bm s_u \times \mathbf e_1 \neq \bm 0 \,(\mbox{or }\bm s_v \times \mathbf e_2 \neq \bm 0),
			\end{cases}
		\end{split}
		\label{eq.PDN.xy}
	\end{equation}
	where $\mathcal N_1({\mathbf {p}})=\{{\mathbf {p}}':||{\mathbf {p}}'-\mathbf {p}||_\infty \le 1\}$, and the smoothness energy components are minimized accordingly:
	\begin{equation}
		\begin{split}
			& E_u({\mathbf {p}}),E_v({\mathbf {p}})=\min\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\left\{||z||_{\text{PD}}\right\}\\
			\mathcal L :
			& \begin{cases}
				\mathbf {p} \to {\mathbf {p}}'
				&,\mbox{if }\bm s_u \times \mathbf e_1 = \bm 0\,(\mbox{or }\bm s_v \times \mathbf e_2 = \bm 0) \\
				\mathbf {p} \to {\mathbf {p}}' \to \mathbf {p} &,\mbox{if }\bm s_u \times \mathbf e_1 \neq \bm 0 \,(\mbox{or }\bm s_v \times \mathbf e_2 \neq \bm 0).
			\end{cases}
		\end{split}
		\label{eq.PDN.C}
	\end{equation}
	The determination of co-planar pixels is, therefore, converted into an energy minimization problem.

	\subsubsection{Energy Minimization Strategy}
	Since depth is discrete, we formulate the energy transfer function $\bm{\mathcal T}$ in (\ref{def.C1}) as follows:
	\begin{equation}
		\begin{split}
			& E_u^{\left( k \right)}({\mathbf {p}}),E_v^{\left( k \right)}({\mathbf {p}})=\min\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\Big\{E^{\left( k-1 \right)}_p({\mathbf {p}}),\\
			& |\hat z_{pp}({\mathbf {p}}'_p)|+\mathbb{I}({\mathbf {p}},{\mathbf {p}}'_p), \\
			& 2 \cdot\left[|\hat z_{oo}({\mathbf {p}}'_o)|+E^{\left( k-1 \right)}_p({\mathbf {p}}'_o)
			\right],\\
			& |\hat z_{oo}({\mathbf {p}}'_o)|+E^{\left( k-1 \right)}_p({\mathbf {p}}'_o)+|\hat z_{pp}({\mathbf {p}}'_d)|+E^{\left( k-1 \right)}_o({\mathbf {p}}'_d)
			\Big\},
		\end{split}
		\label{eq.C}
	\end{equation}
	where the subscripts $p$, $o$, and $d$ respectively denote the variable which is parallel, orthogonal, and diagonal to the given axis (either $u$-axis or $v$-axis); the representations of $E_p$, $E_o$, $\hat z_{pp}$, $\hat z_{oo}$, $\mathbf {p}'_p$, $\mathbf {p}'_o$, and $\mathbf {p}'_d$ are given in the supplement; and the indicator function
		\begin{equation}
		\mathbb{I}({\mathbf {p}},{\mathbf {p}}'_p)=\\
		\begin{cases}
			0,
			& \mbox{if } (\hat z_{pp}({\mathbf {p}}+\bm s_p)\cdot \hat z_{pp}({\mathbf {p}}'_p+\bm s_p) > 0 ) \\
			&\wedge (\bm s_p({\mathbf {p}})=\bm s_p^{\left( k-1 \right)}({\mathbf {p}}'_p))\\
				\infty,
			& \mbox{otherwise }
		\end{cases}
		\label{eq.indicator}
	\end{equation}
	add constraints to the monotonicity and convexity of the smoothness energy transfer function, improving the DP adaptivity to convex surfaces. The four terms in (\ref{eq.C}), in turn, denote the smoothness energy components when we introduce 1) no other pixels, 2) a parallel pixel ${\mathbf {p}}'_p$, 3) an orthogonal pixel ${\mathbf {p}}'_o$, and 4) a diagonal pixel ${\mathbf {p}}'_d$.

	\subsection{Depth Gradient Update Function}
	\label{sec:R}
	This subsection discusses the formulation of the depth gradient update function $\bm{\mathcal R}$. Similar to $\bm{\mathcal T}$, depth gradient update can be divided into two categories: 1) collinear update ($\bm s_u \times \mathbf e_1 = \bm 0$ or $\bm s_v \times \mathbf e_2 = \bm 0$) and 2) non-collinear update ($\bm s_u \times \mathbf e_1 \neq \bm 0$ or $\bm s_v \times \mathbf e_2 \neq \bm 0$). For the collinear update, we use all the collinear pixels which satisfy the constraint in (\ref{eq.indicator}) to estimate depth gradients, while on the other hand, for the non-collinear update, we replace the depth gradient of each given pixel with the ones on other smoother paths using (\ref{eq.PathIntegral.2}).

