text:
PFK CSKA Moskva (ryska: ЦСКА Москва, Центральный Спортивный Клуб Армии, CSKA Moskva, PFK Tsentralnyj Sportivny Klub Armii Moskva) är fotbollssektionen i den ryska idrottsföreningen CSKA Moskva.

Laget vann UEFA-cupen säsongen 2004/2005, efter en final mot Sporting Lissabon. Därmed blev CSKA Moskva det första ryska laget med en europeisk cuptitel. Kända namn i dagens lag är bland andra skyttekungen från säsongen 2008 i Premjer-Liga Vagner Love, den ryska landslagsmålvakten Igor Akinfejev och den stora ryska supertalangen Alan Dzagojev. Laget spelar sina hemmarmatcher på Luzjnikistadion (kapacitet: 84 745 åskådare) i Moskva. Den ryska miljardären Roman Abramovitj har genom sitt oljebolag Sibneft investerat stora summor i laget. Många ryska landslagsspelare spelar för klubben.

CSKA har vid flera tillfällen deltagit i Uefa Champions League, men man har aldrig vunnit turneringen. Säsongen 2009/2010 nådde laget kvartsfinal där Inter blev för svåra.

Meriter

 UEFA-cupen: 1
 2005

 Ryska Premier League: 6
 2003, 2005, 2006, 2013, 2014, 2016

 Ryska cupen: 7
 2002, 2005, 2006, 2008, 2009, 2011, 2013

 Ryska Super Cupen: 6 (rekord)
 2004, 2006, 2007, 2009, 2013, 2014

 Sovjetiska mästare: 7
 1946, 1947, 1948, 1950, 1951, 1970, 1991

 Sovjetiska Cupen: 5
 1945, 1948, 1951, 1955, 1991

 Första Kanalen Cupen: 1
 2007

Spelare

Spelartruppen

Pensionerade nummer
12 – Truppnummer 12 är reserverat för supportrarna, "den tolfte spelaren"

16 –  Serhij Perkhun, Målvakt, 2001

Tränarstaben

Referenser

Noter

Externa länkar
 Officiell webbplats  &
 CSKA Ultras Fansajt
 Videoklipp om CSKA
 Fansajt

Fotbollsklubbar i Ryssland
Sportklubbar bildade 1911
CSKA Moskva
Vinnare av Uefacupen

redpajama_set_name: RedPajamaWikipedia


With Solution-Focused Scaling Questions it is important to use effective scale anchors. This is particularly the case with the 10 position. Scales usually work best when the 10-position is defined not in too idealistic terms (the ideal future) but rather in more realistic terms (the desired situation, the situation you would be satisfied with). Being idealistic in your definition of the 10-position has two disadvantages. The first is that you can be sure that an ideal situation will never be achieved. Problem free, ideal situations don't happen. There are always problems, challenges, and tensions, they belong to life. A second disadvantage of the 10 as ideal is that it will make the client scale the current situation lower. When the 10 represents an ideal situation the client may score the current situation as a 2, while he may score a 4 or a 5 when the 10 would be defined as the situation that would be good enough. A too idealistic 10 can demotivate.
It looks interesting doesn't it?


text:
Home / Companies / Motorola Mobility banks on the success of Moto G and Moto X
Motorola Mobility banks on the success of Moto G and Moto X
3 min read . Updated: 27 Mar 2014, 12:36 PM IST Leslie D'Monte
According to research firm International Data Corporation (IDC), India currently sells around 40 million smartphones annually, a number that is forecast to touch 155 million in 2017. Photo: Nelson Almeida/AFP
Motorola Mobility banks on the success of its Moto G and Moto X to boost its market share in India
Mumbai: Motorola Mobility Llc is banking on the success of its two Android-based mobile phones, Moto G and Moto X, to boost its market share in the highly competitive and rapidly growing smartphone segment in India, said a senior company official on Thursday.
In India, Motorola has tied up exclusively with online retailer Flipkart.com to sell these two models, one of the very few companies using the Internet-only business model. It launched Moto G in January and Moto X this month.
Of India's 1.2 billion population, only around 200 million are Internet users, yet the gamble of selling online is paying off for the company, according to Magnus Ahlqvist, corporate vice-president (strategy) of Motorola Mobility's EMEA and APAC regions.
"We have a very strong proposition at an exceptional price. We don't give out sales numbers but the performance of Moto G has been way beyond our expectations," said Ahlqvist in a phone interview on Wednesday.
Industry analysts agree.
"They've sold about 250,000 Moto G handsets so far. That's a real good number for a product that was launched just six weeks back. I believe the price point is working for them," said an analyst, who did not want to be named.
The 16 GB variant of Moto G is priced at ₹ 13,999 and the 8GB at ₹ 12,499 on Flipkart.
Ahlqvist, who joined Motorola Mobility in August, declined to comment on the veracity of these numbers.
On 6 February, Mint reported that the 16 GB variant of the Moto G phone launched that day was "sold out" within the first 15 minutes, and another set of 16 GB variant offered 12 hours later also ran out of stock within an hour.
A Flipkart spokesperson then said the Motorola exclusivity was a long-term one, which Ahlqvist confirmed.
On 19 March, Motorola introduced Moto X in India, again exclusively on Flipkart, priced between ₹ 23,999 and ₹ 25,999. "Moto X has been very well-received across markets and would continue to add to Motorola's momentum in India," said Ahlqvist.
Motorola Mobility plans to introduce Moto 360—a smartwatch designed by Motorola and powered by Android Wear (extending Android to wearables like Google Glass, smartwatches, etc.)—globally in the second half of 2014, according to a blog post by the company.
Ahlqvist didn't say when the Moto 360 will be introduced in India.
Motorola Mobility's smartphone business is in the process of being acquired by the world's largest personal computer maker China-based Lenovo Corp. On 29 January, Google Inc. said it will sell the business to Lenovo for $2.91 billion. It had acquired Motorola for $12.5 billion in 2012.
According to a 29 January report by Strategy Analytics, the combined entity of Lenovo and Motorola accounted for a 6% share of global smartphone shipments in 2013.
As a result of this new deal—assuming it is approved by the US, Chinese and other authorities—Lenovo-Motorola "becomes instantly the world's third-largest smartphone vendor by volume, behind Samsung Electronics Co. Ltd (32%) and Apple Inc. (15%)".
According to research firm International Data Corporation (IDC), India currently sells around 40 million smartphones annually, a number that is forecast to touch 155 million in 2017.
Lenovo's market share in India is negligible (less than 5%) currently, with the market dominated by Samsung, Micromax and Karbonn, according to research firm Convergence Catalyst. Samsung has more than a 30% share, Micromax has about 22% and Karbonn about 12%. According to the research firm, 41-43 million smartphones were sold in India in 2013.
With the Motorola acquisition, Lenovo will gain in India too where it was a latecomer in the smartphone segment, having introduced its first phones sometime in 2012.
"But for that to happen, Motorola too will have to have launched a few more models. It cannot sustain its success with just two models," said Jayanth Kolla, co-founder and partner at Convergence Catalyst.
Ahlqvist agreed that the online-only sales model "could be questioned even as we remain committed to it" but added that Motorola hopes to "broaden its offerings of models in the three-six months even as we remain extremely disciplined when it comes to launching new models".
In 2014, Convergence Catalyst estimates smartphone sales to reach 60 million, forming close to 35-40% of the total annual mobile handset sales in the country.
Trends such as Flipkart's exclusive partnership with Motorola for selling Moto G and Snapdeal's exclusive partnership with Intex for selling Aqua Octa are expected to intensify in 2014, according to Kolla.
Online retailers have emerged as a strong distribution channel for smart devices, especially smartphones, in the country, he said.
"Their no-frills operations (of e-commerce firms), lean supply chain, direct access to consumers and growing brands (of some of the leading firms) make them the preferred partners for many smartphone OEMs (original equipment manufacturers)," said Kolla.
Leslie D'Monte
Leslie D'Monte has been a journalist for almost three decades. He specialises in technology and science writing, having worked with leading media groups--both as a reporter and an editor. He is passionate about digital transformation and deep-tech topics including artificial intelligence (AI), big data analytics, the Internet of Things (IoT), blockchain, crypto, metaverses, quantum computing, genetics, fintech, electric vehicles, solar power and autonomous vehicles. Leslie is a Massachusetts Institute of Technology (MIT) Knight Science Journalism Fellow (2010-11). In his other avatar, he curates tech events and moderates panels.
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redpajama_set_name: RedPajamaCommonCrawl


What If the District Attorney in Orange County Doesn't File Criminal Charges On Time?
When you get arrested or you bail out you are given a date to appear in court. When you are given a citation you are given a specific date to appear in court. The police are supposed to submit their reports to the prosecutor or city attorney in enough time for them to review the case so that everybody is ready to proceed on the day in question. But what happens if you show up at the court house and your name is not on the calendar?
The case was rejected, which means no charges will be filed against you.
The case was sent back for further investigation by the arresting agency.
The DA's office hasn't received anything from the arresting agency yet, so they have no information yet.
Unless the case was rejected outright, it is still active even though nothing has been filed yet. You will receive a letter in the mail telling you when the new date and time has been set for your arraignment and it will be sent to the address you gave when you were arrested. If you have an attorney, the attorney can be in contact with the DA or City Attorney before the new date is set.
There are occasions when the attorney can be in contact with the DA's office before the case is filed. In the Law Office of Rudolph E. Loewenstein, we check on filings several times a week for our clients. In some cases we can intervene to stop a filing. At the very least we will be aware of when the case will be heard. There have been times when the client never receives the letter from the court in which case a bench warrant will be issued for his arrest. Of course, no one wants that to happen so being proactive can prevent a lot of problems later.


text:
from google.appengine.api import taskqueue
from google.appengine.ext import blobstore
from google.appengine.ext.webapp import blobstore_handlers
from google.appengine.api import memcache
from google.appengine.ext import webapp
from google.appengine.ext.webapp import util
import collection_cache
import os
from google.appengine.ext.webapp import template
import oauth2
import oauth
import logging
from gaesessions import get_current_session
import evernotebookparser
import springpad

