prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What field does Danny work in in Tennessee?

Read the following text and answer briefly.

Context 1: My Aspergers Child: COMMENTS & QUESTIONS [for Feb , 2017]
I emailed you a while back and you mentioned that I could email when I needed to  Thank you  I last wrote you in December that my son became involved in a dispute involving the local police  We have had 3 court dates  It keeps delaying due to not being able to come to an agreement  But the attorney, even though he was just vaguely familiar with Aspergers, has been very good with Craig  He has the compassion and excellence that is needed here  What started out very bad is turning into a good thing  It will probably take another 90 days or more 
But Craig is working hard  Too hard sometimes  He goes to therapy 3 times a week  Doing excellent  He's more focused and can calm down easier  He's got a lot on his plate but has support from his family  From his attorney  From therapy  And from his work 
He has been renting a room from a lady who has a son with ADHD  It is good for him  I'm a little worried though because since she smokes he wants to find his own place  With all the costs he has to balance it out financially  That is good  I can't help him more than I am which is good  He is stepping up and taking responsibility  He is listening much better 
He is going to have an evaluation today to get an accurate diagnosis  I understand that is a little difficult since he is an adult  Also the PTSD may cover it over  The attorney stated it would help to have the diagnosis 
Aware this is a long update, but thanks for reading  I am fighting much guilt still but I have a lot of peace now  My daughter and her 4 year old son also have Aspergers symptoms  So my life chapters may not close for a while  :-)
My name is Mac  I'm sure you're quite busy, so I'll get right to it I just wanted to pass on compliments on My Aspergers Child and your post, How to Implement the GFCF Diet: Tips for Parents of Autistic Children 
Me and my wife absolutely loved it!
I got a facebook message from him today begging to be able to come home saying he misses home and he will change  He says he will follow rules now  I stated to him the simple rules he has to follow which were - No weed in my house, or smoked in my house, coming home at curfew, going to school, no skipping, no drugs at school, and to drop the attitude of I am 17 I can do whatever I want 
I have made it very clear that if I see any drugs in my home I will be calling the police, as well as if I see signs of it being sold by him I will report him  (He has never had selling amounts in my house,     I believe it's being kept at his "friends" which of course I have no proof of    I just know it is not here 
I know my battle is not over by a long shot, I am sure we will have more consequences and possibly another being kicked out, but I am going to think positive and hope that he learned some form of a valuable lesson here 
Thank you so much for the guidance, never in a million years did I ever think I'd be on this side, (the one needing the help, as I am the one who helps )
I am going to go back to the start of the program like I said earlier and keep notes close by for reference 
Thanks for all you do, helping us all with ODD children/teens
I have a small company providing educational support services to a few families who have children with various disabilities in Ohio  One of the families has multiple adopted children of whom several have significant attachment disorders including RAD  As an experienced teacher and foster parent I have some experience in working with children who have extensive trauma backgrounds  However, I could use additional training  Also working with these children are two staff members with minimal background in attachment disorders who would also benefit from training primarily in behavior management  The primary caregiver to the children does a wonderful job managing their needs  In order to further develop team cohesion, I'm hoping to include her in any training as well 
Is it possible to schedule such a training session with you? If so, please let us know what will work for you including time, place, and cost  Thank you for your assistance 
I just listed to your tapes on dealing with an out of control, defiant teen  I'd like to ask your advice on a particular situation we have  Our 15 year old daughter is smoking pot almost every day at school  Because we had no way to control the situation, we told her, fine, go ahead and smoke weed  However, you will no longer receive the same support from us  You will not have your phone, lunch money to go off campus (she has an account at the school for the cafeteria she can use), and you will be grounded until you can pass a drug test  We will not be testing you except for when you tell us you are ready to be tested  She is now saying she's suicidal because she feels so isolated, yet she continues to smoke weed  In fact, she tried to sneak out last night but was foiled by our alarm system  For the particular drug test we have, I read it takes about 10 days of not smoking to pass the test  What would you do? Please advise 
I am having a problem with my 18 year old son, Danny, with high functioning autism  We finally had him diagnosed when he was 16 years old  I always knew something was going on with him but the doctors misdiagnosed him as bipolar  It's been 2 years now and he will not accept his diagnosis  He won't talk about it and when I try to bring it up he gets very angry  I've tried telling him that it's not a bad thing, that there's been many, many very successful people with Aspergers  He won't tell anyone and refuses to learn about managing life with it  He once shared with me that the other kids at school use it as an insult, like saying someone is so autistic when they do something they don't approve of  So he doesn't want anyone to know  He's turned down services that could help him  He has a girlfriend, going on 8 months  He won't tell her and they're having problems arguing a lot and I wonder if it would help for her to know 