	\subsubsection{Depth Gradient Collinear Update with Recursive Polynomial Interpolation}

	The optimum $\nabla \hat z$ can be obtained by interpolating the depth of $n$ collinear pixels into an $(n-1)$-th order polynomial and subsequently computing its derivative. Lagrange or Newton polynomial interpolation has redundant computations, as each pixel is repeatedly used for polynomial interpolation. Furthermore, it is incredibly complex to yield the closed-form solution of the polynomial's derivative. To simplify the depth gradient collinear update process, we introduce a novel RPI algorithm, which can update $\nabla \hat z$ recursively with only adjacent pixels until the farthest pixel is accessed.

	As $\hat z_u$ and $\hat z_v$ can be computed in the same way, we only provide the details on $\hat z_u$ computation in this paper. To compute $\hat z_u$ of a given pixel $\mathbf {p} = [u_0,v_0]^{\top}$ in the $k$-th iteration, we use a $(k+1)$-th order polynomial to interpolate the depth of $(k+2)$ pixels, which have the same vertical coordinates $v_0$. The $(k+1)$-th order polynomial $f_0^{\left( k+1 \right)}(u)$ interpolated by $\{(u_0,z(u_0)),(u_0+1,z(u_0+1)),\dots,(u_0+k+1,z(u_0+k+1))\}$ can be expanded into a recursive form as follows:
	\begin{equation}
		f_0^{\left( k+1 \right)}(u)=\frac{u_0+k+1-u}{k+1}f_0^{\left( k \right)}+\frac{u-u_0}{k+1}f_1^{\left( k \right)},
		\label{eq.interp.1}
	\end{equation}
	where the $k$-th order polynomials $f_0^{\left( k \right)}$ and $f_1^{\left( k \right)}$ are respectively interpolated by the first and the last $k+1$ pixels (the proof of the theorem is provided in the supplement).

	Since $\hat z_{u}^{\left(k-1\right)}(u_0)=\frac{df_0^{\left( k \right)}}{du}(u_0)$ and $\hat z_{u}^{\left(k-1\right)}(u_0+1)=\frac{df_1^{\left( k \right)}}{du}(u_0+1)$ have been computed in the last iteration,  we can update $\hat z_{u}(u_0)$ using:
	\begin{equation}
		\begin{split}
			& \hat z_{u}^{\left(k\right)}(u_0)=\frac{df_0^{\left( k+1 \right)}}{du}(u_0)\\
			& =\hat z_{u}^{\left(k-1\right)}(u_0)-\frac{1}{k+1}\cdot\left[z(u_0)-f_1^{\left( k \right)}(u_0)\right].
		\end{split}
		\label{eq.interp.2}
	\end{equation}
	Substituting $f_1^{\left( k \right)}(u_0)=z(u_0+1)-\frac{df_1^{\left( k \right)}}{du}(u_0+1)=z(u_0+1)-\hat z_{u}^{\left(k-1\right)}(u_0+1)$ into (\ref{eq.interp.2}) yields
	\begin{equation}
		\begin{split}
			& \hat z_{u}^{\left(k\right)}(u_0)=\hat z_{u}^{\left(k-1\right)}(u_0)-\\
			& \frac{1}{k+1}\cdot\left[z(u_0)-z(u_0+1)+\hat z_{u}^{\left(k-1\right)}(u_0+1)\right].
		\end{split}
		\label{eq.interp.3}
	\end{equation}

	\subsubsection{Depth Gradient Non-Collinear Update}
	Replacing the integral in  (\ref{eq.PathIntegral.2}) with summation yields:
	\vspace{5pt}
	\begin{equation}
		\hat z_p({\mathbf {p}}) \pm \hat z_o({\mathbf {p}}'_p)=\hat z_p({\mathbf {p}}'_o) \pm \hat z_o({\mathbf {p}}).
		\label{eq.DepthGradients}
	\end{equation}
	As the pixels ${\mathbf {p}}'_p$ are regarded as outliers in non-colinear update, we replace $\hat z_o({\mathbf {p}}'_p)$ with $\hat z_o({\mathbf {p}}'_d)$ or $\hat z_o({\mathbf {p}})$ based on the state variable. The depth gradient update function $\bm{\mathcal R}$ in (\ref{def.R}) can be rewritten as follows:
	\begin{equation}
		\begin{split}
			& \hat z_{u}^{\left(k\right)}({\mathbf {p}}),\hat z_{v}^{\left(k\right)}({\mathbf {p}}) = \\
			&    \begin{cases}
				\hat z_p^{\left(k-1\right)}({\mathbf {p}}),
				&\!\!\!\!\!\!\!\!\!\!\mbox{if }\bm s_u^{\left( k \right)},\bm s_v^{\left( k \right)}  =  \bm 0\\
				\hat z_p^{\left(k-1\right)}({\mathbf {p}})\pm\frac{1}{k+1}\left[z({\mathbf {p}}'_p)-z({\mathbf {p}})\right]\\-\frac{1}{k+1}\hat z_p^{\left(k-1\right)}({\mathbf {p}}'_p),
				&\!\!\!\!\!\!\!\!\!\!\mbox{if }\bm s_u^{\left( k \right)},\bm s_v^{\left( k \right)}  =  {\mathbf {p}}'_p-{\mathbf {p}}\\
				\hat z_p^{\left(k-1\right)}({\mathbf {p}}'_o),
				&\!\!\!\!\!\!\!\!\!\!\mbox{if }\bm s_u^{\left( k \right)},\bm s_v^{\left( k \right)}  =  {\mathbf {p}}'_o-{\mathbf {p}}\\
				\hat z_p^{\left(k-1\right)}({\mathbf {p}}'_o)\pm\\\left[\hat z_o^{\left(k-1\right)}({\mathbf {p}}) -\hat z_o^{\left(k-1\right)}({\mathbf {p}}'_d)\right],
				&\!\!\!\!\!\!\!\!\!\!\mbox{if }\bm s_u^{\left( k \right)},\bm s_v^{\left( k \right)} = {\mathbf {p}}'_d-{\mathbf {p}}.
			\end{cases}
		\end{split}
		\label{eq.R}
	\end{equation}
	The representations of $\hat z_p$ and $\hat z_o$ are given in the supplement. The four terms in (\ref{eq.R}), in turn, represent 1) the unchanged depth gradient, 2) parallel update via RPI algorithm, 3) orthogonal update, and 4) diagonal update. The orthogonal and diagonal updates are based on the gradient theorem \cite{williamson2004multivariable}.