APPLICATION_KEY = "d21f1c131fb14c769f54872d0d5a8dc6"
APPLICATION_SECRET = "a65db20e8f8844b282a133026a5c639d"

class MainHandler(webapp.RequestHandler):

    def get(self, mode=""):
        templateName = 'templates/index.html'
        data = {}

        session = get_current_session()
        callback_url = "%s/work" % self.request.host_url
        client = oauth2.SpringpadClient(APPLICATION_KEY, APPLICATION_SECRET,
        callback_url)
        #session.clear()
        if session.has_key("token"):
            token = session.get("token")
            secret = session.get("secret")
            results = client._lookup_user_info(token,secret)
            data = results
            session["userInfo"] = results
            templateName = 'templates/worker.html'
            upload_url = blobstore.create_upload_url('/upload')
            data['upload_url'] = upload_url
            if self.request.get('workbook'):
               data['total'] = 'You uploaded %s bytes. Your items will be copied into a new springpad notebook named "%s". Feel free to upload another while you wait.' % (self.request.get('size'),self.request.get('workbook'))

        path = os.path.join(os.path.dirname(__file__),templateName)
        self.response.out.write(template.render(path,data))

class UploadHandler(blobstore_handlers.BlobstoreUploadHandler):
    def post(self):

        session = get_current_session()
        workbook = self.request.get("workbook")
        upload_files = self.get_uploads('content')
        blob_info = upload_files[0]

        token = session.get("token")
        secret = session.get("secret")
        userInfo = session["userInfo"]

        service = springpad.SpringRpcService()
        service.fetcher = springpad.OAuthFetcher(APPLICATION_KEY, APPLICATION_SECRET,access_token=oauth.OAuthToken(token,secret))
        service.set_user_context(userInfo['username'], userInfo['uuid'])

        mutator = service.get_mutator()
        notebook = mutator.create("Workbook")
        mutator.set(notebook, 'name', workbook)
        mutator.commit()

        taskqueue.add(url='/queue', params={'token': token,
                                            'secret': secret,
                                            'username':userInfo['username'],
                                            'uuid':userInfo['uuid'],
                                            'content':blob_info.key(),
                                             'workbook' :notebook.uuid
                                            })
        self.redirect('/?workbook=%s&size=%s'%(workbook, blob_info.size))

class LoginHandler(webapp.RequestHandler):
    def get(self):

        callback_url = "%s/work" % self.request.host_url
        client = oauth2.SpringpadClient(APPLICATION_KEY, APPLICATION_SECRET,
        callback_url)
        return self.redirect(client.get_authorization_url())

class ValidationHandler(webapp.RequestHandler):
    def get(self):
        callback_url = "%s/work" % self.request.host_url
        client = oauth2.SpringpadClient(APPLICATION_KEY, APPLICATION_SECRET,
        callback_url)
        session = get_current_session()
        auth_token = self.request.get("oauth_token")
        results = client.get_user_info(auth_token)
        session["token"] = results["token"]
        session["secret"] = results["secret"]
        return self.redirect("/")

class ImportWorker(webapp.RequestHandler):
    def post(self):
        workbook = self.request.get("workbook")
        token =  self.request.get("token")
        secret =  self.request.get("secret")
        username = self.request.get("username")
        uuid = self.request.get("uuid")
        blobkey = self.request.get("content")
        try:
            service = springpad.SpringRpcService()
            service.fetcher = springpad.OAuthFetcher(APPLICATION_KEY, APPLICATION_SECRET,access_token=oauth.OAuthToken(token,secret))
            service.set_user_context(username, uuid)
            mutator = service.get_mutator()

            blob_reader = blobstore.BlobReader(blobkey)
            parser = evernotebookparser.NotebookParser2(blob_reader)
            parser.get_items(lambda x: createItem(mutator,workbook,x))
            logging.info("Serviced a customer")
            blob_reader.close()
            blobstore.delete(blobkey)
        except Exception,e:
            logging.exception("-------%s"%e.message)
            try:
                blobstore.delete(blobkey)
            except :
                pass
    def handle_exception(self,exception, debug_mode):
        logging.exception("-------%s"%exception.message)
        self.set_status(200)

def createItem(mutator,workbook, item):
    if not item:
        return

    logging.info("Creating an item")
    if "url" in item:
        note = mutator.create("Bookmark")
        mutator.set(note,"url", item['url'])
    else:
        note = mutator.create("Note")
        mutator.set(note, 'rich', True)

    mutator.set(note, 'name', item['title'])
    mutator.set(note, 'tags', item['tags'])
    mutator.set(note,"text", item['content'])
    mutator.set(note, 'workbooks', [workbook,])

    mutator.commit()

def main():
    application = webapp.WSGIApplication([('/', MainHandler),
                                          ('/work', ValidationHandler),
                                          ('/login', LoginHandler),
                                          ('/queue', ImportWorker),
                                          ('/upload', UploadHandler),],
                                         debug=True)
    util.run_wsgi_app(application)

if __name__ == '__main__':
    main()

redpajama_set_name: RedPajamaGithub


Mission Street Parks Conservancy works with the City of Salem to care for the trees, shrubs, and plantings in Bush's Pasture Park and other historic parks in the City.
Your donation of $100.00 or more entitles you to an annual membership with the Conservancy. A gift of $5,000 or more entitles you to Lifetime membership with the Conservancy.
Please contact us at info@missionstreetparks.org or (503) 588-2410 if you would like to make a gift of stock or other securities. And, of course, please feel free to contact us if you have any questions.


Tom and Schuyler are wrapping up their first deploy with the World Bank to Haiti. They've been doing amazing work in connecting the various participants on the ground in sharing data and providing them with lightweight tools and data from the broader web, CrisisMappers, and CrisisCommons community.
Schuyler has been writing up his experiences but due to connectivity and the large amount of work, they've been utilizing Twitter: @schulyer and @buckley_tom, continuing to keep the world up to date on their progress, safety, and insights. They have even managed to get out video discussing the use of OpenStreetMap by UN OCHA and other reconstruction efforts.
The USB Drives and USB Sticks that we sent down have seemed to be incredibly useful. The ability to utilize dynamic information and map interfaces in an offline, or semi-disconnected environment is empowering, while also allowing people to still print their maps with the data. We're now working on deploying these same tools to Afghanistan and other areas that there has been a large amount of data gathered but not broadly distributed.
Tom and Schuyler head back to the U.S. tomorrow, but the job is far from done. Mikel's Humanitarian OpenStreetMap Team (H.O.T.) is looking to deploy at the end of March to provide continue logistics support and begin training Haitians on building and utilizing the maps - much like the MapKibera project. Haiti has perhaps an excellent opportunity to leap-frog the use of open data and technology by empowering the government and people with tools they can use in reconstruction.
The platforms continue to evolve, incorporating lessons learned from user needs, to new data sources, and working in varied environments. We'll continue to provide data and maps through GeoCommons that are usable by everyone, and distribute tools to anyone who needs them.


text: Q: Is there a way to deploy multiple packaged components sharing a certain package version using Boomi platform API? Normally to deploy a packaged component using Boomi platform API, we use a packaged component Id and this end in deploying just one packaged component (whose id is provided). However, I want to deploy a list of packaged components sharing the same package version in one api call, is that possible?

redpajama_set_name: RedPajamaStackExchange


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I made a mold and cast the tunnel portals from plaster and placed them in position. Factory cotton was rolled up and placed in various hollows under the contours of hills, in tunnels and so forth. This, we found, was a satisfactory method of eliminating much of the noise on the B&R. While all this was going on, we had proceeded with plans for our buildings and built them according to scale from drawing board, paper and balsa wood, using Tracolene for our win- R•i33,'9».kf#• 4 the engineer at the control panel sim- uwa/7 4 .:fil:•li 00 a cab control system that even the upstairs president could run on her first AUTOMATIC -••I- LEADEAN HO 9.• 26254Cetat' EQUIPMENT Lights1 Action1 Everything here1 SEMAPHORE SIGNAL Works like a charm! Signal arm moves automatically showing red or green light. Detailed and precision made for trouble-free operation. 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Harvester ......$2.95 1913 Mercer Raceabout ..$3.95 1906 "Old 16" Loco. ._$3.95 Measures 636• in length 1903 Cadillac _036036...........$2.50 1902 Franklin ......................$2.50 \.036*I· 1 1900 Packard ..__..........$2.50 40 --'•• 1903 Rambler -_........_.$2.50 t··9•f•0364•&j•/•••• 0424 •kiaL . Al#Q .•6'p. 1906 Columbia Electric when complete __$2.50 Write to us. Add 250 for postage and handling. No C.O.D.'s please. mmommE//52& ..=..•lifT,"•COG January 1952 ' pike for a night effect. We covered a piece of screening with broken matchsticks to represent pulp, and also utilized tea leaves glued on screening, which resembled very closely the pile of bark seen at most pulp mills in Canada. We were now ready to continue landscaping operations. The next operation found us making a mixture of powdered asbestos and glue. This we applied liberally over the screening, cotton and flat surfaces to give the effect of rough terrain. We slapped it E : 042 r •,1, HO LOST WAX 1 •,Bliti BRASS LOCO (, ZLi5 ACCESSORIES PROTOTYPE DETAILING : No. L-450-Pop Valve Cluster...65c 042Complete assemblies -Precision made l : • No. L-210 --4*0*1 Headlight 65c You'll love 'ein / (••\1 4•• p..A••illimi .--,I....... '••.*••*1•I 1- /./1 . One of 6 K-K Pilots : ==,4422=il. : • : --laimm•r-042 KnilljN•= No. L-102-Commonwealth Swing Door Type Pilot...$ 1.25 See the Complete K-K Line 1 Sold by All Leading Dealers .D 25 -0361 .:..1.,#Al. lili ,/. 7.. I.-- Semaphore, Color Light and Lamp Post 042 e e 42 ..· .


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text:
Greenville County Sheriff's Office deputy shoots person; officer is also injured
A person was shot by a Greenville County Sheriff's Office deputy Sunday at a home on Cliffview Court. The deputy was also injured, officials said.
Greenville County Sheriff's Office deputy shoots person; officer is also injured A person was shot by a Greenville County Sheriff's Office deputy Sunday at a home on Cliffview Court. The deputy was also injured, officials said. Check out this story on greenvilleonline.com: https://www.greenvilleonline.com/story/news/local/south-carolina/2019/10/20/person-shot-greenville-county-sheriffs-office-deputy/4047006002/
Nikie Mayo, The Greenville News Published 3:40 p.m. ET Oct. 20, 2019 | Updated 12:19 p.m. ET Oct. 23, 2019
A person was shot by a Greenville County Sheriff's Office deputy Sunday at a home on Cliffview Court in Greer. The deputy was not shot but was also injured, according to a prepared statement from the agency.
The shooting happened around 12:30 p.m., after the deputy arrived to do a follow-up investigation. The statement from the Sheriff's Office does not address the nature of the follow-up investigation or what happened in the moments before the shooting occurred.
The deputy shot the person "at least once," according to the Sheriff's Office statement. The person's condition is not known.
Here's why... Some police shootings in South Carolina aren't captured on body camera.
The State Law Enforcement Division is investigating the shooting, and the Sheriff's Office is referring additional questions to that agency. SLED typically is called in to investigate shootings involving South Carolina law enforcement officers.
The deputy was put on administrative leave following the shooting, according to Sheriff's Office spokesman Lt. Ryan Flood.
The Sheriff's Office said a "critical incident briefing video" showing what happened would be released Dec. 4.
The Greenville News recently concluded a six-month investigation into the use of lethal force by South Carolina law enforcement officers. Among the investigation's findings is that from 2009 to 2018, the Greenville County Sheriff's Office had more shootings than any other state agency. Greenville County Sheriff's Office had 31 shootings in that period, according to an analysis of public records.
The Greenville County shooting is the second one this weekend that involves Upstate deputies.
Links to multiple shootings: Some Greenville deputies tied to multiple shootings have extensive misconduct backgrounds
The other one, in Pickens County, was the fatal shooting of a man at a home in Easley.
SLED spokesman Tommy Crosby said Sunday that he could confirm his agency is investigating both the Greenville and Pickens County shootings, but he said he had no other information on either of them.
The Greenville News has filed Freedom of Information requests seeking dash camera and body camera footage of both shootings and is also seeking any 911 calls connected to the incidents.
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text:
\section{Introduction}
Visual correspondence lies at the heart of image understanding, being used as a core component for numerous tasks such as object recognition, image retrieval, motion estimation, object tracking, and reconstruction~\cite{forsyth:hal-01063327}.
With recent advances in deep neural networks \cite{he2016deep,hu2017senet,huang2015dense,krizhevsky2012imagenet,simonyan2015vgg}, there has been substantial progress in learning robust feature representation for establishing correspondences.
Despite the effectiveness of deep convolutional features, however, spatial matching with a geometric constraint is still essential to handle image pairs with large variations, \eg, viewpoint and illumination changes, blur, occlusion, lack of texture, etc. In particular, the presence of intra-class variations, \ie, scenes depicting different instances of the same categories, remains a critical challenge for correspondence~\cite{ham2016proposal, han2017scnet, huang2019dynamic, kim2017dctm, lee2019sfnet, liu2020semantic, min2019hyperpixel, min2020dhpf, rocco17geocnn, rocco18weak, rocco2018neighbourhood}.
The process of geometric matching is the de facto solution of choice, which most recent methods adopt in their models.

