Want to purchase a pool table in Melbourne? When you want to make sure that you are getting the best pool table repair service at the best price for your pool table, you can choose a pool table sale. Pool table repair is not an easy task. It takes knowledge and experience to properly repair a pool table that has been in constant use for many years. A pool table repair may also be necessary if your pool table is out of order or it has been damaged. If you find that you do need pool table repair but you don’t know where to start, then there are a few things that you should consider.

 

Basic pool table repair includes cleaning and polishing pool tables that have not been in use for long. Typically, pool tables are not cleaned every year like homes are. The general rule of thumb for pool tables in most homes is that they are cleaned when the weather is warm. If you have had your pool tables for many years and they are sitting winter blank, it is probably time to clean them. When you find that the weather has started to turn cold, you may want to consider a pool table repair service.

 

Many pool tables come with accessories such as chalkboards or billiard balls. Chalkboard chalkboards usually come with a warranty. If your pool table has been used many times and the chalkboards no longer write properly or have crackling sounds, you may want to replace your pool tables. Chalkboards are relatively inexpensive, but you can still purchase replacement chalkboards for your pool tables at a pool table sale.

 

If you have spent many years playing on wooden pool tables and notice that the wood has started to wear, it may be time to have the entire pool table repaired. Many pool table repair companies specialize in repairing pool tables. If you do not know of anyone who can repair your pool table, you can search on the Internet for pool table repair shops. Some of these shops offer to repair any pool table that has been damaged due to sun damage. Others specialize in repairing specific brands of pool tables.

 

When you begin searching for a pool table sale, you should not overlook second-hand pool tables for sale in Melbourne. There is more to finding a top quality pool table than just its price tag. If you plan on playing on your pool table for many years to come, you need to make sure that you find a pool table with low maintenance costs. When you look for pool table sales, take into account the cost of replacement parts. Pool tables are made from different materials, so it will be necessary to choose the right pool table for you.

 

One of the best places to find great prices on pool tables is on the Internet. There are many online pool table repair shops that offer discounts on pool tables. These shops can service all brands of pool tables, so you do not have to leave your home to get your hands on a great pool table. Before you buy a pool table, be sure to check out the various options that are available. You can often save money on pool tables by shopping online.

Are you looking for information on how to install RV wifi? Then this is the right place for you. Your caravan may be one of the most versatile vehicles when it comes to being able to stay connected to the internet. The problem with older forms of caravan internet connectivity is that they are usually very weak and provide very poor signals. This means that you need an updated form of internet to enjoy your travels – for example from a caravan internet modem. In this article we are going to look at some of the options you have for improving your caravan wifi connection and getting a stronger signal.

 

Some of the old school methods of gaining internet access include using public hotspots. Public hotspots are places like public libraries, schools and cafes that offer free internet access. Unfortunately public hotspots tend to be quite crowded and many times are shared by hundreds of other people. This can make your internet connection weak and actually reduce the speed that you can access it at.

 

One of the best alternatives that you have is satellite internet. Satellite broadband has been around for quite some time and is actually just as strong as public hotspots. Not only does it offer a strong signal but it also has excellent streaming services, which means that you can do all your work from virtually anywhere in the world. In this article we are going to take a look at these two great benefits of choosing satellite internet for your caravan holiday.

 

The first two of satellite internet is obviously the ability to go wherever you want and whenever you want. You no longer need to be stuck at a public hotspot with a slow connection that doesn’t meet your expectations. Instead, you can sit anywhere in the world and still access the internet at incredible speeds. It is this ability to move around easily that makes satellite internet such an amazing service.

 

Another of the fantastic things about having a mobile phone is having a wireless connection. This is why having a mobile plan for your caravan can be such a huge benefit. Caravan wifi plans are much more flexible than their home-based counterparts and allow you to move about with your laptop any time you like. You can even use your mobile phone to make calls while you’re on the move. With a mobile plan you never have to worry about getting a fixed-line connection because mobile phone signals are so strong.

 

These are just a few of the top pros of choosing satellite internet for your caravans. There are many others including the fact that you can keep on top of work even when you’re on holiday, the ability to stay connected no matter where you are and the availability of numerous mobile broadband dongles that you can connect to. There are other benefits as well but these are the biggest reasons to choose satellite internet. Once you start using satellite internet for your caravan, it won’t be long before you compare it to your home-based service. With all of these wonderful benefits you should definitely consider satellite internet for your next mobile internet investment.

Baking with a cookie recipe lover in your corner? You have just stumbled on the perfect article! I have so many of my readers asking me how to bake great cookies. I get it, when I am asked that question by my audience I know what to say. It is such a simple question and yet so complex and delicious. We all love cookies, we all eat them, we all make them – we all know someone who does.

Let’s start by describing some of the most basic of baking recipes. They include; angel food cake mix, gingerbread mix, plain sugar, brown sugar, cocoa powder, vanilla and baking soda, cinnamon, and butter. Some of these items can be purchased in a specialty store and in most cases found in a supermarket or mass merchandiser. Baking soda is a very economical and handy item to have in your home.

Some more advanced cookie baking ideas include; decorating cookies, glazing cookies, adding fruits and nuts to cookies, and making sugar cookies. With today’s markets become more competitive cookie baking businesses and blogs are cropping up everywhere. These new blogs and businesses are providing information for people looking for great cookie recipes and ideas for making their own cookies at home. This is great news for consumers, because the people creating these blogs and small businesses are reaching out to the market and competing with the big boys and are giving the consumer another option. If the customer has more options they generally choose a better business and product.