I'm sad that he thinks it's a life sentence to something horrible instead of accepting, embracing it and learning about it more so he maybe can understand why he's struggling  I told him that he doesn't need to shout it out to the whole world but he won't even accept it himself 
I don't know how to help him with it and because he's almost 19 I have limited control now  It's made my life easier knowing what we're dealing with and I think his life would be easier is he accepted it 
Please help me help him 
I am a clinical psychologist in NYC who now has several (!!) children I see who have RAD  In 20 years of practice, I’d seen only one case  Now, I have at least three children with this  I have no training, per se, in working with this children though I know about setting structure, consistency, etc  I do a lot of work with parents about parenting  I work primarily within the school setting in a charter school whose mission is to educate children on the autism spectrum in a mainstream setting  We use Michelle Garcia Winner’s social thinking program with our ASD kids  I also work with gen ed kids in the school who are at-risk; the school is in the inner city from where the majority of our non-ASD kids live 
It would have been so much easier to mention to my adult son that I think (I know he does, but want to ease into the subject)
he has Asperger's when we were living together two years ago  He has since moved to Tennessee working in his field of interest
which is 3-D printing and software development  I am so happy for him that he has found his way into a job that he truly enjoys
even though he's socially isolated 
He's not diagnosed and does not know he has it  How I know is his classic symptoms being sensory issues (fabric feeling like sandpaper)
communication difficulties, meltdowns and much more  Throughout his childhood I just felt he was a bit different  Nothing major stood out and time
just passes, misdiagnosis of ADHD, low frustration, etc  We've talked about his ADHD numerous times (which I now know he doesn't have) 
It's so much easier to communicate with him now that I know he has Asperger's  I keep it "slow and low" in talking, with long moments
of silence and then we connect  It's really too bad that Asperger's got a diagnostic code back in the 90's, yet all the so called doctors,
physiologist's, etc, didn't know how to diagnose it  Too bad 
There seems to be no one answer to "should I tell my adult son he has Asperger's" from a few specialists I asked  He is typical Asperger,
complicated, highly intelligent (high IQ), anxiety at times, socially isolated, hard to make friends  Not knowing how he will react is the hard part 
How will he be better off knowing he has it? Do I wait to tell him in person, or ease into it with him over Skype? He likes direct, honest, concrete communication 
Why is this so hard for me? Maybe because no one know's if he is going to be better off knowing or not  Do you know if people are better off
knowing? I try to get up the courage to just let him know, then I back down 
I have been searching the web looking for advice and came upon your site  I am trying to read blogs, websites, books, and articles to help guide me  I was so happy when you said that I could ask you a question  My husband and I are struggling with my 27 year old son who lives with us 
Kyle is the youngest of 4 sons  He is a college graduate but never could find the "right" job  He has always been quiet and never had a lot of friends  Two years ago, his girlfriend broke up with him  Kyle had an online gambling addiction and was using pot all the time  After the breakup, Kyle was very depressed and started using heroin and finally told my husband he was using  He is now seeing a psychiatrist who has him on suboxone and antidepressants  He is also seeing a psychologist weekly for counseling but it does not seem to be helping 
Last October,, Kyle lost his job, got drunk, and was agitated and came home , fighting with us, damaging our home and being verbally abusive  My other son , age 32, who also lives with us called the police and Kyle got arrested  He is currently in the family court system  He went through an anger management course and now is in substance abuse classes  Kyle continues to verbally abusive to me and blame me for everything  He says he "hates me "and calls me terrible names  At times, he pushes my husband and intimidates me  My husband and I are so upset  We just hired an attorney for him because since he has been going to these classes, he is getting more depressed and not getting better  Kyle continues to drink while taking his meds prescribed by the psychiatrist and then he has his "moods " My husband and I have met once with the psychiatrist just to give him background information when Kyle started with him 
At this point, we do not know what to do  We never thought at this stage of our life, we would be supporting and spending our retirement money on adult children  I do not know why Kyle hates me, I could not have been a better mom  My husband and I have no life and just do not know what it the right path we should take  Kyle does not want anything to do with us  He spends all his time in his room playing football online We have tried tough love versus caring and love and understanding  Do you have any advice for me?