Geometric matching commonly relies on exploiting a geometric consensus of candidate matches to verify relative transformations. In computer vision, RANSAC~\cite{fischler1981ransac} and Hough transform~\cite{hough1962transform} have long been used as geometric verification for wide-baseline correspondence problems with rigid motion models, while graph matching~\cite{cho2013learning, cho2010reweighted, fey2020deep, rolinek2020deep} has played a main role in matching deformable objects with  non-rigid motion.
Recent work~\cite{cho2015unsupervised,han2017scnet,min2019hyperpixel,min2020dhpf} has advanced the idea of Hough transform to perform non-rigid image matching, showing that the Hough voting process incorporated in neural networks is effective for challenging correspondence problems with intra-class variations.
However, their matching modules are neither fully differentiable nor learnable, and weak to background clutter due to the position-invariant global Hough space.

\begin{figure}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures/0_teaser.pdf}
    \end{center}
    \vspace{-5.0mm}
      \caption{Convolutional Hough matching (CHM) establishes reliable correspondences across images by performing position-aware Hough voting in a high-dimensional geometric transformation space, \eg, translation and scaling.}
    \vspace{-5.0mm}
\label{fig:teaser}
\end{figure}

In this work we introduce {\em Convolutional Hough Matching} (CHM) that distributes similarities of candidate matches over a geometric transformation space and evaluates them in a convolutional manner. As illustrated in Fig.~\ref{fig:teaser}, the convolutional nature makes the output equivariant to translation in the transformation space and also attentive to each position with its surrounding contexts, thus bringing robustness to background clutter.
We design CHM as a learnable layer with an semi-isotropic high-dimensional kernel that acts on top of a correlation tensor.
The CHM layer is compatible with any neural networks that use correlation computation, allowing flexible non-rigid matching and even multiple matching surfaces or objects.
It naturally generalizes existing 4D convolutions~\cite{huang2019dynamic, li2020correspondence, truong2020glunet, rocco2018neighbourhood} and provides a new perspective of Hough transform on convolutional matching.
To demonstrate the effect, we propose the neural network with CHM layers that perform convolutional matching in the high-dimensional space of translation and scaling. Our method clearly outperforms state-of-the-art methods on standard benchmarks for semantic correspondence, proving its strong robustness to challenging intra-class variations.
\section{Related Work}

\smallbreak
\noindent \textbf{Hough transformation.}
The Hough transform~\cite{hough1962transform} is a classic method developed to identify primitive shapes in an image via geometric voting in a parameter space.
Ballard~\cite{ballard1981generalhough} generalizes the idea to identify positions of arbitrary shapes with R-table.
Early approaches~\cite{chen2013robust, cho2012progressive} in computer vision widely adopt Hough transform for its effectiveness in extracting features of a particular shape in an image.
As a representative example, Leibe \etal\cite{leibe2003interleaved} introduce a Hough-based object segmentation and detection method by incorporating information about supporting patterns of parts for the target category.
The idea of Hough voting has widely been adopted in diverse tasks including retrieval~\cite{huan2017vehicle}, object discovery~\cite{Gall2009, MILLETARI201792, novotny2018semi, qi2019deep}, shape recovery~\cite{sun2010depth}, 3D vision~\cite{Knopp2011SceneCC, jan2010orientation}, and pose estimation~\cite{kehl2016eccv} to name a few. In geometric matching, Cho \etal\cite{cho2015unsupervised} first extends it to the Probabilistic Hough Matching (PHM) algorithm for unsupervised object discovery.
Recent methods~\cite{ham2016proposal,ham2018proposal,han2017scnet,kawk2015unsupervised,liu2020semantic,min2019hyperpixel,min2020dhpf,sultani2016what} have demonstrated the efficacy of the Hough matching with good empirical performance.
They, however, are all limited in the sense that the geometric voting is carried out to discover a {\em global} offset consensus rather than a {\em local and individual} consensus for a match, which makes it less accurate and weak to clutter.

\smallbreak
\noindent \textbf{Semantic visual correspondence.}
Traditional approaches to the task of semantic correspondence ~\cite{bristow2015dense, cho2015unsupervised, ham2016proposal, ham2018proposal, kim2013deformable, liu2011sift, taniai2016joint, yang2017object} typically use hand-crafted descriptors~\cite{bay2006surf, dalal2005histograms, lowe2004sift}.
Although the classic methods work satisfactorily for some applications, they still suffer apparent disadvantages of such features, \eg, lack of semantic patterns.
Recent approaches~\cite{han2017scnet, jeon2018parn, jeon2020guided, liu2020semantic, min2019hyperpixel, min2020dhpf, rocco17geocnn, rocco18weak, rocco2018neighbourhood, paul2018attentive, truong2020glunet, wang2019learning} build upon features from convolutional neural network (CNN) pretrained on classification task~\cite{deng2009imagenet}.
Han~\etal\cite{han2017scnet} introduce a CNN-based matching model that learns to compute a correlation tensor. Rocco~\etal\cite{rocco17geocnn} propose to learn a CNN regressor that computes a series of 2D convolutions on a dense correlation matrix to predict global geometric transformation parameters, either affine or TPS~\cite{donato2002approximate}.
Seo~\etal\cite{paul2018attentive} improve the framework with offset-aware correlation kernels with attention modules.
Jeon~\etal\cite{jeon2018parn} stack multiple affine transformation networks and compute correspondences in coarse-to-fine manner.
Wang~\etal\cite{wang2019learning} adopt the CNN architecture to estimate translation and rotation parameters to learn correspondences from raw video.
These methods demonstrate that a series of 2D convolutions acting on correlation tensors is effective in capturing geometric information by exploiting local patterns of similarity.

\smallbreak
\noindent \textbf{4D convolution for visual correspondence.}
Rocco~\etal\cite{rocco2018neighbourhood} introduce the neighbourhood consensus network that uses 4D convolution for visual correspondence. They view 4D convolution as an extension of 2D convolution, which learns multiple similarity patterns of local correspondences, and thus use multiple 4D kernels, requiring a large number of parameters to learn.
Following the work, recent methods~\cite{huang2019dynamic, li2020correspondence, rocco2018neighbourhood, truong2020glunet} also adopt 4D convolution in a similar manner. They commonly consume a high computational cost with a large number of parameters in the kernels and only consider translation in space.
In contrast, we extends the idea of Hough matching~\cite{cho2015unsupervised} for high-dimensional convolution and propose an interpretable and light-weight (semi-isotropic) high-dimensional kernel for visual correspondence. In doing so, it naturally generalizes the existing 4D convolution to higher-dimensional ones and achieves superior performance using only a single kernel per layer with a small number of parameters. The results reveal that the role of high-dimensional convolution on a correlation tensor for matching is to learn a reliable voting strategy rather than to capture diverse patterns in the correlation tensor.

Our contributions can be summarized as follows:\vspace{-5px}
\begin{itemize}
\setlength\itemsep{0em}
    \item We introduce a Hough transform perspective on convolutional matching and propose an effective geometric matching algorithm, CHM, which performs high-dimensional Hough voting in a convolutional manner.
    \item We develop CHM into a trainable neural layer with a semi-isotropic high-dimensional kernel, which learns non-rigid matching with a small number of interpretable parameters.
    \item We propose the convolutional Hough matching network (CHMNet) that performs geometric matching in a translation and scaling space using 6D convolution.
    \item The proposed method sets a new state of the art on standard benchmarks for semantic visual correspondence, proving its robustness to challenging intra-class variations across images to match.
\end{itemize}
\section{Convolutional Hough Matching}\label{sec:CHM}

In this section, we revisit the Hough matching method for visual correspondence and then propose its convolutional version as a high-dimension convolutional layer, which is readily trainable in neural networks.

\subsection{Hough matching \& its convolutional extension}

The Hough transform is a powerful detection method for a geometric object, which exploits the duality between parts and parameters of the object~\cite{ballard1981generalhough, hough1962transform}. It performs voting in a parameter space of the target object, called the Hough space, where votes from the object parts are accumulated to form local maxima in the space. The objects are then detected simply by identifying the positions of local maxima.
The Hough matching method~\cite{cho2015unsupervised}, inspired by the Hough transform, detects reliable correspondences by geometric voting from candidate matches. Given two images, it constructs the Hough space of parameters of geometric transformation between the two images and then accumulates votes of candidate matches for plausible transformation.

Let us assume a local region $\mathbf{x}$ on an image, that is represented by its geometric attributes, \ie, pose and shape. In principle $\mathbf{x}$ can be a form of any parameterization, but in this work we simply describe the region $\mathbf{x}$ by its center and scale.
Now let us consider two images, $I$ and $I'$, and two sets of local regions, $\mathcal{X}$ and $\mathcal{X}'$, obtained from the two images, respectively. For any two regions $(\mathbf{x}, \mathbf{x}') \in \mathcal{X} \times \mathcal{X}'$, a correlation function $c$ computes a non-negative similarity $c(\mathbf{x}, \mathbf{x}')$ using appearance features of the regions. The main idea of Hough matching is to create the Hough space $\mathcal{H}$, that is the space of all possible offsets $\mathbf{h}$ between two regions, \ie, translation and scaling, and accumulate votes from candidate matches onto the Hough space as
\begin{align}
    v(\mathbf{h}) =   \sum_{ (\mathbf{x}, \mathbf{x}') \in \mathcal{X} \times \mathcal{X}'} c(\mathbf{x}, \mathbf{x}') k_\text{iso}( \| (\mathbf{x}' - \mathbf{x}) - \mathbf{h} \|_\mathrm{g} ),
\end{align}
where $ \|\cdot\|_\mathrm{g}$ represents a group-wise distance function that computes the distances separately for two groups, center and scale, \ie, $ \| \mathbf{x} \|_\mathrm{g} = [ \| \mathbf{x}_\mathrm{xy} \|; \|  \mathbf{x}_\mathrm{s} \| ]$ (subscripts $\mathrm{xy}$ for center and $\mathrm{s}$ for scale) and $k_\text{iso}$ is a kernel function that computes similarity between the observed offset, $\mathbf{x}' - \mathbf{x}$, and the given offset $\mathbf{h}$ in the Hough space.\footnote{For the kernel function, previous work uses a form of discretized Gaussian~\cite{cho2015unsupervised} or Dirac delta~\cite{han2017scnet} without learning the kernel parameters.} The kernel $k_\text{iso}$ is designed to assign a voting weight for each candidate match according to how close the offset induced by the match $(\mathbf{x}, \mathbf{x}')$ is to $\mathbf{h}$; we use the group-wise distance to differentiate the effects of center and scale in the kernel.
The resultant voting map $v(\mathbf{h})$ over the Hough space $\mathcal{H}$ can be used to find reliable matches $(\mathbf{x}, \mathbf{x}')$ by suppressing spurious ones corresponding to relatively low voting scores $v(\mathbf{x}'-\mathbf{x})$, \eg, updating the match score via $c(\mathbf{x}, \mathbf{x}') v(\mathbf{x}'-\mathbf{x})$~\cite{han2017scnet}.
Despite its good empirical performance~\cite{cho2015unsupervised,ham2016proposal,ham2018proposal,han2017scnet,kawk2015unsupervised,liu2020semantic,min2019hyperpixel,min2020dhpf,sultani2016what}, the global voting map $v(\mathbf{h})$, which is shared for all candidate matches, is limited in the sense that it cannot capture the reliability of a specific candidate match. This global position-invariant Hough space makes the output less accurate and weak to background clutter, \eg, increasing the score of distant outliers that has a similar offset to that of dominant inliers.