If you are looking for great cookie dough try this recipe for homemade cookie dough. All you need is 3 cups of white, whole wheat flour, three eggs, two tablespoons of instant yeast, and 2 teaspoons of lemon juice. Start by taking one tablespoon of the flour and breaking it into a ball. Then take another tablespoon of the flour and break it into a ball. In order to make the dough come alive add about one tablespoon of instant yeast to the mixture after it is completed.

Next add the milk, a quarter cup of sugar, and one tablespoon of vinegar to the dry ingredients. Mix thoroughly and then just pour into a greased bowl, and start baking. If the cookies start to look dry after baking just add more milk and continue to bake until they are done.

How to bake a delicious cookie is not difficult when using cookie sheets. Simply place the cookie sheet on your oven and preheat the oven to 350 degrees. Once the cookie sheet is done, you can add the dry ingredients to the pan along with the tablespoon of instant yeast, the wet ingredients, and a quarter cup of light or dark chocolate chips or coconut flakes, if you prefer. Bake on the middle rack of the oven for approximately ten to twelve minutes or until the cookies start to bounce back to life and pop in the pan.

Homemade cookie dough can be used in a variety of recipes. These include traditional peanut butter and banana nut chocolate chip cookies, sugar cookies, brownies, donuts, cup cakes, and many others. Some people even enjoy making their own version of Christmas sugar cookies, donuts and brownies. The key to success with homemade cookie dough is using all healthy ingredients and utilizing a slow baking process. Baking a whole batch of cookies at a time takes time, but the end results will definitely be worth the effort.

If you are looking for more ideas for your own unique creations, you can also search the internet for websites that provide cookie dough recipes. You can find hundreds of websites that offer tips and advice on how to bake a variety of cookies. This is a great place to get ideas for new shapes, flavors and ingredients to use when making cookies. When you have finished baking your cookies, you can send them off to be packaged and shipped directly from the store. This is a great way to save money and to make special treats for your family and friends.

Try handletheheat.com for baking recipes from Tessa.

“Why You Would Want A Custom Flag?” It’s simple to say, but what is more important than that? When asked that question, many people have several different responses. They may come right out and say “For personal pride” or “For protection from crime.” Although those are good reasons, I am sure there are much more and this article will list some of those reasons.

For years the custom flag has been used to identify an individual, family, company, group or organization. The flag can also be used to express an opinion or give an observation about a country, government, institution or person. It’s basically a way for the community to unite and show support for something. Why is there a need for custom flags?

Well, lets face it; we live in a melting pot. There are many different cultures and beliefs throughout the world. Have you ever been to China? How do they identify their country with the rest of the world? How is that even possible? Without flags, it would be impossible to tell what country someone came from or what country they were from.

If you go to any public place in China, you will see Chinese locals waving their flag at other Chinese. That’s how universal this flag is. We don’t want to disrespect another culture or country. So why not use that as a way to express ourselves? That’s what patriotism is all about isn’t it?

The second reason to have a custom flag is to show your support for your country. In essence you are showing your love for your country. I’m sure that’s not what you had in mind when you ordered your custom flag. Did you think about how great it would be if you won the championship the previous year? If not you know now how much it would mean to win the championship for your country.

There are many people who really don’t understand what the importance of a custom flag is. Some of them say that it’s just a red flag to tell the world that we’re sore losers. Is that really the case though? What happens if the world realizes that the flag isn’t really representing the country? Will they still support that country?

That’s the only problem with using a flag for promotion. There isn’t really a problem with the country using the flag. The problem is with the people using the flag. I’ve seen far too many people with their flag hand out at events and not giving it back to anyone. It’s disgusting.

So there you have it. Those are just two of the many reasons why you would want a custom flag. Just remember that you make it and that it represents your company, organization, or cause. Good luck in all your endeavors.

If you’re not sure about what type of flag you would like to have, take some time to check out some different options. Speak to a flag supplier and printer. The United States flag has been around forever and it never goes out of style. They also keep changing colors, so you’ll want to stay current as well. Try to find something that represents you, your company, or your cause the best.

When looking for a custom flag, consider the size and shape of it. Are you trying to say something with your custom flag? Make sure it stands out and says something about you. Do your research before making your final decision.

As I stated before, it’s important to give the person giving the custom flag a name. Give it something personal so that they know who they are giving it to. Do your research beforehand to see how much postage you will be paying and what the cost will be. This is a great way to make sure you aren’t breaking any rules.

When people think of why you would want a custom flag, these are the types of questions that come to mind. Whether you are a government agency, a business, or a nonprofit organization, you can get a custom flag designed to represent your business or cause. Check out the different websites on the internet for ideas. If you can’t find what you are looking for, there are many different companies that will help you make your custom flag.

Risk Management is not only for corporations. It applies to all businesses including small businesses, partnerships, family farms and literally every business entity you can think of. As such, there are many sectors of risk management that businesses are required to comply with. When it comes to the agricultural risk management for farmers, a very important one is agriculture risk management for farmers’ risk management. Agriculture risk management for farmers involves implementing several risk management strategies, including data collection, risk management programs, and risk management strategy implementation.