	\section{Experiments}
	\subsection{Datasets and Evaluation Metrics}
	In this paper, we followed \cite{3F2N} and conducted comprehensive experiments to qualitatively and quantitatively evaluate the performance of our proposed SDA-SNE. More details on the 3F2N datasets\footnote{\url{sites.google.com/view/3f2n/datasets}} are available in \cite{3F2N}. Moreover, since range sensor data are typically noisy, we add random Gaussian noise of different variances $\sigma$ to the original 3F2N datasets to compare the robustness between our proposed SDA-SNE and other SoTA SNEs.

	Two evaluation metrics are used to quantify the accuracy of SNEs:  the average angular error (AAE) \cite{lu2017symps}:
	\begin{equation}
		e_{\text{A}}(\mathcal{M})=\frac{1}{P}\sum \limits_{k=1}^{P} \phi_k,\,\,\,\,\,\,
		\phi_k=\mathrm{cos^{-1}}\left(\frac{\langle{\mathbf n}_k,\hat{\mathbf n}_k\rangle}{||{\mathbf n}_k||_2||\hat{\mathbf n}_k||_2}\right),
		\label{eq.ea}
	\end{equation}
	and the proportion of good pixels (PGP) \cite{lu2017symps}:
	\begin{equation}
		e_{\text{P}}(\mathcal{M})=\frac{1}{P}\sum \limits_{k=1}^{P}\delta \left(\phi_k,\varphi_k\right),\,\,\,\,\,\,
		\delta=\begin{cases}
			0&(\phi_k>\varphi)\\
			1&(\phi_k\le\varphi)
		\end{cases},
		\label{eq.ep}
	\end{equation}
	where $P$ denotes the total number of pixels used for evaluation, $\varphi$ denotes the angular error tolerance, $\mathcal{M}$ represents the given SNE,  and ${\mathbf n}_k$ and $\hat{\mathbf n}_k$ represent the ground-truth and estimated surface normals, respectively.

	In addition to the above-mentioned evaluation metrics, we also introduce a novel evaluation metric, referred to as cross accuracy ratio (CAR), to depict the performance comparison between two SNEs as follows:
	\vspace{-5pt}
	\begin{equation}
		\vspace{-5pt}
		r_{\text{A}}(\mathcal{M}_1,\mathcal{M}_2)=\frac{e_{\text{A}}(\mathcal{M}_1)}{e_{\text{A}}(\mathcal{M}_2)}.
		\label{eq.ra}
	\end{equation}
	$\mathcal{M}_2$ outperforms $\mathcal{M}_1$ in terms of $e_\text{A}$ if $r_{\text{A}}>1$, and vice versa if $0<r_{\text{A}}<1$.

	\subsection{Implementation Details}
	\label{sec.imp}
	As discussed in Sec. \ref{sec:Methodology}, initial depth gradients can be estimated by convolving a depth image with image gradient filters, \eg, Sobel \cite{Sobel}, Scharr \cite{Scharr}, Prewitt \cite{Prewitt}, \etc. The second-order filters, \eg, Laplace \cite{Laplace}, can be used to estimate the local depth gradient smoothness $\hat z_{uu}$ and $\hat z_{vv}$. We utilize a finite forward difference (FFD) kernel, $\emph{i.e.}, [0,-1,1]$, as well as a finite backward difference (FBD) kernel, $\emph{i.e.}, [-1,1,0]$ to initialize (coarse) $\hat z_u^{\left( 0 \right)}$ and $\hat z_v^{\left( 0 \right)}$, and use a finite Laplace (FL) kernel, \emph{i.e.}, $[-1,2,-1]$ to estimate $\hat z_{uu}$ and $\hat z_{vv}$. We also conduct experiments w.r.t. different number of iterations to quantify the performance of our proposed SDA-SNE. By implementing the optimum maximum iteration, the trade-off between the speed and accuracy of our algorithm can be significantly minimized.