\begin{figure}[t]
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures/1_CHM.pdf}
    \end{center}
    \vspace{-3.0mm}
    \caption{Convolutional Hough matching that carries out geometric voting in 6D space, \eg, translation and scale.}\label{fig:CHM}
    \vspace{-5.0mm}
\end{figure}

As illustrated in Fig.~\ref{fig:CHM}, in order to address the issue, we create a local and individual voting space for each candidate match $(\mathbf{x}, \mathbf{x}')$ by introducing local windows around the regions, $\mathbf{x}$ and $\mathbf{x}'$:
\begin{align}
    v(\mathbf{x}, \mathbf{x}', \mathbf{h}) = \!\!\!\! \sum_{ (\mathbf{p}, \mathbf{p}') \in \mathcal{P}(\mathbf{x}) \times \mathcal{P}'(\mathbf{x'})} \!\!\! c(\mathbf{p}, \mathbf{p}') k_\text{iso}( \| (\mathbf{p}' - \mathbf{p}) - \mathbf{h} \|_\mathrm{g} ),
\end{align}
where $\mathcal{P}(\mathbf{x})$ denotes the set of neighbor regions within the local window centered on $\mathbf{x}$.
Since this local voting space is now dedicated to $(\mathbf{x}, \mathbf{x}')$,
we can simply assign a match score $v$ for the candidate match by taking the vote value at the bin with offset zero:
\begin{align}
    \label{eqn:houghmatching_1}
   v(\mathbf{x}, \mathbf{x}') &= \sum_{ (\mathbf{p}, \mathbf{p}') \in \mathcal{P}(\mathbf{x}) \times \mathcal{P}'(\mathbf{x'})} c(\mathbf{p}, \mathbf{p}') k_\text{iso}( \| \mathbf{p}' - \mathbf{p} \|_\mathrm{g} ).
\end{align}
With a slight abuse of notation, let us use $k( \mathbf{z}, \mathbf{z}')$ to represent the kernel value corresponding to two positions, $\mathbf{z}$ and $\mathbf{z}'$, each representing a local region in the parameter space of regions, \ie, 3D space of center and scale in our case. The equation above then can be generalized to a form of 6D convolution with an arbitrary kernel $k$:
\begin{align}
   c_{\text{HM}}(\mathbf{x}, \mathbf{x}') &= \sum_{ (\mathbf{p}, \mathbf{p}') \in \mathcal{P}(\mathbf{x}) \times \mathcal{P}'(\mathbf{x}')} c(\mathbf{p}, \mathbf{p}') k( \mathbf{p} - \mathbf{x}, \mathbf{p}' - \mathbf{x}') \nonumber  \\ &= (c \ast k) (\mathbf{x}, \mathbf{x}'),
\end{align}
which becomes equivalent to Eq.~\ref{eqn:houghmatching_1} when the group-wise isotropic kernel $ k_\text{iso}$ is used.

Note that this convolutional extension of Hough matching has a generic form; it reduces to a similar form of 4D convolutions in~\cite{huang2019dynamic, li2020correspondence, rocco2018neighbourhood, truong2020glunet} when the Hough space is restricted to center translation, and generalizes to higher dimensions beyond 6D when additional transformation dimensions is introduced such as rotation, shear, and others.

\subsection{Convolutional Hough matching layer}

We design the convolutional Hough matching (CHM) as a learnable convolution layer:
\begin{align}
   c_{\text{HM}}(\mathbf{x}, \mathbf{x}'; k, b) = b +  (c \ast k) (\mathbf{x}, \mathbf{x}'),
\end{align}
where $b$ is a bias term for the layer and $k$ represents a kernel with a specific type of weight sharing.
The group-wise isotropic kernel $k_{\text{iso}}$, which is directly derived from Hough matching, can be implemented by weight sharing among parameters with the same offset $|\mathbf{z}-\mathbf{z}'|$ in $k( \mathbf{z}, \mathbf{z}')$.
While it is a reasonable choice, the fully isotropic kernel assigns the same importance to the matches of the same offset regardless of their distances from the kernel position $(\mathbf{x}, \mathbf{x}')$. It may be an excessive constraint in the sense that the distance of an object from the center of focus is likely to be relevant to the importance.

We thus relax the isotropy and propose the position-sensitive isotropic kernel $k_\text{psi}( \|\mathbf{p}' - \mathbf{p}\|_\mathrm{g}; \|\mathbf{p} - \mathbf{x}\|_\mathrm{g}, \|\mathbf{p}' - \mathbf{x}'\|_\mathrm{g} )$ that differentiates the distances from the kernel position, $\|\mathbf{p} - \mathbf{x}\|_\mathrm{g}$ and $\|\mathbf{p}' - \mathbf{x}'\|_\mathrm{g}$. The kernel $k_{\text{psi}}$ is implemented by sharing parameters whose triplets, $(\|\mathbf{p}' - \mathbf{p}\|_\mathrm{g}, \|\mathbf{p} - \mathbf{x}\|_\mathrm{g}, \|\mathbf{p}' - \mathbf{x}'\|_\mathrm{g})$, are the same.

The CHM layer is compatible with any neural network layer that computes correlations between images, and can be stacked multiple times to improve the performance. As a result of substantial parameter sharing, the 6D kernels, $k_{\text{iso}}^{\mathrm{6D}}$ and $k_{\text{psi}}^{\mathrm{6D}}$ in $\mathbb{R}^{H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k} \times H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k}}$, contain only a small number of parameters, thus making CHM resistant to overfitting in training; \eg, the kernels with $H_\mathrm{k}=W_\mathrm{k}=5$ and $S_\mathrm{k}=3$ contains only 45 and 220 parameters, respectively, while the full kernel has 5,625.
More importantly, the perspective of Hough matching on convolution provides the interpretability of the learned kernel: each element in the kernel is a voting weight of the corresponding neighbor match in the local offset space. Based on this perspective, Figure~\ref{fig:CHM_kernels} visualizes the kernel $k_{\text{psi}}^{\mathrm{6D}} \in \mathbb{R}^{H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k} \times H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k}}$ of size $H_\mathrm{k} = W_\mathrm{k}=5$ and $S_\mathrm{k}=3$ trained in our experiment.
For the ease of visualizing 6D tensor, we decompose it into multiple (four in case of $k_{\text{psi}}^{\mathrm{6D}}$) 4D tensors in which each of the map shows parameter values of the kernel with the same offset, where the arrows represent the offset vectors relative to the kernel position $(\mathbf{x}, \mathbf{x}')$, and the circles mean zero offset. The maps reveal that weights for matches with smaller offsets and closer distance are learned to be higher (darker), which appears to be a reasonable voting strategy. For more information, refer to our Appendix.

\begin{figure}[t]
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures/2_kernel_vis.pdf}
    \end{center}
    \vspace{-3.0mm}
    \caption{Visualization of learned CHM kernel ($k_{\text{psi}}^{\mathrm{6D}}$). Refer Appendix~\ref{sec:appendix_A} for the visualization method.}\label{fig:CHM_kernels}
    \vspace{-6.0mm}
\end{figure}
\section{Convolutional Hough Matching Networks}

Based on CHM, we develop a family of image matching models, dubbed {\em Convolutional Hough Matching Networks (CHMNet)}, which consists of three parts: (1) high-dimensional correlation computation, (2) convolutional Hough matching, and (3) flow formation (and keypoint transfer).
Figure~\ref{fig:architecture} illustrates the overall architecture.

\subsection{High-dimensional correlation computation}
Following other recent methods~\cite{huang2019dynamic, li2020correspondence, min2019hyperpixel, min2020dhpf, rocco2018neighbourhood}, we also use as a CNN feature extractor pretrained on ImageNet classification~\cite{deng2009imagenet}.
Given an input image $I$, the feature extractor outputs a feature map in $\mathbb{R}^{C \times H \times W}$.
We construct feature maps of multiple scales $\{{\mathbf{F}}_{s}\}_{s=1}^{S}$ by resizing the output for $S-1$ times by the scaling factor of $\sqrt{2}$, followed by $3 \times 3$ conv layers with parameters $\{\theta_{s}\}_{s=1}^{S}$, reducing channel dimensions of input feature map by $1/\rho$.
The $S$ different conv layers learn to capture effective semantic information of receptive fields with different scales for the subsequent multi-scale (6D) correlation computation.
The same is done for $\{{\mathbf{F}}'_{s}\}_{s=1}^{S}$ given image $I'$.
We set $S=3$, \ie, $\{1/\sqrt{2}, 1, \sqrt{2} \}$, and $\rho=4$ in our experiments.

Given a set of feature pairs from multiple scales $\{({\mathbf{F}}_{s}, {\mathbf{F}}'_{s})\}_{s=1}^{S}$, we compute all possible 4D correlation tensors placed on the $S \times S$ grid:
\begin{align}
    \mathbf{C}^{(0)}_{mn}(\mathbf{x}_{m}, \mathbf{x}'_{n}) = \text{ReLU}\Bigg(\frac{{\mathbf{F}}_{m}(\mathbf{x}_{m}) \cdot {\mathbf{F}}'_{n}(\mathbf{x}'_{n}) }{\norm{{\mathbf{F}}_{m}(\mathbf{x}_{n})} \norm{{\mathbf{F}}'_{n}(\mathbf{x}'_{n})}} \Bigg),
\end{align}
where $\mathbf{x}_{m} \in \mathcal{X}_{m}$ and $\mathbf{x}'_{n} \in \mathcal{X}'_{n}$ are spatial positions of feature map at scale $m$ and $n$, respectively, and ReLU clamps negative correlation scores to zero.
To process it in the subsequent 6D CHM layer, we interpolate each 4D correlation $\mathbf{C}^{(0)}_{ij}$ to have the same spatial size to build 6D correlation tensor $\mathbf{C}^{\mathrm{(1)}} \in \mathbb{R}^{H \times W \times S \times H \times W \times S}$ such that ${\mathbf{C}}_{::i::j}^{\mathrm{(1)}} = \zeta_{1}({\mathbf{C}}_{ij}^{\mathrm{(0)}})$
where $\zeta_{1}(\cdot)$ is a function that interpolates input 4D tensor to the size ${H \times W \times H \times W}$.