The agricultural risk management process Melbourne starts by defining hazards and risks. These are the factors that make things happen. For example, if a farmer discovers a chemical spill, it’s time-sensitive and can have catastrophic results. By identifying risks early on, a risk management team can minimize the effects of these events by controlling the risk at the earliest stages possible. In this case, the aviation risk management team may focus their efforts on chemical spills and working with affected farms to safely dry out the area.

In order to apply risk analytics in the agricultural risk management context, a team needs to have its own analytics suite. There are several open standard databases that can be used for risk analysis in the agricultural industry. One open standard database is The Agriculture Risk Intelligence (ARIS). ARIS contains comprehensive and timely information on risk profiles of almost all the factors that affect the agricultural industry. This database was developed by the United States Department of Agriculture (USDA) and is managed by the National Agriculture Information Center (NACE).

N ACE maintains a list of agriculture risk management standards and energy risk management strategies that are required to comply with the policies of the USDA. Along with risk profile information, N ACE maintains a list of business objectives that are needed to meet the goals of the organization. For example, the business objectives of the USDA may include the reduction of risks to public health and safety, economic stability of the agricultural sector, food supply security, environmental quality, product quality, animal welfare and animal health, among others.

In addition to the above mentioned policies, risk management standards may also include the implementation of controls and risk management systems in a given environment. For example, biological hazards may be present in an agricultural setting. A risk management standard will identify how the presence of such hazards should be controlled. A control system will then be implemented to address potential dangers. Similarly, biological hazards may occur due to the use of chemicals in the agricultural setting. A risk management standard will identify the types of chemicals commonly used and their effects on the environment and human health.

Besides the policies and procedures that relate to risk management, there are two important concepts that are crucial to this practice and they are crisis risk management and threat management. Crisis risk management is concerned with reducing risks that could lead to catastrophic events or other catastrophic outcomes. Examples of catastrophic outcomes include floods, storms, earthquakes and civil conflicts. Crisis risk management takes place in different fields such as medicine, finance, infrastructure and food markets. Threat management takes place in areas such as nuclear proliferation, terrorism, weapons proliferation and ethnic and religious conflicts.

These concepts are interrelated and need to be understood properly for the risk management process to be successful. When an error is made in the assessment process, it can lead to disastrous consequences. For instance, if one’s risk assessments fail to take into account the possible impact of price increases or decreases, the resulting risk level can be much higher than one might initially assume. Likewise, if one’s risk assessments do not take into account the probability of natural disasters, human errors can also have disastrous effects. Therefore, before any risk management activities take place, it is imperative that risk management plans Melbourne are in place that address each and every aspect of the various aspects of risk. The various risk management activities include identification, assessment, prioritization, monitoring, corrective and control.

The risk management process also needs to incorporate climate change risk management, which is concerned with incorporating climate change into the overall risk management plan in order to decrease costs and reduce damage. This type of risk management seeks to identify the vulnerabilities associated with weather conditions and the associated environmental changes that are likely to affect natural disasters and events. Climate change risk analytics was introduced to help managers and agencies to address the issues associated with climate change and are based on the assumption that Earth’s average temperature is rising. This risk management focuses on three aspects: adaptation, mitigation and risk management.

If you are planning to have an online presence, a website development melbourne firm can help you achieve the same. They provide cost-effective solutions and expert services that will enhance your website. Before signing on a non-disclosure contract (NDA), however, you should first check whether the web development company you are considering has the required skill sets, knowledge and expertise required for your project. You should also be able to determine how professional and courteous the staff members of the web development firm are and whether they respond promptly and efficiently to your queries and requests.

 

While most web development companies are usually open to hiring from a pool of freelancers, it is still best to choose a team that has a track record of delivering quality work. Look for testimonials and reviews of the company from its past clients and preferably talk to the current clients to find out how well the service was. Ideally, the team should have several in-house programmers who are experienced and can be deployed on any project at any time. The best teams have a number of experienced developers who work together to deliver consistent and quality web applications.

When selecting a team, like https://www.foxandlee.com.au, it is important to evaluate what each member brings to the table. Each member must have some skill set or capability which can contribute positively to the completion of your web development project. It is equally important that each member is able to contribute in a meaningful way to the overall success of the web development project. Your web developers should be skilled and comfortable working together in an environment where communication and feedback are easily provided. Communication channels such as phone and email should be available to all team members during the course of the project.

In addition to having a quality code of conducts, you should also hire a dedicated web development team that you know is committed to continuing to support your business. Most companies hire web developers or designers who leave within one or two months after the initial contract due to being unable to meet their deadlines. When you hire a dedicated team, you ensure that you are not making a mistake by retaining a developer or designer who will disappear just as soon as you receive payment. The most dedicated web developers will continue to work with you even though they are not receiving any payment for their services.

It is also very important that you select a web design and development firm that will be flexible enough to meet your evolving business needs. It is possible to find web developers and designers who are experienced in both client-side programming and server-side programming. As your business grows, it is often necessary to switch developers or designers to accommodate growth. As your needs change, it is advisable to hire web development teams that are able to adapt to your changing needs.

It is also important to hire a team that has soft skills. Good soft skills means that the development team has the ability to work with a client without feeling threatened or intimidated. If you want to hire web developers and designers that are capable of working with all types of clients, it is advisable to search for firms that have worked with various industries, including those that are into internet and ecommerce.