	\subsection{Algorithm Convergence and Computational Complexity}
	Our proposed SDA-SNE converges when the state variable and surface normal estimation remains stable. We can obtain the lower bound of the smoothness energy components as follows:
	\begin{equation}
		E_u({\mathbf {p}}),E_v({\mathbf {p}}) \ge\min\limits_{{\mathbf {p}}'\in \mathcal N_1({\mathbf {p}})}\Big\{|\hat z_{uu}({\mathbf {p}}')|,|\hat z_{vv}({\mathbf {p}}')|\Big\}
		\ge 0.
		\label{eq.C.lb}
	\end{equation}
    The smoothness energy decreases monotonously after each iteration. Therefore, all state variables stop transferring after finite iterations (the smoothness energy converges to a global minimum).

	The computational complexity of our proposed PRI algorithm is $\mathcal{O}(n)$ ($n$ denotes the number of interpolated pixels), when computing the depth gradient using (\ref{eq.R}). Therefore, RPI is much more efficient than Lagrange or Newton polynomial interpolation whose computational complexity is $\mathcal{O}(n^2)$.

	Furthermore, when the image resolution is $M\times N$ pixels, the computational complexity of our proposed SDA-SNE is $\mathcal{O}(lMN)$, where $l$ denotes the number of iterations in DP. In most cases, $l\ll M,N$ because DP  automatically stops when $|\hat z_{uu}|$ and $|\hat z_{vv}|$ no longer decrease. If we set a limit on $l$, SDA-SNE's computational complexity can be reduced to $\mathcal{O}(MN)$, which is identical to the computational complexities of 3F2N \cite{3F2N} and CP2TV \cite{CP2TV}.

	\subsection{Performance Evaluation}
	As our proposed multi-directional DP algorithm mainly aims at improving the performance of depth gradient estimation, it is compatible with any depth-to-normal SNEs. The AAE scores of SoTA depth-to-normal SNEs and such SNEs using our depth gradient estimation strategy are given in Table \ref{table.r}. It can be observed that by using our proposed depth gradient estimation strategy, the AAE scores of such depth-to-normal SNEs decrease by about 20-60\%. Since SDA-SNE based on CP2TV performs better than SDA-SNE based on 3F2N, we use CP2TV to estimate $n_z$ in the following experiments. The qualitative comparison among these SNEs is shown in Fig. \ref{fig.error}. It can be observed that our proposed SDA-SNE significantly outperforms 3F2N and CP2TV near/on discontinuities.

	\begin{table}[htb]
		\setlength\tabcolsep{6pt}
		\begin{center}
			\vspace{0in}
			\footnotesize
			\caption{Comparison of $e_\text{A}$ (degrees) between 3F2N and CP2TV w/ and w/o our depth gradient estimation strategy leveraged.}
			\label{table.r}
			\setlength\arrayrulewidth{0.7pt}
			\begin{tabular}{l|cc|cc}
				\hline
				\multicolumn{1}{c|}{\multirow{2}*{Datasets} }
				& \multicolumn{2}{c|}{3F2N}           & \multicolumn{2}{c}{CP2TV}                                                 \\

				\cline{2-5}

				& w/o SDA                                          & w/ SDA                               	& w/o SDA                                      & w/ SDA                            \\

				\hline

				Easy
				& 1.657    & \textbf{0.782}
				& 1.686    & \textbf{0.677}

				\\
				Medium
				& 5.686    & \textbf{4.535}
				& 6.015    & \textbf{4.379}

				\\
				Hard
				& 15.315   & \textbf{9.237}
				& 13.819   & \textbf{8.098}

				\\

				\hline
			\end{tabular}
		\end{center}
	\end{table}

	\begin{table*}[!t]
		\setlength\tabcolsep{5pt}
		\begin{center}
			\vspace{0in}
			\footnotesize
			\caption{ Comparison of $e_\text{A}$ and $e_\text{P}$ (with respect to different $\varphi$) among SoTA SNEs on the 3F2N datasets \cite{3F2N}.}
			\label{table.eleven_methods}
			\setlength\arrayrulewidth{0.7pt}
			\begin{tabular}{l|ccc|ccc|ccc|ccc}
				\hline
				\multicolumn{1}{c|}{\multirow{3}*{Method}} & \multicolumn{3}{c|}{$e_\text{A}$ (degrees) $\downarrow$} &  \multicolumn{9}{c}{$e_\text{p}$ $\uparrow$}                                                                                \\
				\cline{2-13}