\subsection{Convolutional Hough Matching}
A CHM layer takes the 6D correlation tensor ${\mathbf{C}}^{\mathrm{(1)}}$ to perform convolutional Hough voting in the space of translation and scaling:
${\mathbf{C}}^{\mathrm{(2)}} = \text{CHM}({\mathbf{C}}^{\mathrm{(1)}}; k_{\mathrm{psi}}^{\mathrm{6D}})$, where $k_{\mathrm{psi}}^{\mathrm{6D}} \in \mathbb{R}^{H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k} \times H_\mathrm{k} \times W_\mathrm{k} \times S_\mathrm{k}}$ is a 6D position-sensitive isotropic kernel.
In our experiments, we set $H_\mathrm{k} = W_\mathrm{k} = 5$ and $S_\mathrm{k} = 3$ with stride 1 for all dimensions and use zero-padding to the input to retain the same size at the output.
We then perform max-pooling on ${\mathbf{C}}^{\mathrm{(2)}}$ to select the most dominant vote among candidate match scores in the scale space, reducing the tensor dimension down to 4D: ${\mathbf{C}}_{ijkl}^{\mathrm{(3)}} = \max_{m,n}{\mathbf{C}}_{ijmkln}^{\mathrm{(2)}}$.
We proceed another CHM with a 4D kernel $k_{\mathrm{psi}}^{\mathrm{4D}} \in \mathbb{R}^{H_\mathrm{k} \times W_\mathrm{k} \times H_\mathrm{k} \times W_\mathrm{k}}$: ${\mathbf{C}} = \text{CHM}(\zeta_{2}(\sigma({\mathbf{C}}^{\mathrm{(3)}}));k_{\mathrm{psi}}^{\mathrm{4D}})$,
where $\sigma(\cdot)$ is the sigmoid activation function and $\zeta_{2}(\cdot)$ is the upsampling function that resizes input 4D tensor to the size of $\bar{H} \times \bar{W} \times \bar{H} \times \bar{W}$ for fine-grained localization.
We set $\bar{H} = 2H$ and $\bar{W} = 2W$ in our experiment.

\subsection{Flow formation \& keypoint transfer}

\smallbreak
\noindent \textbf{Flow formation.}
The output ${\mathbf{C}}$ can easily be transformed into a dense flow field by applying kernel soft-argmax~\cite{lee2019sfnet}. We  first normalize the raw correlation scores with softmax:
\begin{align}
    \hat{\mathbf{C}} = \frac{\exp{(\mathbf{G}^{\mathbf{p}}_{kl} {\mathbf{C}}_{ijkl})}} {\sum_{(k',l') \in \bar{H} \times \bar{W}} \exp{(\mathbf{G}^{\mathbf{p}}_{k'l'} {\mathbf{C}}_{ijk'l'})}},
\end{align}
where and $\mathbf{G}^{\mathbf{p}} \in \mathbb{R}^{\bar{H} \times \bar{W}}$ is 2-dimensional Gaussian kernel centered on $\mathbf{p} = \argmax_{k,l}{{\mathbf{C}}_{ijkl}}$.
Using the estimated probability map $\hat{\mathbf{C}}$, we then transfer all the coordinates on dense regular grid $\mathbf{P} \in \mathbb{R}^{\bar{H} \times \bar{W} \times 2}$ of image $I$ to obtain their corresponding coordinates $\hat{\mathbf{P}}' \in \mathbb{R}^{\bar{H} \times \bar{W} \times 2}$ on image $I'$:
$\hat{\mathbf{P}}'_{ij:} = \sum_{(k,l) \in \bar{H} \times \bar{W}} \hat{\mathbf{C}}_{ijkl} \mathbf{P}_{kl:}$.
We now can construct a dense flow field at sub-pixel level using the set of estimated matches $(\mathbf{P}, \hat{\mathbf{P}}')$.

\smallbreak
\noindent \textbf{Keypoint transfer.}
As in~\cite{lee2019sfnet}, one simplest way of assigning a match $\hat{\mathbf{k}}$ to some keypoint $\mathbf{k}=(x_{k}, y_{k})$ is to pick a single, discrete sample of a transferred coordinate such that $\hat{\mathbf{k}} = \hat{\mathbf{P}}_{y_{k}x_{k}}'$.
However, this may cause mis-localized keypoints as the discrete sampling under sub-pixel level hinders fine-grained localization.
To this end, we define a soft sampler $\mathbf{W}^{(\mathbf{k})} \in \mathbb{R}^{\bar{H} \times \bar{W}}$ for given keypoint $\mathbf{k}=(x_{k},y_{k})$ as follows
\begin{align}
    \mathbf{W}_{ij}^{(\mathbf{k})} = \frac{\max{(0, \tau - \sqrt{(x_{k} - j)^{2} + (y_{k} - i)^{2}})}} {\sum_{i'j'} \max{(0, \tau - \sqrt{(x_{k} - j')^{2} + (y_{k} - i')^{2}})}},
\end{align}
such that $\sum_{ij}\mathbf{W}_{ij}^{(\mathbf{k})} = 1$ where $\tau$ is a distance threshold.
We assign a match to the keypoint $\mathbf{k}$ by $\hat{\mathbf{k}} = \sum_{(i,j) \in \bar{H} \times \bar{W}} \hat{\mathbf{P}}_{ij:} \mathbf{W}_{ij}^{(\mathbf{k})}$.
The soft sampler $\mathbf{W}^{(\mathbf{k})}$ effectively samples each transferred keypoint $\hat{\mathbf{P}}_{ij}$ by giving weights inversely proportional to the distance to $\mathbf{k}$.

\subsection{Training objective}
We assume that keypoint match annotations are given for each training image pair, as in~\cite{choy2016universal,han2017scnet,li2020correspondence,min2019hyperpixel,min2020dhpf}; each image pair is annotated with a set of coordinate pairs $\mathcal{M}=\{(\mathbf{k}_m, \mathbf{k}'_m)\}_{m=1}^{M}$, where $M$ is the number of annotations.
Following the aforementioned keypoint transfer scheme, we obtain a set of predicted and ground-truth keypoint pairs on image $I'$: $\{(\hat{\mathbf{k}}_m', \mathbf{k}_m')\}_{m=1}^{M}$ by assigning a match $\hat{\mathbf{k}}_m'$ to each $\mathbf{k}_m$.
Our objective in training is formulated as $\mathcal{L} = \frac{1}{M}\sum_{m=1}^{M}\|\hat{\mathbf{k}}_{m}' - \mathbf{k}_{m}'\|$, which minimizes the average Euclidean distance between the predicted keypoints and the ground-truth ones.

\section{Experimental Evaluation}
In this section we evaluate the proposed method, compare it with recent state of the arts, and discuss the results.

\smallbreak
\noindent \textbf{Implementation detail.}
For the feature extractor network, we employ ResNet-101~\cite{he2016deep}, truncated after the \texttt{conv4\_23} layer, pre-trained on ImageNet~\cite{deng2009imagenet}.
Both input and output channel sizes of all the CHM layers are set to 1.
We set spatial size of the input image to $240 \times 240$, thus having $H = W = 15$ and $\bar{H} = \bar{W} = 30$.
Due to parameter sharing structure of $k_{\mathrm{psi}}^{*}$ and $k_{\mathrm{iso}}^{*}$, magnitudes of the loss gradient with respect to the shared weights are unevenly distributed during training time.
To resolve the numerical instability, the shared weights are normalized before the convolution by dividing by the number of times being shared.
The network is implemented in PyTorch~\cite{pytorch} and optimized using Adam~\cite{kingma2015adam} with a learning rate of 1e-3. We finetune the backbone network by setting its learning rate 100 times smaller than CHM layers, \eg, 1e-5.

\smallbreak
\noindent \textbf{Datasets.}
We evaluate the proposed network on three standard benchmark datasets of semantic correspondence: SPair-71k~\cite{min2019spair}, PF-PASCAL~\cite{ham2018proposal}, and PF-WILLOW~\cite{ham2016proposal}.
SPair-71k~\cite{min2019spair} is a highly challenging, large-scale dataset, which contains 70,958 pairs from 18 categories with large variations in view-point and scale.
PF-PASCAL~\cite{ham2018proposal} and PF-WILLOW~\cite{ham2016proposal} respectively contain 1,351 pairs from 20 categories and 900 pairs from 4 categories with small variations in view-point and scale.
Each pair in the datasets consists of keypoint match annotations for semantic parts.

\smallbreak
\noindent \textbf{Evaluation metric.}
We adopt the standard evaluation metric, percentage of correct keypoints (PCK), for the evaluation.
Given a set of predicted and ground-truth keypoint pairs $\mathcal{K}=\{(\hat{\mathbf{k}}'_{m}, \ \mathbf{k}'_{m})\}_{m=1}^{M}$, PCK is measured by $\mathrm{PCK}(\mathcal{K}) = \frac{1}{M}\sum_{m=1}^{M} \mathbbm{1} [\|\hat{\mathbf{k}}'_{m} - \mathbf{k}'_{m}\| \leq \alpha_{\tau} \cdot \max{(w_{\tau}, h_{\tau})}]$ where $w_{\tau}$ and $h_{\tau}$ are the width and height of either an entire image or an object bounding box, \eg, $\tau \in \{\text{img}, \text{bbox}\}$, and $\alpha_{\tau}$ is a tolerance factor.

\subsection{Results and analysis}
On the SPair-71k dataset, following~\cite{min2019hyperpixel, min2020dhpf}, we evaluate two versions for each model: a finetuned model (F), which is trained on SPair-71k, and a transferred model (T), which is trained on PF-PASCAL.
On PF-PASCAL and PF-WILLOW, following the common evaluation protocol~\cite{choy2016universal, han2017scnet, huang2019dynamic, kim2018recurrent, li2020correspondence, min2019hyperpixel, min2020dhpf, rocco18weak, rocco2018neighbourhood}, our network is trained on the training split of PF-PASCAL~\cite{ham2018proposal} and evaluated on the test splits of PF-PASCAL and PF-WILLOW.
We use the same training, validation, and test splits of PF-PASCAL used in~\cite{han2017scnet}.
The quantitative results are summarized in Tab.~\ref{tab:main_table};
we note different levels of supervision for each method in the first column to ensure fair comparison.
The proposed model finetuned on SPair-71k (F) clearly surpass current state of the art by a significant margin, outperforming \cite{min2020dhpf} by 9\%p of PCK ($\alpha_{\mathrm{bbox}}=0.1$), \ie, 24.1\% relative improvement.
On PF-PASCAL, our model achieves 4.4\%p and 0.9\%p improvement with $\alpha_{\mathrm{img}} \in \{0.05, 0.1\}$.
Robust performance on SPair-71k (T) and PF-WILLOW verifies reliable transferability of our model.
Figure~\ref{fig:qualitative_spair} visualizes example qualitative results on SPair-71k.