If you’re a beginning singer, you have likely heard the term “vocal coach” or “singing instructor”. What is a vocal coach and how does he or she differ from a singing instructor? There are actually a few basic differences. First of all, a vocal coach will focus on teaching techniques for vocal technique and not just improving your singing ability. A singing instructor focuses on improving the quality of your voice so you can sing better. A vocal coach and singing instructor will also use different resources and methods to teach their students and these methods may be very different from one another. The first main difference between a vocal coach in Nashville TN and a singing teacher is the primary focus of their lessons. Vocal coaches work primarily on vocal technique. They will do vocal exercise routines, vocal technique drills and even single notes, designed specifically to help you improve your ability or to refine your technique in a certain area of your range. A singing teacher, on the other hand, will spend more time showing the singer’s the proper way to use their voice as well as teaching students various aspects of vocal technique. The teacher may use music or even computer programs to help the students with the techniques. Both Vocal Coaches and Voice Teachers have basic educational goals. They want their students to be fully prepared to hit their registers with maximum power and consistency and both Vocal Coaches in Nashville TN and Voice Teachers have specific educational requirements to meet before they can formally start lessons. Some Vocal coaches teach only one certain technique whereas some teachers are well versed in many different areas of vocal performance, such as resonators, head phones, voice projection and even bass lines. Many aspiring singers believe that it is possible to become a professional vocal coach or at least a better vocal coach than what they are already. This is much harder than it actually sounds. To get the best results out of either a singing teacher or a vocal coach, you need to have certain characteristics in common. First, both a singing teacher or a vocal coach should know how to motivate a student. A student who shows up for vocal lessons and does not get the results he or she is hoping for will look for a new teacher. A good singer will always seek motivation from a singing teacher or a vocal coach. Secondly, both teachers and motivational speakers will know when a student needs to listen and take notes. A good singer will be able to take notes themselves and understand what a professional singer is saying to them in order to better sing. One characteristic that you should look for in a vocal coach or a singer teacher is an ear for when a child needs to adjust a technique during a song. If a singer has a natural tendency to over-do a technique during a song then the vocal coach may teach the student how to properly modify the technique. An example of this would be mapping between syllables. If the student hears that his or her breath is overlapping another voice, then it will be a good idea to let the girl breathe evenly through the chest instead of letting the breath hit the lower lungs. An additional characteristic of a good vocal coach or singing teacher is that they will teach both upper and lower larynx muscles. The most common problem among singers is that their vocal chords are not positioned correctly. The singer may have a perfectly aligned larynx but the problem is that the voice does not sound clear because of the clashing sounds caused by the upper larynx and the lower larynx. A good vocal coach will help a singer align his or her larynx properly and allow the person to sing clearer and louder helping when music producers in Nashville TN are looking for new talent.. Some of the more advanced vocal techniques such as gargling, humming and vibrato can also be taught to a singer by a gifted vocal coach or singing teacher. The best vocal coaches and teachers will always tell their students the right technique to use regardless if they are beginners or seasoned professionals. The technique they use for one person might not work well for another. Therefore, the vocalist must always ask questions and listen to what the instruction entails for his or her own individual needs. A good vocal coach or teacher always listens carefully to the lessons that their students take and gives good and constructive feedback to ensure that the students understand the lessons and proceed on the artist development services in Nashville TN.

With so many different types of producing videos Melbourne options, how do you know which professionals in Melbourne are going to provide you with the best service and how can you make an impact by choosing the right videographer? This is a question that many people often ask but don’t always have the answer to. There are several factors that go into hiring the best video and audio professional for your business events, meetings, seminars, conventions, promotions, product launch events and social media promotions.

 

The first step on how to hire a video production company Melbourne is making a list of potential professionals in your area. When making this list, you want to include everyone from the best videographers in Melbourne to the most affordable videographers. If you want to hire an amazing videographer who charges top dollar then it is likely you will not find anyone within your price range. However, if you are willing to spend some time doing research then you should be able to find a great professional who is willing to work within your budget and deliver a quality job.

 

Once you have your list, you need to start contacting each videographers on your list to see how they can help you with your event. Once you have established a relationship or rapport with one of the videographers from Melbourne then you should continue to communicate with them until you are comfortable with them. You need to establish what type of videography you would like to have for your event. For example, if you are having a grand opening or other big event then you may want to look at live action videography. If you are having a product launch then you may want to look at product videos. It is important to know what type of service you need so that you can communicate that clearly to the videographer.

 

Once you have spoken with a few videographers on your list and you are comfortable with them then it is time to make your decision. The first thing that you want to do is find out how much experience each of the professionals have. A good way to determine this is to ask how long they have been working with this type of service and how many awards they have received. Also take into consideration how long they have been in their field and how successful they have become. There is no point in hiring the best videographer in Melbourne if they have never won an award or been recognized for their work.

 

Once you have all of this information that is necessary for you to make a well informed decision, you are ready to begin your search for the best videographer in Melbourne. The only thing left to do at this point is to contact them and tell them how you feel about their work, how long you want the video to be and what you want to be shown during the video. You want to keep in mind that some videographers may be willing to work quickly, but others will need a little more time to complete your project. Finding the right videographer for you is going to be a very personal decision, but once you have made it you can rest assured knowing that you have hired the best in the business.

 

Once you have a short list of professionals then you are ready to begin your search online. You will find a website for each videographer that will give you all of the information that you need to make your decision and once you have chosen which videographer you want to work with, you can get started on your video project. No matter what the size of your video, or what type of videographer you are looking for, you are going to find exactly what you need online.