				& \multicolumn{1}{c}{\multirow{2}*{ Easy}}              & \multicolumn{1}{c}{\multirow{2}*{ Medium}}    & \multicolumn{1}{c|}{\multirow{2}*{ Hard}}         &  \multicolumn{3}{c|}{\multirow{1}*{Easy}} & \multicolumn{3}{c|}{\multirow{1}*{Medium}} & \multicolumn{3}{c}{\multirow{1}*{Hard}}                                                                                \\
				\cline{5-13}
				&
				&                                             &                                              & $\varphi$=10$^\circ$                       & $\varphi$=20$^\circ$                       & $\varphi$=30$^\circ$ & $\varphi$=10$^\circ$ & $\varphi$=20$^\circ$ & $\varphi$=30$^\circ$
				& $\varphi$=10$^\circ$                       & $\varphi$=20$^\circ$                       & $\varphi$=30$^\circ$\\
				\hline

				PlaneSVD \cite{klasing2009realtime}
				& 2.07
				& 6.07
				& 17.59
				& 0.9648
				& 0.9792
				& 0.9855
				& 0.8621
				& 0.9531
				& 0.9718
				& 0.6202
				& 0.7394
				& 0.7914

				\\

				PlanePCA \cite{jordan2014quantitative}
				& 2.07
				& 6.07
				& 17.59
				& 0.9648
				& 0.9792
				& 0.9855
				& 0.8621
				& 0.9531
				& 0.9718
				& 0.6202
				& 0.7394
				& 0.7914

				\\

				VectorSVD \cite{klasing2009comparison}
				& 2.13
				& 6.27
				& 18.01
				& 0.9643
				& 0.9777
				& 0.9846
				& 0.8601
				& 0.9495
				& 0.9683
				& 0.6187
				& 0.7346
				& 0.7848

				\\

				AreaWeighted \cite{klasing2009comparison}
				& 2.20
				& 6.27
				& 17.03
				& 0.9636
				& 0.9753
				& 0.9819
				& 0.8634
				& 0.9504
				& 0.9665
				& 0.6248
				& 0.7448
				& 0.7977

				\\

				AngleWeighted \cite{klasing2009comparison}
				& 1.79
				& 5.67
				& 13.26
				& 0.9762
				& 0.9862
				& 0.9893
				& 0.8814
				& 0.9711
				& 0.9809
				& 0.6625
				& 0.8075
				& 0.8651

				\\
				FALS \cite{badino2011fast}
				& 2.26
				& 6.14
				& 17.34
				& 0.9654
				& 0.9794
				& 0.9857
				& 0.8621
				& 0.9547
				& 0.9731
				& 0.6209
				& 0.7433
				& 0.7961

				\\

				SRI \cite{badino2011fast}
				& 2.64
				& 6.71
				& 19.61
				& 0.9499
				& 0.9713
				& 0.9798
				& 0.8431
				& 0.9403
				& 0.9633
				& 0.5594
				& 0.6932
				& 0.7605

				\\

				LINE-MOD \cite{hinterstoisser2011gradient}
				& 2.64
				& 6.71
				& 19.61
				& 0.8542
				& 0.9085
				& 0.9343
				& 0.7277
				& 0.8803
				& 0.9282
				& 0.3375
				& 0.4757
				& 0.5636

				\\

				SNE-RoadSeg \cite{fan2020sne}
				& 2.04
				& 6.28
				& 16.37
				& 0.9693
				& 0.9810
				& 0.9871
				& 0.8618
				& 0.9512
				& 0.9725
				& 0.6226
				& 0.7589
				& 0.8113

				\\

				3F2N \cite{3F2N}
				& 1.66
				& 5.69
				& 15.31
				& 0.9723
				& 0.9829
				& 0.9889
				& 0.8722
				& 0.9600
				& 0.9766
				& 0.6631
				& 0.7821
				& 0.8289

				\\
				CP2TV \cite{CP2TV}

				& 1.69
				& 6.02
				& 13.82
				& 0.9740
				& 0.9843
				& 0.9899
				& 0.8512
				& 0.9554
				& 0.9755
				& 0.6840
				& 0.8099
				& 0.8562
				\\
				\hline
				\rowcolor{gray!20}
				SDA-SNE (iteration = $\infty$)
				& 0.68
				& \textbf{4.38}
				& \textbf{8.10}
				& \textbf{0.9947}
				& 0.9982
				& 0.9991
				& \textbf{0.9075}
				& \textbf{0.9868}
				& \textbf{0.9939}
				& \textbf{0.8035}
				& \textbf{0.9254}
				& \textbf{0.9461}
				\\

				\rowcolor{gray!20}
				SDA-SNE (iteration = $3$)
				& \textbf{0.67}
				& \textbf{4.38}
				& 8.14
				& \textbf{0.9947}
				& \textbf{0.9983}
				& \textbf{0.9992}
				& \textbf{0.9075}
				& 0.9867
				& 0.9938
				& 0.8027
				& 0.9174
				& 0.9453
				\\
				\hline
			\end{tabular}
		\end{center}
	\end{table*}

	\begin{figure}[!t]
		\centering
		\includegraphics[width=0.46\textwidth]{fig/fig_base.png}
		\caption{CAR comparison between CP2TV-based and 3F2N-based SDA-SNEs w.r.t. different maximum iterations.}
		\label{fig.base}
	\end{figure}