\smallbreak
\noindent \textbf{FLOPs, running time, and memory.}
We collect publicly available codes of some recent methods~\cite{huang2019dynamic, li2020correspondence, liu2020semantic, min2020dhpf, rocco2018neighbourhood} to measure their FLOPs, inference time\footnote{Some inference time results are retrieved from~\cite{min2020dhpf}, which is measured on a machine with an Intel i7-7820X and an NVIDIA Titan-XP. For fair comparison, inference time and memory footprint of all the methods are measured on a machine with the same CPU and GPU and includes all the pipelines of a model: from feature extraction to keypoint prediction.}, and memory footprint and compare them with ours in Tab.~\ref{tab:main_table}.
Although the proposed method demands larger memory than some 4D conv based models~\cite{li2020correspondence, rocco2018neighbourhood}, smaller channel sizes of CHM (6D) layers (\{1,1\} vs. \{16,16,1\}) provide noticeable efficiency in terms of GFLOPs (19.6 vs. 44.9).
To achieve faster inference time, we further improve the original implementation of 4D conv~\cite{rocco2018neighbourhood} and develop an efficient nD conv which enables real-time inference (54ms) without increasing FLOPs and memory.
See Appendix~\ref{sec:appendix_B} for details on our implementation of nD convolution.

\begin{figure}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures/5_qualitative.pdf}
    \end{center}
    \vspace{-5.0mm}
      \caption{Qualitative results on SPair-71k dataset. Our model predicts reliable matches under deformations, and large changes in view-point and scale.}
    \vspace{-6.5mm}
\label{fig:qualitative_spair}
\end{figure}

\smallbreak
\noindent \textbf{Robustness to background clutter.}
Recent methods for semantic correspondence~\cite{ham2016proposal, han2017scnet, huang2019dynamic, kim2018recurrent, kim2017dctm, lee2019sfnet, min2019hyperpixel, min2020dhpf, rocco17geocnn, rocco18weak, rocco2018neighbourhood} predict matching scores for all candidate matches but rarely evaluate their robustness to background clutters.
Here, we compare some recent methods~\cite{li2020correspondence, min2019hyperpixel, min2020dhpf, rocco2018neighbourhood} and ours in terms of robustness to background clutter based on the predicted matching scores.
Each method, however, exploits its correlation tensor differently from others with its own flow formation (keypoint transfer) scheme.
Therefore, given all possible candidate matches in correlation tensor, simply defining matches with top-$k$ scores as positive matches may yield biased estimates.
To ensure fair comparison, for each model, we define a set of coordinates on a regular grid on the input pair of images and assign their best matches using its own keypoint transfer method, thus providing the same number of (fairly collected) candidate matches to every model that we compare.
For each candidate match, we define its match score as a score nearest to spatial position in the correlation tensor.
Given top-$k$ matches according to their matching scores, we define true positives (TPs) as matches falling inside object segmentation masks (bounding box)\footnote{We use object seg. masks and bounding boxes for SPair-71k and PF-PASCAL respectively due to absence of mask annotation in PF-PASCAL.} and false positives (FPs) as those lying outside object masks (boxes).
Precision and recall are measured by  $\frac{N_{\mathrm{TP}}}{N_{\mathrm{TP}} + N_{\mathrm{FP}}}$ and $\frac{N_{\mathrm{TP}}}{N_{\mathrm{mask}}}$, respectively, where $N_{\mathrm{TP}}$ and $N_{\mathrm{FP}}$ are respectively the number of TPs and FPs while $N_{\mathrm{mask}}$ is the number of all candidate matches that fall in the object segmentation masks.
In defining TPs and FPs, we use masks and boxes only due to the absence of dense flow annotation in SPair-71k and PF-PASCAL, but we find that they are good approximation enough to distinguish inliers from outliers in our experimental setup.

Figure~\ref{fig:pr_curve} plots precision-recall curves for the recent methods~\cite{li2020correspondence, min2019hyperpixel, min2020dhpf, rocco2018neighbourhood} and ours.
The proposed method clearly outperforms other methods, indicating our model effectively discriminates between semantic parts and background clutters as seen in the last row of Fig.~\ref{fig:outlier_qualitative} which visualizes sample pairs with top 300 confident matches.
When CHM is either removed (w/o CHM) or replaced with global matching module (CHM $\xrightarrow{}$ RHM), predicted matches become unreliable, being mostly scattered on the background and even hardly regularized.
For our model evaluated on SPair-71k, precision and recall have inverse relationship in most cases.
Although initial growth in our PR curves on SPair-71k indicates that some true matches have in fact low match scores, it still surpasses the other models, revealing the reliability of our approach under large variations.

\subsection{Ablation study and analysis}

\begin{figure}
    \begin{center}
        \includegraphics[width=0.95\linewidth]{_arxiv_figures/6_maxvotes.pdf}
    \end{center}
    \vspace{-5.0mm}
    \caption{\label{fig:freq_scale} Frequencies over the maxpooled positions in scale-space on SPair-71k, PF-PASCAL, and PF-WILLOW.}
    \vspace{-3.0mm}
\end{figure}

\begin{table}[t]
    \begin{center}
    \scalebox{0.625}{
        \begin{tabular}{lccccccc}

                \toprule

                \multirow{2}{*}{\shortstack{Kernel\\type}} & \multicolumn{2}{c}{SPair-71k PCK ($\alpha_{\text{bbox}}$)} & \multicolumn{2}{c}{PF-PAS. PCK ($\alpha_{\text{img}}$)} & \# params. & FLOPs & time \\
                & 0.05 & 0.1 & 0.05 & 0.1 & in CHM & (G) & ({\em ms}) \\

                \midrule

                $k_{\mathrm{psi}}^{\text{6D-4D}}$    & \textbf{27.4}$_{\pm 0.16}$ & \textbf{46.4}$_{\pm 0.34}$ & \textbf{80.4}$_{\pm 0.28}$ & \textbf{91.6}$_{\pm 0.23}$ & 275 & \underline{19.6} & 54 \\\\[-1.8ex]
                $k_{\mathrm{full}}^{\text{6D-4D}}$   & 25.9$_{\pm 0.74}$ & 44.8$_{\pm 0.65}$ & \underline{79.8}$_{\pm 0.67}$ & 90.7$_{\pm 0.19}$ & 6,250 & \underline{19.6} & 43 \\\\[-1.8ex]
                $k_{\mathrm{iso}}^{\text{6D-4D}}$    & 24.5$_{\pm 0.28}$ & \underline{44.9}$_{\pm 0.16}$ & 76.5$_{\pm 0.29}$ & 90.2$_{\pm 0.40}$ & \underline{60} & \underline{19.6} & 46 \\

                \midrule

                $k_{\mathrm{psi}}^{\text{4D-4D}}$    & \underline{26.4}$_{\pm 0.25}$ & 44.5$_{\pm 0.34}$ & 79.3$_{\pm 0.25}$ & \underline{91.1}$_{\pm 0.32}$ & 110 & \textbf{15.9} & 32 \\\\[-1.8ex]
                $k_{\mathrm{full}}^{\text{4D-4D}}$   & 26.1$_{\pm 0.33}$ & 43.9$_{\pm 0.53}$ & 78.4$_{\pm 0.82}$ & 90.3$_{\pm 0.43}$ & 1,250 & \textbf{15.9} & \textbf{26} \\\\[-1.8ex]
                $k_{\mathrm{iso}}^{\text{4D-4D}}$    & 21.0$_{\pm 0.54}$ & 39.7$_{\pm 0.73}$ & 71.8$_{\pm 0.99}$ & 88.0$_{\pm 0.49}$ & \textbf{30} & \textbf{15.9} & \underline{27} \\
                \midrule

                \rowcolor{mygray} $k_{\mathrm{psi; sparse}}^{\text{6D-4D}}$    & 26.3$_{\pm 0.18}$ & 45.2$_{\pm 0.41}$ & 80.3$_{\pm 0.86}$ & 91.1$_{\pm 0.05}$ & 275 & - & 55 \\
                \bottomrule
        \end{tabular}}
    \vspace{-2.0mm}
    \caption{\label{tab:ablation_kernel} Ablation study of CHM kernels over multiple runs.}
    \vspace{-9.0mm}
    \end{center}
\end{table}

\smallbreak
\noindent \textbf{Analyses on CHM kernel.}
We conduct ablation study on CHM kernel by replacing position-sensitive isotropic kernels with $k_{\mathrm{full}}^{\mathrm{nD}}$\footnote{Note $k_{\mathrm{full}}^{\mathrm{nD}}$ is a n-dimensional kernel without any parameter sharing. The number of parameters in $k_{\mathrm{full}}^{\mathrm{nD}}$ is proportional to $k^{n}$.} and full isotropic ones $k_{\mathrm{iso}}^{\mathrm{nD}}$.
For the ease of notation, we denote by $k_{\mathrm{psi}}^{\text{6D-4D}}$ a model with two CHM layers whose kernels are $k_{\mathrm{psi}}^{\mathrm{6D}}$ and $k_{\mathrm{psi}}^{\mathrm{4D}}$.
Table~\ref{tab:ablation_kernel} shows average PCK, its standard deviations, parameter sizes, FLOPs, and average inference time of our model with different kernels over five runs.
Despite a huge difference in the number of parameters (110 vs. 1,250),
the proposed semi-isotropic kernel $k_{\mathrm{psi}}^{\text{4D-4D}}$ outperforms $k_{\mathrm{full}}^{\text{4D-4D}}$ on Spair-71k (44.5 vs. 43.9) and extending its voting space to 6D, \eg,  $k_{\mathrm{psi}}^{\text{6D-4D}}$, further improves PCK to 46.4 on SPair-71k, which clearly shows efficacy of 6D convolution in scale-space\footnote{To verify the efficacy of the proposed kernel even with sparse match information, we further limit the set of potential matches in $\mathbf{C}^{(0)}$ using $K$ nearest neighbors without using MinkowskiEngine~\cite{choy2019fully} as it does not provide high-dim. kernel customization.
As seen in shaded row in Tab.~\ref{tab:ablation_kernel}, our model with the sparse correlation is comparably effective to $k_{\mathrm{psi}}^{\text{6D-4D}}$, which is consistent to the results of~\cite{rocco2020sparse}. We set $K=10$ in our experiment.}.
The comparable performance of $k_{\mathrm{iso}}^{\text{6D-4D}}$ to $k_{\mathrm{full}}^{\text{6D-4D}}$ reveals that full isotropic parameter sharing can also be a reasonable choice for reducing the large capacity of $k_{\mathrm{full}}^{\text{6D-4D}}$.

In Figure~\ref{fig:freq_scale}, we also plot frequencies over the maxpooled positions in scale-space after 6D CHM layer ($k_{\mathrm{psi}}^{\text{6D-4D}}$).
The maximum votes on both PF-PASCAL and PF-WILLOW are mostly concentrated on the center scale whereas they are distributed over different scales on SPair-71k;
this is a reasonable voting strategy as objects in PF-PASCAL and PF-WILLOW hardly vary in scale while those in SPair-71k show large variations in both scale and view-point.