Using a seattle web design agency to build a business website is a good decision for many people. This type of company can often provide clients with a customized website that is designed to meet the needs of the individual business. Using a professional will help ensure that the site is not only effective but professional as well. In addition to this, the website will be created in a manner that will keep with current market trends. When considering all of these benefits, it becomes easier to see why people are turning to agencies to create their website. These pros and cons should help anyone come to a conclusion as to whether or not to use a web design company for their next project.

One of the pros is that most companies are affordable. These companies are not big corporations that charge exorbitant fees for their services. Therefore, even if a person has limited funds for a project, they will likely be able to get a reasonable plan set up for how to go about using a website. Another pro is that these companies are very organized. Most have several different designers working on different aspects of the company’s website at one time. Because of this, no two websites will be exactly alike.

The main con when using a web design agency to build a business website is that most companies charge a lot of money. This cost is generally associated with a monthly or annual fee. The average price for a business website can run from as little as two hundred dollars per month to over one thousand dollars per year. Before deciding upon a specific company, individuals should make sure to do some research so that they will know what companies are comparable in terms of price.

Another of the pros when using a company for the creation of their website is that it will likely be less complicated to use than building the website from scratch. Some people are not good at creating websites. Using a service will give them a professional looking website without having to learn any programming or HTML code. Most people do not have the technical skills necessary to create a website without knowing a lot about either. Most of the companies that offer web design services also offer other types of website development services and marketing services such as ppc seattle (Google Pay Per click).

One con of using a web design agency to build a business website is that the end product may not be as high quality as a website that was designed by an individual. Many companies do not offer the same amount of expertise as an individual. When a person is hired to design a website, they will likely know how to optimize the site for search engines. They also will have knowledge of marketing tactics that can help the business get more traffic.

The pros and cons of using a web design agency to build a business website is something that should be carefully considered. If a person feels that they do not have the skill required to build a quality website, then it may be best to hire a company that offers these services. It is important to choose a company that has experience in this type of web site creation. A good way to find out is to contact them and ask what kind of experience they have. By doing this, a person can determine if they will be able to handle the job or not.

On sunday is the 2010 Int. Symp. on Voronoi Diagrams (ISVD). We shall present: Jensen-Bregman Voronoi diagrams and centroidal tessellations

Here are the slides of the talk.
The Jensen-Bregman divergence is a distortion measure defined by the Jensen difference provided by a strictly convex function. Jensen-Bregman divergences extend the well-known Jensen-Shannon divergence by allowing to choose an arbitrary convex function generator instead of the standard Shannon entropy. This class of Jensen-Bregman divergences notably includes the squared Euclidean distance. Although Jensen-Bregman divergences are symmetric distances by construction, they are not metrics as they violate the triangle inequality. We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of information-theoretic divergences. We show that those Jensen- Bregman divergences appear naturally into two contexts: (1) when symmetrizing Bregman divergences, and (2) when computing the Bhattacharyya distances of statistical distributions. The Bhattacharyya distance measures the amount of overlap of distributions, and is popularly used to provide both lower and upper bounds in machine learning: the Bayes? misclassification error. Since the Bhattacharyya distance of popular distributions in statistics called the exponential families (including familiar Gaussian, Poisson, multinomial, Beta/Gamma families, etc.) can be computed equivalently as Jensen-Bregman divergences, the Jensen-Bregman Voronoi diagrams allow one also to study statistical Voronoi diagrams induced by an entropic function. Keywords-Jensen?s inequality, Jensen difference, Bregman divergences, Jensen-Shannon divergence, Bhattacharyya distance, Burbea-Rao divergence, Hellinger-Kakutani-Matusita divergences, f-divergence, -divergence, exponential families, information geometry.

The following work has been presented at CVPR 2010. It considers orthogonal projection onto a tangent plane instead of vertical projection when defining a distance using a convex generator.

Shape database search is ubiquitous in the world of biometric systems, CAD systems etc. Shape data in these domains is experiencing an explosive growth and usually requires search of whole shape databases to retrieve the best matches with accuracy and efficiency for a variety of tasks. In this paper, we present a novel divergence measure between any two given points in Rn or two distribution functions. This divergence measures the orthogonal distance between the tangent to the convex function (used in the definition of the divergence) at one of its input arguments and its second argument. This is in contrast to the ordinate distance taken in the usual definition of the Bregman class of divergences [4]. We use this orthogonal distance to redefine the Bregman class of divergences and develop a new theory for estimating the center of a set of vectors as well as probability distribution functions. The new class of divergences are dubbed the total Bregman divergence (TBD).We present the L1-norm based TBD center that is dubbed the t-center which is then used as a cluster center of a class of shapes The t-center is weighted mean and this weight is small for noise and outliers. We present a shape retrieval scheme using TBD and the t-center for representing the classes of shapes from the MPEG-7 database and compare the results with other state-of-the-art methods in literature..