This whole ODD and ADHD is killing me as a parent  I work in the field of adult psych and addictions so I am well educated  I have been dealing with my teen being like this for almost 3 years and I totally lost my cool today with my 17-year-old son to the point I told him he is out of the house  He can never simple rules, comes and goes as he pleases sometimes doesn't come home, just recently back in school from several suspension for drug related    I am just so exhausted  He has made me hate life, hate being a parent and sometimes I just feel like not even being here  I bought your program in hopes to it would help, I am at week three and I feel things are getting worse    what am I doing wrong??
My partner hasn't been diagnosed yet but I know he has aspergers   day to day is a struggle   I feel I'm going crazy with how he makes me feel Feel let down constantly  He lies alot but I've been told they can't but I know he does I just feel trapped and unloved We have a 4yr old daughter together and my main worry with how he is that it will effect our daughter ; (his skills as a parent are so weak He can't disapline at all Feel so alone  he hides it well too I just wondered if things will get worse? He's angry so quick in arguments Scares me etc I can't leave as he's the main bread winner and our daughter loves him to bits Don't know why I'm writing this  Sorry if I'm going on and not making sense :(
I wanted to let you know about a research opportunity for children, teens, and young adults with autism  I am studying the effects of Brazilian Jiu Jitsu, and psychotherapy on helping people with autism develop subjective awareness of others 
I am writing you to see if this might help someone in your practice, or to see if you might know of someone with autism who may benefit from participating in this study  The requirements of the study will be:
1  A participant should be between 7-21 years of age and have a diagnosis of Autism Spectrum Disorder 
2  The participant should enroll in an approved Jiu Jitsu Academy and attend at least two sessions a week for a period of six months 
3  The participant should enroll in social skills groups, provided by my office or be in a steady psychotherapeutic relationship in your office, at least once a week, or minimally two to three times a month 
4  The participant will be given a SRS (Social Responsiveness Scale) test at the beginning of the study, at three months, and again at six months 
If you know of anyone who might benefit from this novel approach to helping to develop social awareness in autism, please do not hesitate to contact me for further information 
I have a 10 year old daughter who has outbursts with prolonged crying almost like tantrums that 2 year olds have when they cannot express themselves 
I had her in therapy from age 6-8 years old for the same thing but I feel that the sessions didn't really help much 
She has severe sensitivities to light, sound, vibration, frequencies which trigger irritability and crying 
We changed her diet and tried getting her involved with activities but she is anti-social and prefers reading than being social  She is terrified of change even in daily routine (even that will trigger prolonged crying) 
It frustrates me because I don't know what else to do with her behavior 
I've tried acupuncture (she refused at the first session); she refuses massage too 
She is an honor-roll student at school and has very minimal issues at school but if she has had a bad day it does result in a tantrum or crying and defiance 
How can I get her tested for Asperger's Syndrome?
Last night our 24 year old son with Aspergers told his dad and I that he is pulling out of the 4 college classes that he recetnly enrolled in because he has not been attending class or turning in his assignments  He paid $2800 (his own money) for tuition and I reminded him of this when he told us but it did not seem to bother him 
This is the 3rd time he has started college courses and has not completed them  (He also took some concurrent college classes while he was in high school that he failed)  This is a son who basically had a 4 0 grade point average through 10th grade and got a 34 on the ACT the first time he took it 
With the news that he was once again not sticking with college courses I did not sleep well  When I got up this mornning I began looking online for help in how to deal with his situation  I found your "Launching Adult Children With Aspergers" and purchased it  Most of what is included are things we have done or did with our son throughout his life  I was hoping for more help so I am emailing you now in hopes of more specific ideas 
We noticed some things with our son, Taylor, as a yound child but as we had not heard of Aspergers at that time we just did what we thought would help him  As a toddler and a child at pre-school he generally went off on his own to play  When I talked to his pre-school teacher about my concerns (that I was worried he would end up a hermit) she said she did not see him being a loner and that he seemed to interact fine with others in many situations  We worked with him on making eye contact when talking with others  We explained different emotions in people's faces and mannerisms to help him know how to interact with others  We discussed the fact that people would say things that did not mean what they souneded like - such as "I'm so hungry I could eat a horse"  As we did these things he worked hard to better understand communication with others 
During his 4th grade year I had a teacher from the gifted program ask me if I had ever heard of Aspergers  I told her that I had not heard of it  She proceeded to read me some of the charateristics and so many of them described my son  So we had him tested by the school district during the summer between 4th and 5th grade and they did find that he had Aspergers but that he was high functioning  We then set him up with and EIP which stayed with him until his sophomore year  We pulled him from it at that time because we had moved and the new district was requiring him to take one class a day that was a study class  This reduced the number of required classes he could take and he was doing fine with his studies at the time 