\noindent \textbf{Ablation study on matching modules.}
We analyze the effect of CHM, by either removing or replacing them with the matching module of~\cite{min2019hyperpixel}.
Figure~\ref{fig:outlier_qualitative} and Table~\ref{tab:ablation_study} summarize qualitative and quantitative results, respectively.
The output of global offset voting (CHM $\xrightarrow{}$ RHM) includes many outliers from the background, showing its weakness to the background clutter.
Without the last CHM layer (w/o last CHM), the model fails to effectively refine upsampled correlation scores.
The model prediction is severely damaged without any matching modules (w/o CHM) as seen in second row of Fig.~\ref{fig:outlier_qualitative}.
For keypoint transfer, kernel $\mathbf{G}$ and soft sampler $\mathbf{A}^{(\mathbf{k})}$ help our model find reliable matches by suppressing noisy match scores in $\mathbf{C}$ and effectively aggregating neighborhood transfers, respectively.

\begin{figure}
    \begin{center}
        \includegraphics[width=0.95\linewidth]{_arxiv_figures/7_ablation_qual.pdf}
    \end{center}
    \vspace{-5.0mm}
    \caption{\label{fig:outlier_qualitative}Ablation study on matching modules.}
    \vspace{-2.0mm}
\end{figure}

\begin{table}[t]
    \begin{center}

    \scalebox{0.78}{
        \begin{tabular}{lcccc}

                \toprule
                \multirow{3}{*}{Method} & \multicolumn{2}{c}{SPair-71k} & \multicolumn{2}{c}{PF-PASCAL}  \\

                & \multicolumn{2}{c}{PCK ($\alpha_{\text{bbox}}$)} & \multicolumn{2}{c}{PCK ($\alpha_{\text{img}}$)} \\
                 & $0.05$ & $0.1$ & $0.05$ & $0.1$ \\

                \midrule

                CHMNet$_{\mathrm{res101}}$    & 27.2 & 46.3 & 80.1 & 91.6 \\

                \midrule

                CHM $\xrightarrow{}$ RHM & 21.8 & 38.2 & 77.1 & 89.6 \\

                w/o last CHM layer ($k^{\mathrm{4D}}_\mathrm{psi}$) & 24.9 & 43.1 & 79.5 & 89.7  \\

                w/o CHM & 10.1 & 21.6 & 61.6 & 78.5  \\

                \midrule

                w/o kernel $\mathbf{G}$ & 26.6 & 45.5 & 79.5 & 91.3 \\

                w/o soft sampler $\mathbf{A^{(\mathbf{k})}}$ & 23.1 & 43.8 & 78.9 & 89.6 \\

                \bottomrule
        \end{tabular}}

    \vspace{-2.0mm}
    \caption{\label{tab:ablation_study}Ablation study of core modules in our model.}
    \vspace{-11.0mm}
    \end{center}
\end{table}

\smallbreak
\noindent \textbf{Effect of channel size.}
To study the effect of channel size, we train our model\footnote{We use the models in the middle section of Tab.~\ref{tab:ablation_kernel}, \eg, $k_{*}^{\text{4D-4D}}$.} using three different kinds of kernels ($k_{\mathrm{psi}}^{\text{4D-4D}}$, $k_{\mathrm{iso}}^{\text{4D-4D}}$, and $k_{\mathrm{full}}^{\text{4D-4D}}$) with different channel sizes, \ie, different number of kernels.
Table~\ref{fig:channel_ablation} summarizes the results, showing that increasing the channel size rarely brings performance gain and typically harms the quality of prediction for kernels $k_{\mathrm{psi}}^{\text{4D-4D}}$ and $k_{\mathrm{full}}^{\text{4D-4D}}$.
We train the models on the training split of PF-PASCAL and evaluate on test splits of PF-PASCAL and SPair-71k.
For $k_{\mathrm{iso}}^{\text{4D-4D}}$, although increasing channel size improves performance up to certain amount due to its small capacity, it eventually exhibits similar patterns to other kernels after all.

These experiments imply that the high-dimensional convolution on a correlation tensor may play a different role from 2D convolution on an image feature tensor; the role of convolutional matching is to learn a reliable voting strategy rather than to capture diverse patterns in the correlation tensor. This is consistent with the Hough matching perspective, but previous 4D convolution methods~\cite{huang2019dynamic, li2020correspondence, rocco2018neighbourhood, truong2020glunet} with a different perspective commonly use multiple full kernels ($k_{\mathrm{full}}^{\mathrm{4D}}$) for layers.
To verify our result, we have conducted a similar experiment using the model of~\cite{rocco2018neighbourhood} and obtained the consistent result; the original model, which uses channel sizes of $\{16, 16, 1\}$ for three layers of 4D convolution, achieves 76.2\% PCK on our machine while the model with reduced channels of $\{1, 1, 1\}$ achieves 76.4\% PCK.
Note that in terms of the number of parameters in a layer, our CHM layers ($k^{\text{6D-4D}}_{\mathrm{psi}}$) have $247\sim654$ times smaller number of parameters than the 4D convolution layers used in previous methods~\cite{huang2019dynamic, li2020correspondence, rocco2018neighbourhood, truong2020glunet}.
This light-weight layer design is particularly important in practice, since the use of multiple channels, \ie kernels, for high-dim convolution quickly increases the cost both in computation and memory.

For additional results and analyses, we refer the readers to the Appendix.

\begin{figure}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures/8_ablation_ch.pdf}
    \end{center}
    \vspace{-5.0mm}
    \caption{\label{fig:channel_ablation}PCK performance on SPair-71k and PF-PASCAL with different channel sizes of 1, 2, 4, 8, and 16.}
    \vspace{-6.0mm}
\end{figure}

\section{Conclusion}
We have introduced the convolutional Hough matching (CHM) and proposed the  powerful matching model, CHMNet, that leverages CHM in a high-dimensional geometric transformation space for establishing reliable visual correspondence.
The extensive experiments on several standard benchmarks for semantic visual correspondence demonstrate the benefits of our approach. In particular, our method generalizes existing 4D convolutions and also provides the perspective of Hough transform for geometric matching with interpretable high-dimension kernels.
We believe further research on this direction can benefit a wide range of other problems related to correspondence.

\smallbreak
\noindent \textbf{Acknowledgements.}
This work was supported by Samsung Advanced Institute of Technology (SAIT), the NRF grants (NRF-2017R1E1A1A01077999, NRF-2021R1A2C3012728), and the IITP grant (No.2019-0-01906, AI Graduate School Program - POSTECH) funded by Ministry of Science and ICT, Korea.

\section{Additional results and analyses}
\label{sec:appendix_A}

\smallbreak
\noindent \textbf{Analysis on scale-space maxpool.}
To further analyze the results in Fig.~\ref{fig:freq_scale}, we visualize maxpooled positions of predicted matches on sample pairs of SPair-71k~\cite{min2019spair}, PF-PASCAL~\cite{ham2018proposal}, and PF-WILLOW~\cite{ham2016proposal}. Figure~\ref{fig:scalemax_qual} shows the results and describes how we visualize them.
Due to large scale-variations in pairs of SPair-71k, our model collects winners of scale-space vote, \ie, $\text{CHM}(\cdot; k_{\mathrm{psi}}^{\text{6D}})$, from diverse positions in scale-space.
In contrary, objects in PF-PASCAL and PF-WILLOW exhibit relatively small scale-variations, thus encouraging our model to collect winners of the vote mostly from the original scales.
We observe that the maxpooled positions typically depend on scales of object's parts as seen in Fig.~\ref{fig:scalemax_qual}.

\smallbreak
\noindent \textbf{Learned CHM kernels.}
Figure~\ref{fig:kernel_vis_demo} describes how we visualized Fig.~\ref{fig:CHM_kernels}.
For straightforward visualization of high-dimensional geometry on 2D plane, we use tesseracts and their arrangement on a 2D grid to represent 4D and 6D tensors respectively.
Learned kernels of $k_{\mathrm{psi}}^{\text{6D-4D}}$ (ours), $k_{\mathrm{full}}^{\text{6D-4D}}$, and $k_{\mathrm{iso}}^{\text{6D-4D}}$ are respectively visualized in Figs \ref{fig:k_psi}, \ref{fig:k_full}, and \ref{fig:k_iso}.

Interestingly, the weight patterns of kernels $k_{\mathrm{psi}}^{\text{6D-4D}}$ and $k_{\mathrm{full}}^{\text{6D-4D}}$ are remarkably similar;
the weights for matches with large offsets and closer distance are learned to be higher (darker) while those with small offsets and far distance are learned to be lower (brighter).
Moreover, learned weight patterns of 4D maps in second, fourth, sixth, and eighth rows of $k_{\mathrm{full}}^{\text{6D}}$ in Fig.~\ref{fig:k_full} are noticeably similar to each other.
We also observe that patterns in first and last rows, and patterns in third and seventh rows of $k_{\mathrm{full}}^{\text{6D}}$ are similar to each other as well.
In contrast, $k_{\mathrm{iso}}^{\text{nD}}$ is unable to express diverse weight patterns due to its parameter-sharing constraint that enforces full isotropy.
This observation reveals that our kernel $k_{\mathrm{psi}}^{\text{nD}}$ in CHMNet clearly benefits from its reasonable parameter-sharing strategy, in terms of both efficiency and accuracy as demonstrated in Tab.~\ref{tab:ablation_kernel}.

\section{More implementation details}
\label{sec:appendix_B}

\smallbreak
\noindent \textbf{Coordinate normalization.}
Following the work of~\cite{lee2019sfnet}, we use height and width normalized coordinates to ensure numerical stability of loss gradients such that
\begin{align}
    \begin{bmatrix}
        -1 \\
        -1
    \end{bmatrix} \leq \mathbf{P}_{ij:} \leq
    \begin{bmatrix}
        1 \\
        1
    \end{bmatrix},
\end{align}
where $\mathbf{P}$ is a set of coordinates on a dense regular grid used for flow formation.
This normalization gives spatial bounds $[-1, 1]$ to the intermediate output coordinates $\hat{\mathbf{P}}'$, $\mathbf{k}$, and $\hat{\mathbf{k}}'$.

\smallbreak
\noindent \textbf{Hyperparameters.}
During training, the learning rates of CHM layers and backbone feature extractor are set to 1e-3 and 1e-5, respectively with batch size of 16.
The distance threshold $\tau$ in Eqn.11 is set to $0.1$.
We set the standard deviation of Gaussian kernel $\mathbf{G} \in \mathbb{R}^{30 \times 30}$ to 17.

\smallbreak
\noindent \textbf{Implmentation of high-dimensional convolution.}
As PyTorch~\cite{pytorch} supports only upto 3D convolution, we must manually implement (dense) high-dimensional convolutions.
We first demonstrate the original implementation of 4D convolution~\cite{rocco2018neighbourhood}, and how we efficiently re-implemented the same 4D convolution and improved it for high-dimensional convolution.
Given $B$ correlation tensors in a minibatch\footnote{We omit channel sizes of the tensor for brevity.} $\mathbf{C} \in \mathbb{R}^{B \times H \times W \times H' \times W'}$ and a 4D kernel $\mathbf{K} \in \mathbb{R}^{k \times k \times k \times k}$, we denote each 4D piece of $\mathbf{C}$ by $\mathbf{C}_{i} \coloneqq \mathbf{C}_{:i:::} \in \mathbb{R}^{B \times W \times H' \times W'}$ and each 3D tensor in $\mathbf{K}$ by $\mathbf{K}_{i} \coloneqq \mathbf{K}_{i:::} \in \mathbb{R}^{k \times k \times k}$.
The work of \cite{rocco2018neighbourhood} implements 4D convolution $f_{\mathrm{4D}}$ by performing $H$ times of following operation:
\begin{align}
    \label{eqn:rocco4d}
    f_{\mathrm{4D}}(\mathbf{C})_{i} = &f_{\mathrm{3D}}(\mathbf{C}_{i-p}, \mathbf{K}_{1}) + f_{\mathrm{3D}}(\mathbf{C}_{i-p+1}, \mathbf{K}_{2}) \\ \nonumber &+ ... + f_{\mathrm{3D}}(\mathbf{C}_{i+p}, \mathbf{K}_{k}) + b
\end{align}
where $f_{\mathrm{3D}}$ is a function that performs 3D convolution on $\mathbf{C}_{*}$ across the batch given 3D kernel $\mathbf{K}_{*}$, $p$ is a padding size, and $b$ is a bias term.
We set the padding size $p=\lfloor k / 2 \rfloor$ in our experiment.
As a result, $f_{\mathrm{4D}}$ in Equation~\ref{eqn:rocco4d} performs $kH$ times of 3D convolutions.