	Additionally, we compare the performance of our proposed SDA-SNE w.r.t. a collection of maximum iterations. Fig. \ref{fig.base} shows the CAR scores achieved by CP2TV-based and 3F2N-based SDA-SNEs on the 3F2N easy, medium, and hard datasets. It can be observed that the performance of SDA-SNE saturates after only several iterations. The accuracy increases by less than 5\% when the maximum iteration is set to infinity. Therefore, the maximum iteration of multi-directional DP can be set to 3 to minimize its trade-off between speed and accuracy. Moreover, it can be observed that our proposed SDA-SNE converges (the CAR score reaches the maximum) with more iterations on the hard dataset than on the easy and medium datasets. This is probably due to the fact that the hard dataset possesses more discontinuities than the easy and medium datasets. As a result, more distant pixels are required to be introduced to yield better depth gradients.

	\begin{figure}[!t]
		\centering
		\includegraphics[width=0.46\textwidth]{fig/fig_noised.png}
		\caption{AAE comparison among 3F2N, CP2TV, and SDA-SNE on the 3F2N datasets with different levels of Gaussian noise added.}
		\label{fig.noise}
	\end{figure}

	\begin{figure*}[!t]
		\centering
		\includegraphics[width=0.960\textwidth]{fig/fig_error.pdf}
		\caption{Examples of experimental results: Columns (\romannumeral1)-(\romannumeral5) show the depth images, surface normal ground truth, and the experimental results obtained using 3F2N, CP2TV, and our proposed SDA-SNE, respectively. Rows (a)-(c) show the results of the easy, medium, and hard datasets, respectively.}
		\label{fig.error}
	\end{figure*}

	Moreover, we compare our proposed SDA-SNE with all other SoTA SNEs presented in Sec. \ref{sec:RelatedWork}.  $e_\text{A}$ and $e_\text{P}$ of all SNEs on the 3F2N easy, medium, and hard datasets are given in Table \ref{table.eleven_methods}. The ${e_\text{A}}$ scores achieved by SDA-SNE are less than $1^\circ$ (easy), $5^\circ$ (medium), and $9^\circ$ (hard), respectively. The ${e_\text{P}}$ scores (tolerance: $10^\circ$) achieved by SDA-SNE are about 100\% (easy), 90\% (medium), and 80\% (hard), respectively. These results suggest that our proposed SDA-SNE performs significantly better than all other SoTA SNEs, no matter whether an iteration limit is added or not.

	Furthermore, we compare the robustness (to random Gaussian noise) of our proposed SDA-SNE with two SoTA depth-to-normal SNEs (3F2N \cite{3F2N} and CP2TV \cite{CP2TV}), as shown in Fig. \ref{fig.noise}. The logarithms of AAE scores and Gaussian variances are used to include the results of different datasets into a single figure. It is evident that (1) the $e_\text{A}$ scores achieved by all SNEs increase monotonically with the increasing noise level, and (2) our proposed SDA-SNE outperforms 3F2N and CP2TV at different noise levels. In addition, the compared SNEs' performance on the easy dataset degrades more dramatically than the medium and hard datasets, as the original easy dataset contains much fewer discontinuities. Although the performance of these depth-to-normal SNEs becomes remarkably similar, with the increase in noise level, our proposed SDA-SNE consistently outperforms 3F2N and CP2TV. This further demonstrates the superior robustness of our algorithm over others.

	\section{Conclusion}
	This paper presented a highly accurate discontinuity-aware surface normal estimator, referred to as SDA-SNE. Our approach computes surface normals from a depth image by iteratively introducing adjacent co-planar pixels using a novel multi-directional dynamic programming algorithm. To refine the depth gradient in each iteration, we introduced a novel recursive polynomial interpolation algorithm with high computational efficiency. Our proposed depth gradient estimation approach is compatible with any depth-to-normal surface normal estimator, such as 3F2N and CP2TV. To evaluate the accuracy of our proposed surface normal estimator, we conducted extensive experiments on both clean and noisy datasets. Our proposed SDA-SNE achieves the highest accuracy on clean 3F2N datasets ($0.68^\circ$, $4.38^\circ$, $8.10^\circ$ on the easy, medium, and hard datasets, respectively), outperforming all other SoTA surface normal estimators. It also demonstrates high robustness to different levels of random Gaussian noise. Additional experimental results suggest that our proposed SDA-SNE can achieve a similar performance when reducing the maximum iteration of multi-directional dynamic programming to $3$. This ensures the high efficiency of our proposed SDA-SNE in various computer vision and robotics applications requiring real-time performance.

	\section{Acknowledgements}

	This work was supported by the National Key R\&D Program of China, under grant No. 2020AAA0108100, awarded to Prof. Qijun Chen. This work was also supported by the Fundamental Research Funds for the Central Universities, under projects No. 22120220184, No. 22120220214, and No. 2022-5-YB-08, awarded to Prof. Rui Fan.