Simplification and hierarchical representations of mixtures of exponential families just got out at Signal Processing
Abstract

A mixture model in statistics is a powerful framework commonly used to estimate the probability measure function of a random variable. Most algorithms handling mixture models were originally specifically designed for processing mixtures of Gaussians. However, other distributions such as Poisson, multinomial, Gamma/Beta have gained interest in signal processing in the past decades. These common distributions are unified in the framework of exponential families in statistics. In this paper, we present three generic clustering algorithms working on arbitrary mixtures of exponential families: the Bregman soft clustering, the Bregman hard clustering, and the Bregman hierarchical clustering. These algorithms allow one to estimate a mixture model from observations, to simplify such a mixture model, and to automatically learn the ?optimal? number of components in a simplified mixture model according to a resolution parameter. In addition, we present jMEF, an open source JavaTM library allowing users to create, process and manage mixtures of exponential families. In particular, jMEF includes the three aforementioned Bregman clustering algorithms.

Keywords: Mixtures models; Kullback?Leibler divergence; Bregman divergence; Exponential family; Mixtures of exponential families; Clustering algorithms

Yet another report by Olivier who attended RFIA 2010

RFIA 2010 — Reconnaissance de formes et intelligence artificielle

RFIA (which stands for Reconnaissance de formes et intelligence artificielle) is a French-speaking conference about pattern recognition and artificial intelligence. The 17th edition held in Caen from January 20 to January 22 2010.

There was four parts in the workshop: an invited speaker session, a short talk session, a poster session and a demo session.

The short talk session was divided into two tracks: pattern recognition and artificial intelligence. I mainly followed the Pattern Recognition track. Unfortunately I was only able to attend the first two days of the conference, so I will present only talks from these days. Sang Ly presented a method to estimate moves of a stereoscopic vision system. Hervé Jégou presented a new compact representation of bag-of-words for image retrieval. Thierry Germa presented some results to follow people with a mobile platform, using information from image and RFID tags. Pierre Lébraly explained an extrinsic calibrating method for multi-camera system, using a planar mirror.

The three invited talks I attended were given by Olivier Teytaud (INRIA Saclay, France) speaking about artificial intelligence methods for Go game, Schlomo Zilberstein (University of Massachusetts, USA) presenting challenges and directions for decentralized decision making and by Nicu Sebe (University of Amsterdam, Nederlands) detailing some perspectives for human centered computing.

Frank.

Olivier attended the VISAPP conference where he presented results on

Reranking with Contextual dissimilarity measures from representational Bregman k-means

Here is the abstract followed by a short report of the conference.

We present a novel reranking framework for Content Based Image Retrieval (CBIR) systems based on con-textual dissimilarity measures. Our work revisit and extend the method of Perronnin et al. (Perronnin et al., 2009) which introduces a way to build contexts used in turn to design contextual dissimilarity measures for reranking. Instead of using truncated rank lists from a CBIR engine as contexts, we rather use a clustering algorithm to group similar images from the rank list. We introduce the representational Bregman divergences and further generalize the Bregman k-means clustering by considering an embedding representation. These representation functions allows one to interpret ?-divergences/projections as Bregman divergences/projections on ?-representations. Finally, we validate our approach by presenting some experimental results on ranking performances on the INRIA Holidays database.

VISAPP 2010 — International Conference on Computer Vision Theory and Applications

VISAPP is part of VISIGRAPP – The International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications. The purpose of this joint conference is to bring together researchers and practitioners on the areas of computer vision, imaging, computer graphics and information visualization, interested in both theoretical advances and applications in these fields. Computer Vision, Imaging, Computer Graphics and Information Visualization are well known areas which are becoming more and more interrelated with important interdisciplinary work, often as a result of an iterative combined process of image analysis and synthesis with models created in one of the fields being used to improve models created in another.

The VISIGRAPP component conferences are specialized in the following topics: GRAPP is structured along four main tracks, covering different aspects related to Computer Graphics, from Modelling to Rendering, including Animation and Interactive Environments, IMAGAPP covers theory, applications and technologies related to image display, colour coding, medical imaging, remote sensing, business document processing, digital fabrication, printing and electronic devices, VISAPP has also four main tracks, namely: Image Formation and Processing, Image Analysis, Image Understanding and Motion, Tracking and Stereo Vision and IVAPP structured along several topics related to Information Visualization.

The 2010 edition of VISIGRAPP was held in Angers (France) from May 17th to May 21th. There were 4 invited speakers: Pascal Fua, from the École Polytechnique Fédérale de Lausanne (Switzerland), spoke about different ways of modeling deformable surfaces from monocular video sequences; Ali Mohammad-Djafari, from CNRS and Supéléc (France) spoke about methods for solving inverse problems, mainly regularization and bayesian estimation approaches, with applications in imaging and computer vision. The talk by Gabriela Csurka, from Xerox Research (France), was about Fisher kernel representation, describing Bag-Of-Features representation, Fisher vector and some applications. The last keynote speaker, Brian Barsky from University of California Berkeley, described recent research for modeling the human vision process, mainly for simulating eye diseases and consequences of surgery cures.