In this work, we implement a fast version of the 4D convolution which performs significantly smaller number of 3D convolutions compared to the original one.
We first reshape the correlation tensor of a minibatch as $\mathbf{C} \in \mathbb{R}^{BH \times W \times H' \times W'}$ and make $k$ copies of it.
Using the 3D kernels $\{\mathbf{K}_{i}\}_{i=1}^{k}$, we apply 3D convolution $f_{\mathrm{3D}}$ on each copy and denote its output by $\mathbf{\hat{C}}^{i}=f_{\mathrm{3D}}(\mathbf{C}, \mathbf{K}_{i})$.
We again reshape the tensors $\{\mathbf{\hat{C}}^{i}\}_{i=1}^{k}$ to have size $B \times H \times W \times H' \times W'$ and perform the following:
\begin{align}
    f_{\mathrm{4D}}(\mathbf{C})_{i} = \mathbf{\hat{C}}_{i-p}^{1} + \mathbf{\hat{C}}_{i-p+1}^{2} + ... + \mathbf{\hat{C}}_{i+p}^{k} + b.
\end{align}
Note that the number of 3D convolution operations in our implementation is $H$ times smaller compared to that in the original implementation~\cite{rocco2018neighbourhood} ($k$ (ours) vs. $kH$ \cite{rocco2018neighbourhood}).
Given a 4D correlation tensor $\mathbf{C} \in \mathbb{R}^{16 \times 30 \times 30 \times 30 \times 30}$, our implementation takes about 0.7 ms while the implementation of \cite{rocco2018neighbourhood} takes about 150 ms on a machine with an Intel i7-7820X CPU and an NVIDIA Titan-XP GPU.
A high-dimensional convolutions ($\geq$ 5D) are implemented in a similar manner; our implementation of 6D convolution with input in $\mathbb{R}^{16 \times 15 \times 15 \times 3 \times 15 \times 15 \times 3}$ takes about 180 ms on the same machine.

We also manually implement parameter-sharing kernels $k_{\mathrm{psi}}^{\text{nD}}$ and $k_{\mathrm{iso}}^{\text{nD}}$:
Before applying convolution, we instantiate high-dimensional kernel filled with zeros and assign parameters to their corresponding indices by addition.

\section{Qualitative results}

The proposed convolutional Hough matching allows a flexible non-rigid matching and even multiple matching surfaces or objects.
To demonstrate the ability of the CHM in matching multiple objects, we visualize some qualitative results of our method (CHMNet) on some toy images with multiple instances in Fig.~\ref{fig:multiple}.
Top 300 confident matches predicted by our model (CHMNet) are mostly on {\em common} instances in the input pairs of images.
Replacing convolutional Hough matching (learnable local voting layer) to regularized Hough matching~\cite{cho2015unsupervised,min2019hyperpixel} (non-learnable global voting layer) severely damages the model predictions;
the confident matches become noisy and unreliable, mostly being scattered on background.
Without CHM layers, the model fails to localize common instances in the images.
Figure~\ref{fig:pr_qualitative} also visualizes sample pairs of PF-PASCAL with top 300 confident matches predicted by each model.
Our model effectively discriminates between semantic parts and background clutters as seen in the second row of Fig.~\ref{fig:pr_qualitative}.
The absence of CHM layers severely harms the model predictions as seen in the third and last rows of Fig.~\ref{fig:pr_qualitative}.
These results reveal that the proposed CHM layers effectively find reliable matches between common instances across different images while being robust to background clutter even in presence of multiple instances.

The qualitative comparisons to the recent semantic correspondence approaches~\cite{huang2019dynamic,li2020correspondence,min2019hyperpixel,min2020dhpf,rocco2018neighbourhood} are visualized in Figs.~\ref{fig:pascal_qual}, \ref{fig:spair_qual_viewpoint}, and \ref{fig:spair_qual}.
We warp source images to target images using predicted correspondences:
Given source keypoints, each model predicts their corresponding positions in target image by using its own keypoint transfer scheme, \eg, nearest neighbor assignment~\cite{min2019hyperpixel,min2020dhpf}, hard-assignment by taking mostly likely match~\cite{huang2019dynamic,li2020correspondence,rocco2018neighbourhood} or soft argmax (ours).
Using the keypoint correspondences, we compute thin plate spline (TPS) transformation parameters~\cite{donato2002approximate} and apply the transformation to source image to align target image.
Figure~\ref{fig:pascal_qual} shows the results on PF-PASCAL.
Figures~\ref{fig:spair_qual_viewpoint} and \ref{fig:spair_qual} show the results on SPair-71k.
Our model effectively warp the source images to align the source objects to the target ones based on predicted correspondences even in presence of large view-point, illumination, and scale differences.
Representative failure cases of our model are shown in Fig.~\ref{fig:failure_cases}.

\begin{figure}[t]
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/multiple_match.pdf}
    \end{center}
    \vspace{-5.0mm}
       \caption{Multiple instance matching with top 300 confident matches.}
    \vspace{-2.0mm}
\label{fig:multiple}
\end{figure}

\begin{figure}[t]
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/failure_cases.pdf}
    \end{center}
    \vspace{-3.0mm}
       \caption{Failure cases on SPair-71k~\cite{min2019spair} dataset in presence of extreme changes in view-point, large intra-class variation, and deformation. We show the keypoints of ground-truth correspondences in circles and the predicted keypoints in crosses with a line that depicts matching error.}
    \vspace{-5.0mm}
\label{fig:failure_cases}
\end{figure}

\clearpage

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/scalemax_qual.pdf}
    \end{center}
    \vspace{-5.0mm}
       \caption{Visualization of maxpooled position in scale-space. In each image pair, we show source keypoints (given) and their corresponding target keypoints (predicted) in circles in left and right images respectively. The size (large, medium, and small) of each circle indicates maxpooled position in scale-space. If both circles of a match are large, its match score is pooled from position $(\sqrt{2}, \sqrt{2})$ in scale space. If the size of one circle is medium and that of the other is small, its match score is from position $(1, 1/\sqrt{2})$ and so on. We show ground-truth target keypoints in crosses with a line that depicts matching error. Best viewed in electronic form.}
    \vspace{-3.0mm}
\label{fig:scalemax_qual}
\end{figure*}

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/kernel_vis_demo_bg.pdf}
    \end{center}
    \vspace{-5.0mm}
       \caption{Description of visualizing learned weights of high-dimensional kernels: (Left) The arrows represent the offset vectors relative to the kernel position $(\mathbf{x}, \mathbf{x}')$, and the circles mean zero offset. (Right) For straightforward visualization, we decompose a high-dimensional kernel into multiple 4D kernels (tesseracts) and visualize learned weights of each 4D kernel as a set of maps consisting of offset vectors. Darker offsets mean larger weights while brighter ones mean smaller weights.}
    \vspace{-3.0mm}
\label{fig:kernel_vis_demo}
\end{figure*}

\clearpage

\begin{figure*}[h]
    \begin{center}
        \includegraphics[width=0.9\linewidth]{_arxiv_figures_appendix/pas_psi_bg.pdf}
    \end{center}
    \vspace{-2.0mm}
      \caption{Learned $k_{\mathrm{psi}}^{\text{6D-4D}}$ used in CHMNet. The 6D kernel ($k_{\mathrm{psi}}^{\text{6D}}$) consists of {\em four} 4D kernels each of which has 55 parameters.}
    \vspace{-3.0mm}
\label{fig:k_psi}
\end{figure*}

\begin{figure*}[h]
    \begin{center}
        \includegraphics[width=0.9\linewidth]{_arxiv_figures_appendix/pas_full_bg.pdf}
    \end{center}
    \vspace{-2.0mm}
      \caption{Learned $k_{\mathrm{full}}^{\text{6D-4D}}$. The 6D kernel ($k_{\mathrm{full}}^{\text{6D}}$) consists of {\em nine} 4D kernels each of which has 625 parameters.}
    \vspace{-3.0mm}
\label{fig:k_full}
\end{figure*}

\begin{figure*}[h]
    \begin{center}
        \includegraphics[width=0.9\linewidth]{_arxiv_figures_appendix/pas_iso_bg.pdf}
    \end{center}
    \vspace{-2.0mm}
      \caption{Learned $k_{\mathrm{iso}}^{\text{6D-4D}}$. The 6D kernel ($k_{\mathrm{iso}}^{\text{6D}}$) consists of {\em three} 4D kernels each of which has 15 parameters.}
    \vspace{-3.0mm}
\label{fig:k_iso}
\end{figure*}

\clearpage

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/supp_comparison.pdf}
    \end{center}
    \vspace{-2.0mm}
       \caption{Sample pairs with top 300 confident matches. TP and FP matches are colored in blue and red respectively.}
    \vspace{-3.0mm}
\label{fig:pr_qualitative}
\end{figure*}

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/pascal_qualitative.pdf}
    \end{center}
    \vspace{-2.0mm}
       \caption{Example results on PF-PASCAL~\cite{ham2018proposal}: (a) source image, (b) target image (c) CHMNet (ours), (d) DHPF~\cite{min2020dhpf}, (e) ANC-Net~\cite{li2020correspondence}, (f) HPF~\cite{min2019hyperpixel}, (g)
       DCCNet~\cite{huang2019dynamic}, and (h) NCNet~\cite{rocco2018neighbourhood}.}
    \vspace{-3.0mm}
\label{fig:pascal_qual}
\end{figure*}

\clearpage

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/spair_qualitative_viewpoint.pdf}
    \end{center}
    \vspace{-2.0mm}
       \caption{Example results with large view-point differences from SPair-71k~\cite{min2019spair}: (a) source image, (b) target image (c) CHMNet (ours), (d) DHPF~\cite{min2020dhpf}, (e) ANC-Net~\cite{li2020correspondence}, (f) HPF~\cite{min2019hyperpixel}, (g)
       DCCNet~\cite{huang2019dynamic}, and (h) NCNet~\cite{rocco2018neighbourhood}.}
    \vspace{-3.0mm}
\label{fig:spair_qual_viewpoint}
\end{figure*}

\clearpage

\begin{figure*}
    \begin{center}
        \includegraphics[width=1.0\linewidth]{_arxiv_figures_appendix/spair_qualitative.pdf}
    \end{center}
    \vspace{-2.0mm}
       \caption{Example results with large illumination and scale differences, and truncation from SPair-71k~\cite{min2019spair}: (a) source image, (b) target image (c) CHMNet (ours), (d) DHPF~\cite{min2020dhpf}, (e) ANC-Net~\cite{li2020correspondence}, (f) HPF~\cite{min2019hyperpixel}, (g)
       DCCNet~\cite{huang2019dynamic}, and (h) NCNet~\cite{rocco2018neighbourhood}.}
    \vspace{-3.0mm}
\label{fig:spair_qual}
\end{figure*}
\end{appendices}

redpajama_set_name: RedPajamaArXiv


text: Q: FTP via intermediary computer or network binding? Linux Server 18.04 installed vsftpd as my FTP server. This machine is also running openVPN, but when VPNs tun0 is active obviously my IP changes. Is there anyway i can route FTP traffic to connect to another machine on my network and direct it to the server like when i ssh via a jumpbox, or is there anyway to bind vsftpd to use my actual network and ignore tun0?