	\clearpage
	\balance
	{\small
		\bibliographystyle{ieee_fullname}

redpajama_set_name: RedPajamaArXiv


My mother passed away on Saturday. Norma Jean Bishop was 87 years old.
Probably the single most important lesson mom taught me was this: Those who are least loveable, are the most in need of love.
I have lived my life by this lesson. It's why I joined the peace corps. It's why I became a teacher. It's why I chose to run for office.
I am infinitely grateful for how long I had my mother in my life. Her memory lives on in all those who knew her, especially her family.
Thank you all for your support in this challenging time, and you are invited to join my family and me in a celebration of her life.


Meaning and definitions of paramount, translation in Samoan language for paramount with similar and opposite words. Also find spoken pronunciation of paramount in Samoan and in English language.
What paramount means in Samoan, paramount meaning in Samoan, paramount definition, examples and pronunciation of paramount in Samoan language.


Over the last few years we've had a few different plots on different allotment sites, but even now we are still learning and I think we are still considered new to the world of growing our own. We often find ourselves getting super excited when we have our first success, especially when it's something we have tried and failed with for several years. But one thing we've learnt from our allotment is not to give up and to always try again!
This year, with our plot now fenced in and winter dug we tried yet again with brassicas, something which we've grown every year but never quite managed to get from the plot to the kitchen. At the moment it looks like we might have our own Brussels sprouts and red cabbage ready for Christmas, which I am so excited about as they are my favourite winter veg!


text:
How will traditional holiday sales days fare this year?
Caroline Champlin Nov 29, 2021
https://www.marketplace.org/2021/11/29/how-will-traditional-holiday-sales-days-fare-this-year/
New data shows that in-store sales on this year's Black Friday were down compared with pre-pandemic levels. Above, the festive Galleria mall in Houston on Black Friday. Brandon Bell via Getty Images
In-store shopping was down on Black Friday compared with pre-pandemic times, according to data from Sensormatic Solutions. That was also true for e-commerce spending on the big sale day, based on analysis from Adobe.
On Cyber Monday, there was evidence that shoppers have been spreading their spending over more than just a few days.
Kinshuk Jerath, a digital marketing professor at Columbia Business School, said his email inbox was relatively quiet on Cyber Monday. "I don't feel like I'm being blasted by ads for the last three or four days about how cheap things have become," he said. "There's been some of that, but not as much as usual."
Retailers may not see the need for sales, Jerath said, because inventory is limited and many consumers seem to be heeding retailers' calls to shop early over multiple days.
This November has had more big online shopping days than last year, according to Vivek Pandya, an analyst with Adobe.
"This year, when we look at the amount of days that have seen $3 billion or more spent online, we've seen about 21 of those. When we compare it to the same period last year, we only saw about eight," he said.
That's a lesson retailers may keep in mind for next year as they weigh whether to launch promotions around traditional sale days.
"They're going to try to keep these days still there for consumers who like to shop on them, but then open up the impact to the broader season so that they can maximize revenues during that time too," Pandya said.
Some retailers may have the luxury of taking the emphasis off deal days. But that's not the case for others, like mall merchants and some clothing retailers.
"Cyber Monday is still very, very important for a lot of companies who have not been able to crawl out of the hole that the pandemic has put them in," said Sucharita Kodali with Forrester.
So for those companies, she said, Cyber Monday is still a lifeline.
How's the holiday shopping season starting to shape up?
Retail sales fell 1.1% in November, biggest drop in 7 months
From gifts to flights, it's best to buy early this holiday season
Feeling pressured? Americans pushed to overspend this holiday season
Holiday surge in e-commerce could cause delivery crunch
Holiday shopping will look different this year, with discounts coming early


By Rich Birch This weekend, all across the country, leaders are going to get up in front of their churches and talk about upcoming events and opportunities to connect with the community. They want to move people to action, but in reality a large portion of those in the room will simply tune out for that part of the service and then tune back in when something more interesting comes along. You know it's true … because you've done it!
Click here to read this article in the original publication on Leading Ideas.


Consiglieri Private Limited is conducting the Rapid Market System Analysis (RMSA) of Hilsa Market System in Southern part of Bangladesh. The study is primarily focused on Hilsa extractors and their households from 26 upazilas of 9 districts. This is a part of the USAID Project "Enhanced Coastal Fisheries (ECOFISH) Project.


text:
St. Valentine's Day Massacre, è un EP degli Headgirl pubblicato nel 1981 dall'etichetta discografica Bronze Records.

Il disco
Il disco è stato registrato quando il batterista dei Motörhead, Phil Taylor, si stava riprendendo da una ferita al collo.

Le due band (Motörhead e Girlschool), hanno deciso di collaborare usando il nome unico Headgirl. Il risultato è questo ep contenente tre tracce; una dei Motörhead suonata dalle Girlschool (Bomber); una delle Girlschool suonata dai Motörhead (Emergency); e una suonata da entrambi, ovvero la cover di Johnny Kidd & The Pirates Please Don't Touch.

Philty Animal è comunque presente nella copertina del disco che, ha raggiunto la posizione nº5 nelle classifiche inglesi.