It was during the 2nd half of his Junior year that we noticed some of his grades going down  Then during his Senior year is when he started skipping classes and not doing assignments  We had not realized it before then but we soon became aware that he was addicted to gaming  He would go to the library or somewhere else on campus and play games on the computer rather than go to class  It was also at this time that he began lying about his actions (so as not to get in trouble) 
Based on his grades and his ACT score he received offers from colleges for full tuition scholarships  He chose the college where he had taken concurrent classes during his high school years  But he proceeded to skip class and not turn in assignments so he lost his scholarship and quit attending college  During this time he was only able to find employment through an employment agency where he was mostly sent to manuel labor type jobs (which is not something he enjoys but he did it anyway)  It was during this time that at one place had gone to on numerous occasions he was told if he came late one more time they would tell the emplyment agency they did not want him to come there anymore  (This seemed to make an impression on him because he has continued to be reliable and responsbile at his places of employment) 
At 19 1/2 he left to serve a 2 year full-time mission for our church  He completed his mission successfully  (I don't think it was without some struggle, stress and depression, but he was able to pick himself up and move on from those times) 
When he came home he started working for the employment agency again but began looking for employment elsewhere  He got a job at a local Chick Fil-A where he has worked for 3 years  He started college again shortly after he came home but as before it was short lived  He did finish out the semester but failed most of the classes due to his skipping class and not turning in assignments  When he skipped class he would usually sleep in his car 
Taylor's life consists of working (where to the best of our knowledge) he does well, he is reliable and his employer likes him  When he comes home from work he either sleeps or plays video games or other games - such as kakuro  He spendes most of his time in the basement where his bedroom is and this is where he games  Taylor owns his own car, bought his own laptop and very rarely spends money  He pays us $200 /month to still live at home, unloads the dishwasher on a regular basis and does the weekly garbage  However, his room is a mess and he only cleans his bathroom when I tell him he needs to clean it 
Taylor used to read quite a bit and loved to learn  It has just been in his adult years that he has not read as much - I think because of his gaming addiction  Taylor goes to church on a regular basis but sleeps through the main meeting  In Sunday class room settings he stays awake - I think because he is able to particpate in discussions 
Taylor has only had 2 real friends since entering Junior High school  And as of now he only keeps in contact with one of them who still lives in Georgia  We have lived in Utah since the summer of 2007 and he has never had a friend to do things with since we have lived here  He has two younger siblings, a brother 22 and a sister 20  They love Taylor and spend time with him when they are home  They are both at college and doing well 
Throughout Taylor's school years he has seen a counsleor on a fairly regular basis  One summer during junior high he attended a weekly class where he interacted with other kids with Aspergers  We did see a lot of change in him from this group  After he returned from his mission he went to see a counselor for a short period - this counselor tried to help him with some social skills  His dad and I went with him the first 3 or 4 times but we found out that after we quit going with him he only went a few more times and then scheduled appointments but did not show a couple of the times  We only found this out when a bill came for a "no show" appointment 
I don't know if this is too much information but were are in dire need of help for him  In the information that we purchased from you you mentioned that you do coaching for Aspergers adults  I don't know if you can help us but I thought I would check with you just in case 
Alas I think I have found your information too late to save my marriage but I am hoping to save myself 
I am currently going through a very very painful separation after a 27 year relationship with my husband whom I am convinced has aspergers syndrome  It is a long and painful story and I am desperately trying to process it all alongside dealing with a very conflictual separation  My partner is angry non communicative and totally dismissive of me and our long shared history 
He walked out last year after I discovered he had been visiting massage parlours and developed a relationship with an illegal Chinese escourt whom he subsequently moved in with  He had been seeing this woman behind my back for over 18 months  The pain of all this indescribable and his dismissal of my pain and very existence beyond belief 
Leading up to this I had been battling anxiety and depression which my husband found very hard to cope with 
Over the years of our relationship I knew something was off but I just could not put my finger on it  I often felt a complete lack of validation and empathy  Communication was also difficult as my husband was defensive and unwilling to look at issues in our marriage 
Please Mark could you help me validate some of this pain and try and make dense of 27 years of my life without drowning in fear guilt and despair about my future 