There were a lot of parallel sessions (2 or 3 only for VISAPP) which led to difficult choices between interesting talks. I will just a say a few words about some talks from the session I attended: Nuria Ortigosa presented the paper “Disparity maps for free path detection” describing an algorithm to detect obstacles and free path using stereo images. Sang Min presented the paper “Object retrieval based on user-drawn sketches”, describing a similarity measure between sketches based on tensorial fields. Duan-Yu Chen, presenting the paper “Real-time gender recognition for uncontrolled environment of real-life images” spoke about a way to combine different approaches in order to get a robust algorithm. Laura Papaelo presented TOPMESH, a powerful library to build models of non-manifolds objects. Codruta Orniana-Ancuti, presenting “Robust grayscale conversion for vision-substitution systems”, described a method to convert color images to grayscale images suitable for an auditory-vision system. Roland Moerzinger presented the paper “Improving person detection in videos by automatic scene adaptation” which introduces a way to improve person detection quality in the case of a static camera and a single planar ground. Zhaolin Su, for the paper “Real-time enhancement of image and video saliency using semantic depth of field”, presented a way to change the depth of field of an image using saliency information. The talk of Svenja Kahn, “Time-of-flight based scene reconstruction with a mesh processing tool for model based camera tracking”, described a way to build 3D models using a time-of-flight camera without the need of a prior knowledge of the model, for application in augmented reality.

Even if the goal to allow meetings between people from different fields is very interesting, it was really difficult to attend sessions from other fields without missing interesting talks in one’s own field.

The Jensen-Bregman divergence is a distortion measure defined by the Jensen difference provided by a strictly convex function. Jensen-Bregman divergences extend the well-known Jensen-Shannon divergence by allowing to choose an arbitrary convex function generator instead of the standard Shannon entropy. This class of Jensen-Bregman divergences notably includes the squared Euclidean distance. Although Jensen-Bregman divergences are symmetric distances by construction, they are not metrics as they violate the triangle inequality. We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of information-theoretic divergences. We show that those Jensen- Bregman divergences appear naturally into two contexts: (1) when symmetrizing Bregman divergences, and (2) when computing the Bhattacharyya distances of statistical distributions. The Bhattacharyya distance measures the amount of overlap of distributions, and is popularly used to provide both lower and upper bounds in machine learning: the Bayes? misclassification error. Since the Bhattacharyya distance of popular distributions in statistics called the exponential families (including familiar Gaussian, Poisson, multinomial, Beta/Gamma families, etc.) can be computed equivalently as Jensen-Bregman divergences, the Jensen-Bregman Voronoi diagrams allow one also to study statistical Voronoi diagrams induced by an entropic function.

We study the centroid with respect to the class of information-theoretic distortion measures called Burbea-Rao divergences. Burbea-Rao divergences generalize the Jensen-Shannon divergence by measuring the non-negative Jensen difference induced by a strictly convex and differentiable function expressing a measure of entropy. We first show that a symmetrization of Bregman divergences called Jensen-Bregman distances yields a natural definition of Burbea-Rao divergences. We then define skew Burbea-Rao divergences, and prove that skew Burbea-Rao divergences amount to compute Bregman divergences in asymptotical cases. We prove that Burbea-Rao centroids are always unique, and we design a generic iterative algorithm for efficiently estimating those centroids with guaranteed convergence. In statistics, the Bhattacharyya distance is widely used to measure the degree of overlap of probability distributions. This distance notion is all the more useful as it provides both upper and lower bounds on Bayes misclassification error, and turns out to be equal at the infinitesimal level to Fisher information. We show that Bhattacharyya distances on members of the same exponential family amount to calculate a Burbea-Rao divergence. We thus get as a byproduct an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods that were mostly limited to univariate or “diagonal” multivariate Gaussians.

Here is the arxiv report

Well, the title of the blog is computational information geometry. It includes the word “computational”. Of course, one aspects is to bring computational geometry vistas to information geometry by fostering algorithmic techniques. Another aspect, is to ponder what can be done with computations in the field of information geometry. The basic elements being numbers, let us realize that computational reals are of measure zero in the set of reals.
A real is said computationable if we can enumerate all its bits one by one (up to infinity). Emire Borel in the 1920’s already felt strange with this notion (that came in time for Turing later): consider all YES/NO questions (say, in French -:) ) and let each bit of the Borel real B answer the corresponding question.

Being able to compute such a real would yield insights to all questions, including all yes/no questions of science, technology in the future, etc. We expect B by contradiction thus to be uncomputable. Otherwise, imagine having this oracle (Is there advanced life in other planets, will human have wings and fly someday, etc.)

Two major dawbacks of course is the notion of infinity in digital expansion and the notion of time in languages. Actually, to construct a non computable real, one uses Cantor diagonal arguments…

Anyway, in Computational science, one has from time to time to think of the basic notion of computable numbers. There are the basic atoms of any computational science.
Frank.

I recently read articles on paleomagnetism. It is common to make the assumption of antipodal symmetry for the distribution of the dispersion of the directions. Two such spherical distributions (directional statistics) with such an antipodal symmetry are the Bingham and the Kent distributions. It will be worth writting the jMEF files for those ones.

Frank.

Olivier was kind enough to provide this report trip on the French computational geometry days, as known as JGA 2010. The plenary speaker slides are online and worth a check.

The Journées de Géométrie Algorithmique are the main French-speaking meeting about computational Geometry. It is an informal workshop for the members of the French Computational Geometry community. The 2010 edition hold in Marseille in March 2010.

The meeting was split into two parts: lectures from invited speakers and short talks by meeting participants. We had the great pleasure to listen to 5 high-quality lecturers: Alexander Zvonkin, from University Bordeaux 1, spoke about Belyl functions and combinatorial maps; Claire Mathieu, from Brown University, made a very interesting course about linear programming and semi-definite programming methods; Francis Clarke, from University of Lyon 1, gave a lecture about non-smooth analysis and its applications. Albert Cohen, from University Paris 6, presented some results about theory and algorithms for anisotropic triangulations, with application to images. The lectures session ended with Jonathan R. Shewchuk, from the University of California Berkeley, who spoke on theoretically guaranteed mesh generation, making a survey of the whole literature, from basic Delaunay triangulations to last results on the field.