Tracce

 "Please Don't Touch" (Heath, Robinson)
 "Bomber" (Lemmy Kilmister, Phil Taylor, Eddie Clarke)
 "Emergency" (Kim McAuliffe, Kelly Johnson, Enid Williams, Denise Dufort)

Formazioni
 Denise: batteria
 Kim: chitarra ritmica
 Kelly: chitarra, voce
 Eddie: chitarra, voce
 Enid: basso
 Lemmy: basso, voce

Note

Collegamenti esterni

EP dei Motörhead
EP delle Girlschool


text:
Infrastructure: Two bidders shortlisted to deliver East West Link
The Victorian Coalition Government today announced that two consortia had been shortlisted to continue to the final stage of bidding to deliver the eastern section of the East West Link.
Treasurer Michael O'Brien and Roads Minister Terry Mulder said that two stand out bids have been shortlisted to build and operate the first stage of the East West Link.
The shortlisted bidders are:
East West Connect; and
Inner Link Group.
"The Coalition Government is pleased with the high standard of bids to deliver what is one of the world's largest infrastructure projects out to tender at this time," Mr O'Brien said.
"The two consortia selected to proceed to the final stage include some of the world's most experienced infrastructure companies. The highly competitive nature of the tendering process will deliver the best value for money for Victorian taxpayers.
"The Government thanks the Momentum Infrastructure consortium for its high quality engagement and participation in the tendering process for the eastern section of the East West Link," Mr O'Brien said.
Mr Mulder said the eastern section of the East West Link is a critical project for not only Melbourne, but Victoria.
"The project will ease congestion for commuters, provide better access for freight and connect our freeway network," Mr Mulder said.
"The Coalition Government is on track to announce the successful tenderer in October, with construction to commence in late 2014," Mr Mulder said.
The next milestone for the project will be the planning approval decision. The Planning Minister must release the Assessment Committee's recommendations, and his response to those recommendations, by 30 June. The bidders must comply with any changes approved by the Minister arising from the Assessment Committee report.
Source: News Release Premier of Victoria Denis Napthine, 23 June 2014
http://www.premier.vic.gov.au/media-centre/media-releases/10276-two-bidders-shortlisted-to-deliver-east-west-link.html
August 12th, 2019 /
From Art Series Hotels to the art of the deal – a sold out evening with William Deague
Over 160 people came out on a chilly winter's evening for a 'fireside chat' with Deague Group CEO, William Deague and Ironfish CEO, Joseph Chou. Spilling out of the Ironfish North Sydney boardroom, attendees gathered to hear unique insights into the life and success of a prominent business-leader, w


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Personal injury can happen to anyone. Among the most common causes of personal injury are car and motorcycle accidents, slips and falls, and defective products. While many of us default to an "it won't happen to me" mindset, the very nature of these incidents being "accidents" means they are unpredictable and therefore have the potential to affect us all.
Some preventative measures can be taken to decrease one's chances of experiencing personal injury. Such actions include observing and obeying traffic laws and practicing proactive rather than strictly reactive driving; wearing bright or reflective clothing when walking or cycling in traffic areas; wearing appropriate footwear to suit various weather and walking conditions; taking note of one's physical surroundings and observing potential hazards such as spills or poor lighting; and exercising caution in alcohol-fueled situations in which violent acts are more likely to occur.
That being said, proactive behaviours are not a foolproof defense against personal injury. Recent examples of personal injury in Cambridge span a wide spectrum. A 45-year-old woman recently sustained life-threatening injuries when her motorcycle was struck by a pickup truck just south of Cambridge. An assault outside a Burger King in nearby Waterloo left a 24-year-old man with non-life-threatening injuries. A young man from Kitchener exited his vehicle after being rear-ended when the offending driver suddenly took off, trapping the victim between the two vehicles and causing severe bruising to his legs. Several Ontarians filed claims against a testosterone replacement therapy provider for failing to warn customers of increased cardiovascular risks.
Personal injury, in addition to physical suffering, can affect one's ability to earn income, keep up with household tasks and expenses, care for oneself and dependents, and receive satisfaction from life at a level comparable to before the accident. The psychological ramifications of personal injury and the subsequent changes to one's everyday existence are not to be ignored, either. As though an injury and its continuing consequences are not agony enough, negotiating the complexities of an insurance claim often cause more headache than healing. Many victims of personal injury do not receive the full compensation to which they are entitled simply because they lack the necessary knowledge of claims proceedings. For these reasons, hiring a personal injury lawyer to represent an injured party expedites the claims process and typically results in speedier, more substantial compensation.
The lawyers at the Injury Lawyers of Ontario (ILO) in Cambridge are experienced in handling a diverse range of personal injury claims. Our areas of practice include car and motorcycle accidents, automobile accident benefits, catastrophic injury, product liability, brain and spinal cord injury, disability claims, slip and fall injury, and wrongful death. Working with only the most trusted firms in Cambridge and across Ontario, the ILO group uses our collective expertise to ensure those suffering from personal injury receive their deserved compensation in full.