Thank you for listening and your site 
I have had problems with drunkenness, being late for school, not handing in school work, buying pot from a dealer etc  I chose to focus on the drinking and did the grounding then (grounding happened 3 times)  I also stopped sleep overs at friends 100%  I have stopped handing out money for no reason or even buying treats like chocolate 
I did lose it one evening (and didn't do the poker face) when I was trying to unplug the internet at midnight on a school night (she’s always late for school so I am trying to get her to sleep at a reasonable hour)  I was physically stopped and pushed around so I slapped my daughter (it was not hard)  This ended up with her saying she didn’t want to come home (the next day after school)  By this stage, I also had enough and didn’t go get her  I thought I am not begging  You will run out of money soon  It was quite a relief to have some peace  Daughter’s Dad was in town (from another country) and called a family meeting with the counsellor  To cut a long story short, daughter and her counsellor put it on the table that daughter wants to go live somewhere else (with her friends family) because of the stress at home with me (we live on our own) (i e  stricter rules and her bucking up against it) 
I didn’t really want this but made a compromise that daughter would go there Tues morning – Friday afternoon as the friend is an A student whereas my daughter is failing  They do the same subjects  I made the decision at the end of the day based on what is good for me – some time away from the daughter  I also thought of your book when the child went to live with the grandparents – daughter will dig her own hole over at the friend’s house  They have a week day no going out policy which made me think it is OK  I went and discussed with them the problems experienced (drinking, pot, late nights, not handing in work)
I am also trying to follow the let go of school thing per your book  I find it really difficult to remain calm when I can see daughter on her phone and watching series (when I have her on the weekends) when I know there are projects due  I hired her a private tutor once a week for help with a subject  The tutor has just fired my daughter for not handing in work and being not committed  It’s not the first time private tutoring has not been appreciated  The school give me a report back on a Friday as to whether everything is handed in  The deal is – if the work is not handed in – no pocket money and no Friday night out)  Her school is a "progressive" school and there are no repercussions for her being late or not handing in work  I would change schools if I could but there are only 8 months left of school (she turns 18 in August) 
We have just completed the first week and beginning week two of your material  We are agreeing with your take and see our son and ourselves in most of what you are saying  Prior to finding your material and starting your program we had been having extreme out of control behaviors and had to call the police because he was breaking things in our house and pushed my husband  This happened three weeks ago  After that incident we took away privileges ie  PS4, phone (which had already been taken for a few days), and friends  So, last week while doing your program he already didn’t have privileges and has continued with poor behavior – name calling, throwing things, slamming doors  We are not sure when to give privileges back  He has been given the privilege of playing with friends on occasion  His 13th birthday is tomorrow  This past weekend, for his birthday my husband and he went boar hunting  Of course we debated about it but decided to go ahead since it was his bday  We are cooking some of the meet on the grill tomorrow night for his bday and inviting a couple of his friends over for a cookout  No more gifts other than cards and balloons  We are wondering if we should go ahead and give him his privileges back and not sure how to do it  Last Friday morning we attempted to talk giving him a date to return privileges and that conversation ended with him getting angry but he gathered from our conversation that he is getting his stuff back on his bday  We are starting week 2 assignments today but not sure how to handle what was already in place  Of course, we aren’t seeing the respect and responsibility we are looking for but realize it has been a long time  We were wanting him to pay for his phone and thought it might be a good time to introduce that idea  Allowing him to earn his phone  We expect that he will be angry with this idea and not sure how to implement 
My son and myself are interested in a inpatient Aspergers program  We line in Calif which is preferable  My son is very high functioning and was diagnosed dry late  He was eight years old  He has never been in or attended a full day of class  Partially due to depression,anxiety, and trouble with his ADHD also his aversion and being bullied and of course his Aspergers  He will not attend his freshmen year due to surgery on both Achilles' tendons from walking on his toes  With physical therapy he should be ready by his sophomore year! We all feel he needs in patient therapy to give him the tools on how to work with his issues in a structured setting and a place that will give him tools for the rest of his life 
In my utter desperation to find a way to get some help for my daughter's increasingly challenging behaviour I trawled the internet to see if I could find some strategies that would provide specific methods on dealing with teenagers with Asperger's syndrome  When I came across your website, I couldn't believe that every statement you made was exactly what I have been going through with my daughter  She has just turned 14 last week, and was diagnosed with Asperger's/ Autism Spectrum Disorder 15 months ago  I have already been seeing a child psychologist for the past five months, however the methods she has been advising have not been very effective 