The talks covered a large variety of Computational Geometry related topics and I will simply give a few words about some talks that appeared particularly interesting to me. Primoz Skraba, from the INRIA Geometrica team in Saclay, described a clustering algorithm using persistent homology which allows, using some stability properties of the persistent diagram, to recover the right number of clusters with good spatial stability. Jean-Daniel Boissonnat, from the Geometrica group in Sophia, presented a certified algorithm for manifold reconstruction which has a linear complexity in the ambient dimension. It uses tangential Delaunay complexes and techniques of sliver removal to achieve this result. Julie Digne, from the CMLA lab of the ENS Cachan, showed a new strategy for segmentation of the meshing of a data point set. Given a scalar function, like the mean curvature, defined on the meshing, it uses a Maximally Stable Extremal Regions method to extract level set on the meshing. Thomas Iwaszko presented his work on a new variant of the Voronoi diagrams: he generalizes the empty-ball property of the classical diagrams to any shape (actually, a union of spheres) and study the resulting diagrams.

The next JGA will be held in 2012, due to the SoGC 2011 conference taking place in France.
Frank.

Tensors are square symmetric positive-definite matrix. They are surprisingly in 1-to-1 mapping with symmetric matrices through matrix exponentiation (exp.log computed on the diagonal elements of the spectral decomposition). I said surprisingly because tensors are symmetric and therefore a proper subset of the set they are in bijection with. Yes, nobody knows what is going on at infinity, and Cantor set theory and life was not so happy due to this… Anyway, obviously symmetric matrices are closed by addition and scalar multiplication, so that it is a vector space. Now using the bjiection tensor-symmetric matrix, we deduce a vector space for tensors. Euclidean distance (or other norm induced distance) hold, and we can transpose that to distances on tensors… These are called log-euclidean metrics, for distance metrics on matrix logarithm. And yes the metric with triangle inequality is derived from Cauchy-Schwarz inner product ineq.

Spaces are beautiful lands to explore!
Frank.

Shannon entropy is said additive in the sense that the entropy of the joint distribution

H(X*Y)

is the sum of the entropies:

H(X*Y) = H(X)+H(Y).

This property is not true for the quadratic entropy (sum of squares). The Java program below checks experimentally those entropy functional properties. Frank.

Here is the Java code

/*
 (C) March 2010, Frank Nielsen
 Demonstrates that Shannon entropy is additive
 and that the quadratic entropy is not
*/
public class ShannonAdditiveEntropy
{
 //quadratic entropy
public static double Q(double [] p)
{
double res=0.0d;
for(int i=0;i < p.length;i++)
    res+= p[i]*p[i];
    
return res;     
}   
 //Shannon entropy
public static double H(double [] p)
{
double res=0.0d;
for(int i=0;i < p.length;i++)
    res+= -p[i]*Math.log(p[i]);
    
return res;     
}   
    
public static void main (String [] args)
{
int n=10,i,j;
double sump=0.0, sumq=0.0;
double [] p=new double[n];
double [] q=new double[n];
double [] pq=new double[n*n];

for(i=0;i < n;i++)
{
p[i]=Math.random(); q[i]=Math.random(); 
sump+=p[i]; sumq+=q[i];
}
// normalize to densities
for(i=0;i < n;i++)
{
    p[i]/=sump;  q[i]/=sumq;
}

for(i=0;i < n;i++)
    for(j=0;j < n;j++)
     pq[i*n+j]=p[i]*q[j];
     
System.out.println("Show that Shannon entropy is an additive entropy:");
System.out.println("H(p)="+H(p)+" H(q)="+H(q)+" H(pq)="+H(pq) );
double delta=H(pq)-H(p)-H(q);
System.out.println("Additive entropy:"+delta);
System.out.println("Show that the quadratic entropy is NOT an additive entropy:");
System.out.println("Q(p)="+Q(p)+" Q(q)="+Q(q)+" H(pq)="+Q(pq) );
double deltaq=Q(pq)-Q(p)-Q(q);
System.out.println("Not an ddditive entropy:"+deltaq);
     
}
    
}

Frank.

How do we visualize relationships in the jungle of (statistical) distances? I tried to give insights at a glance with this poster.

A long time ago, well in 1996, I investigated output-sensitive algorithms. I then designed an algorithm for peeling iteratively the convex hulls of a 2D point set. This yields the notion of depth of a point set, and is a useful index in statistics as attested by recent papers published in this area.

Here is the reference work:
Output-sensitive peeling of convex and maximal layers

The center of mass (=centroid) is defined as the center point minimizing the squared of the Euclidean distances (=variance). If one of the source point is an outlier corrupting your dataset, and if that outlier goes to infinity, then your centroid follows it can is clearly not a robust centerpoint. The median on the contrary is defined as the center point minimising the sum of Euclidean distances. It is robust as it breaks only if n/2 outlier points go to infinity. However, there is no closed form solutions.

Doing some survey, I found that since Fermat (who allegedly first ask the question for 3 points), it has been studied and rediscovered in many communities. The current labeling of this point should be Fermat-Fagnano-Weber-Torricelli-Steiner point, and I forgot many names…

History is full of insights!
Frank.