Our main difficulty with our daughter is her overwhelming obsession to use her cell phone (and to a lesser extent her laptop) constantly  Without any restriction, she will be on it every minute of the day, and will be awake until the early hours every day  We have tried to incorporate her input around rules as to when she has to give in her phone, but she is unwilling to compromise on a time that she should give it to us, believing that she should have unlimited use  I believe she is unable to do any adequate study or homework, as she is constantly having to look at the phone  We have tried to put rules in place that she has to give in her phone and laptop on school nights at 22:15  If she is able to do this then she is given rewards, and if she doesn't then she knows that there will be consequences  The consequence has been restricted use the following day  However, this is usually where we fail, because taking her phone away from her results in tantrums, screaming, and even threatening to harm herself  This behaviour is relentless to the point where the whole family becomes deeply distressed, and inevitably results in her getting the phone back 
This obsession is affecting her schoolwork, and more severely her eyesight  She has become very shortsighted, and her eyesight continues to deteriorate as a result of holding the phone or laptop very close, and mostly in the dark without any lights on  My husband and I have a constant battle on our hands daily, in all areas of discipline with our daughter, but our main concern is that we have been unable to find a way to minimise this obsessive behaviour centred around her phone and laptop  Please can you provide some strategies that can help us specifically with this problem 
First of all, I thank you for developing this program and I am only at the first stage of assignment 1  I have loads of books I have bought, attended psychiatrists for my son and myself, family therapy, occupational therapy, begged and prayed for change but have been dealing with behavioural issues for so long I am definitely exhausted and resentful 
I am a mum to a 15 yr old boy with ASD, dyslexia, OCD and ODD  Sorry to focus on the labels but just to give you an idea of what I am dealing with  I also have a 13 yr old son whom finds his brother’s behaviours difficult, embarassing and challenging  My husband whom is not in great health ( he had a cerebral aneurysm clamped two years ago and has two further aneurysms that are inoperable so endures fatigue, headaches and stress)  We have however a pet cat that is very social and a calming influence in the home! I was fortunate enough to have loving parents but I lost both my mum and dad in 2008 and 2015  My inlaws are elderly and quite directly say they are too old to help us so it feels we are alone in dealing with the issues we have 
I am desperate for change as the household is one of stress and anger and I feel all the control lies in my son Patrick’s hands  I am hopeful your programme can make life better for all of us but I wonder if it is too early to ask you two questions?
The first lies with what to do when Patrick goes into my other son Brendan’s room and will either turn on a light when he is sleeping, yell when he is on his phone or create some disturbance  He will not leave the room when asked to do so and the situation always escalates into yelling and Brendan attempting to physically remove him  This happens regularly and always ends badly with doors slamming, my husband being woken and myself in tears feeling the lack of control and also I admit I seem to think “Why me?” which rationally I know is of no help 
The second problem is leaving the house for school  Patrick refuses personal hygiene (either morning or night) and any request to even brush his teeth is fraught with swearing and abuse  If I can get him to shower, he will watch the water roll down the drain and turn up the water really high temp (mu husband has had to turn down the thermostat on the hot water service) without so much as getting wet  My husband leaves for work at 6am but I leave at 745 to work as a nurse in a busy outpatients department in the Alfred Hospital (Melbourne)  My work is my sanity as it is a paid break from home but most days I am late which is causing considerable stress and anxiety not to mention my responsibility to do my job  Patrick simply refuses to leave the house and as much as I am tempted to just walk out and leave I know the house would be left unlocked and wonder if Patrick would even attend school  The time I need to leave is not negotiable but Patrick uses this to his advantage and seems to delight in stressing me out and subsequently speeding to work in a frazzled mess 
The interesting and frustrating element in all of this is that although he is socially isolated at school (he has no friends) and academically challenged his behaviour at school is not a problem  He is quiet and his teachers report he does his best and is compliant and well mannered  It is like a Jekyll and Hyde situation where another side of him at home is so angry and abusive yet at school this behaviour does not happen 
I’m Jackie, I now work primarily as a freelance tech writer, after starting my career in software development and moving on to teach IT to young adults at a variety of colleges and schools 
My freelance work is pretty varied and looks at many aspects of the computer industry as a whole, and I’ve just recently completed a piece which gives help and advice to anyone wanting to become a game designer, which you can read here: http://www gamedesigning org/become-a-game-designer/  It highlights the hard work and effort it takes to get into such a role, and also how you can further your career and continue to learn and improve as you go  I hope you’ll agree it shows that starting work in the industry takes dedication and skill and that becoming a game designer isn’t just a fly-by-night job!
If you’d be interested in sharing a quick mention of my work on your blog that would be really wonderful and I’d appreciate the chance to get my work out there to a wider audience  Alternatively, I’d be happy to write a short blurb or paragraph or two (or a longer piece - just let me know) highlighting the key points because I think some of your readers might get a lot of value from it 
My son just turned 15 and is a freshman in high school  Although this is his first year in a general ed environment, he is struggling with behaviors in school  He has meltdowns and does not express why he would have them until much later  Once we all know what caused it, the school will accommodate him and try to "change up" things so as not to cause his meltdown  Once that is resolved, another issue comes up and causes him to melt down  He is a high functioning and academically does well, when he wants to do the work  We battle at home over homework  He does not care how it is done, as long as he hands it in  He thinks failing a test is ok, at least he took the test  Homework is never on his mind when he gets home from school  If I never prompt him, he would never open is backpack  He can be aggressive but is never intentionally trying to hurt anyone  He may push over a chair in school, but it is not directed at anyone  We know how that in itself could hurt someone who gets hit by it though  He is defiant in that he only wants to do what interests him  He does not go out by himself (still immature), or abuse alcohol or drugs and never curses  He is a very funny kid and very talented  His main problems are task avoidance and seeking attention  He can be disrespectful to adults in that he is "cheeky" with them, trying to be funny or cute  And he has no "filters" 
I’ve just finished reading your Living with an Aspergers Partner ebook  I found it so informative, thank you 
You offered some personal advise, and i wanted to run a situation past you and seek your input as to a strategy for what to do next 
I’ve been seeing a guy for about 7 months now who I believe has Aspergers  I came to this conclusion months ago and I don’t think he realizes, (or acknowledges) although he is aware he has some traits 
He’s highly intelligent and successful, a pattern seeker, has a tendency to focus on the project to hand to the total exclusion of all else for as long sit takes (work or home) socially awkward (has learned coping strategies), sensitive to loud noise, high anxiety with control strategies, black and white thinking etc  He’s currently not working and I’ve seen a slow withdrawal over the last 6 weeks, including the need to ‘escape’ and leave a situation at least once 
He also has a bipolar ex overseas who has primary custody one daughter where there has been ongoing patterns of drama which has recently increased 
Over the past couple of months (since stopping work and drama increase) I’ve gone from being ‘wonderful’ in his eyes to him now being sorry and not having the ‘urge’ to spend close/intimate time with me and offering friendship  Since he shared that with me in a message he’s stonewalled and has retreated to the safety of minimal messages and talks about not knowing what best to say and not being able to find the right words somehow 
He’s a good kind man who I feel is struggling  I’m concerned about his anxiety and possibly the risk of depression  I’m fairly resilient and whilst i’m disappointed he doesn’t want to pursue a relationship with me, i’m concerned for him and his well being  One of his very few close friends is also just leaving the country to live overseas 
The strategy I’ve used so far is simply to back off and give him space  I’ve asked to take him up on an original offer he made to talk but haven’t pushed it  I also haven’t been aggressive or accusatory in the few messages i’ve sent 
Any advise you could give would be greatly appreciated,
Carli who is 10 years old and has had behavioral issues her whole life  The other night she came home very upset after having a conflict with a friend  She was at her friend's house and her and her friend wanted to get on the computer and the older sister was using it  Carli made up a story that someone was at the door to get the older sister off the computer  Her friend didn't understand that she was making up a story to get the sister off the computer  She got excited that someone was at the door and ran downstairs to answer the door  In the process of getting the door, she fell and yelled at Carli  Carli became extremely upset  She was able to control her feelings at her friend's house, but when she came home, she proceeded to cry extremely loudly for over an hour  Her dad spent most of that time with her, talking to her and trying to calm her down  After an hour, I asked him if he could please tell her to be more quiet because the other members of the household were trying to go to sleep 
My question is    how do I as the girlfriend, handle this? He did not like that I asked her to be quiet  We have a rule that if she is having bad behavior, and can't calm down in 5 minutes, he takes her out of the house because her yelling doesn't stop for a long time and is very upsetting to everyone in the household  I would like to ask him to do this with this kind of situation as well  Is this a reasonable request? His thought was that she shouldn't be made to calm down, because everyone handles being upset in a different way  But, she was literally sobbing and wailing very loudly 
My other question is should she have been told that if she wouldn't have lied, this wouldn't have happened? She has a history of lying and of not accepting responsibility for her actions  My boyfriend became very upset with me when I brought this up  He was being very sympathetic and understanding to her  I feel like he was giving her negative attention, and being an over indulgent parent by not putting his foot gown and saying, "you can't carry on like this, even though you are upset"  Please let me know how we can handle these situations better 
I am contacting you for help with adult AS  I am taking initiative to pre screen potential therapists to help my current boyfriend get therapy and help with Adult AS 
He has seen many therapists, but it seems like they aren’t really helping him with his problems  They don’t seem to understand how his (undiagnosed) AS would affect therapy approaches  For example, he may not share enough in therapy session and I’m assuming an AS therapist would recognize that is part of the AS and employ strategies to get information from him that helps with treatment  Sometime he tunes out when he is processing something heavy or that he doesn’t want to hear necessarily, or he gets distracted and I’m hoping an As therapist would recognize that and get that he may need repeated something for example, if this is happening 
He is currently suffering from depression that appears clinical in nature as well as reoccurring negative thoughts about something specific that has been worrying him about our relationship  Today he told me these reoccurring thoughts happen during all waking hours unless he watches TV, he never gets a break from them and they make him feel like he is going crazy  As his girlfriend, I am extremely concerned that he cannot get relief from these thoughts and that the therapists he is seeing are unable to help him with his problems  Therefore, I am taking initiative to try and help him find better therapy options, because I want to see him someone who can better help him get to the bottom of things and help him with the challenges he is facing  He really needs an advocate that will help him go deep to figure things out and not just assume therapies are working well, without seeing changes or getting supporting feedback from him in that regard 
Here are some questions I am trying to ask in advance to find the right people to help us with this  As you may know, insurance for these therapies are not often available  We don’t have a lot of money to go from therapist to therapist to find the right person and are hoping prescreening will help 
I recently downloaded your e-book and listened to your talks and your information is by far the most helpful I have been able to find to date  It's very accurately describes my situation as an NT wife married to a very probable AS husband  I think you for taking the time to write this and sharing your insights as well as the experiences of many of your clients  It has really helped me understand the last 32 years of our marriage and get a grasp on how to move forward 
One area that is of primary concern to me, that I did not see addressed, is stimming  I believe that is the behavior my husband is showing through constant vocal singing, repetition of words, shouting out, as well as slapping himself in the chest and general nervous activity  It is very loud and disruptive to our household and it is often a relief when he is not at home  I think there may be a level of Tourette's syndrome as well 
I did some searches on the Internet and could not find anything that really describes his behavior  Most of what I found was flapping or children's behavior  I understand that it is a release of nervous tension but I am really trying to find some strategies to help him stop this behavior as it is extremely frustrating and builds my resentment in dealing with it daily  A lot of it is embarrassing as well and sounds childish to me 
He usually does this when close family members are around and will reign himself in if he is around other people besides us  When we are home it is constant  He also has a lot of anger, mostly at himself, and blows up at unimportant things, it is as if he has a ton of negative energy inside him that need to get out and stimming is one outlet 
I will try to build my acceptance of it, but I also would just like him to stop especially the loudest and most annoying portions  Would you have any resources you could point me to?
Context 3: Paper Info

Title: CONTOUR COMPLETION USING DEEP STRUCTURAL PRIORS
Publish Date: 9 Feb 2023
Author List: Ali Shiraee, Morteza Rezanejad, Mohammad Khodadad, Dirk Walther, Hamidreza Mahyar

Figure

Figure 1: Just by looking at subfigure (a), we, as humans, can easily perceive a shape like the one in subfigure (b) This is an extraordinary capability of our human brain and in this paper, we tried to see whether convolutional neural networks can show such capabilities 
Figure 2: The trajectory from random noise X N to the incomplete image X I in image space The network will pass a completed version of the image, X C , throughout this trajectory 
Figure 4: This figure shows our iterative process to complete the fragmented contours of an image given as input to our pipeline 
Figure 5: This example shows how different scores change throughout a single run All three scores change in the range of [0, 100] Our goal is to maximize reconstruction_score and minimize the overfit_score, but we should consider that the minimization lower bound is data dependent and is not zero 
Figure 6: Evolutionary process of the deep structure prior The right column shows the incomplete shapes given to the model and the rest of the columns show how the model is overfitting gradually to produce the incomplete shapes In each column, we are showing an intermediate iteration of this process The loss-term setup enables our pipeline to let the completed image appears during this iterative process 
Average MSE and IoU values between the incomplete (Raw) images, the output of DIP and DSP methods, and ground truth for each image are provided in this table 
For this experiment, we ran the model over a subset of complex dataset with 500 incomplete images at various levels of alpha for 250 iterations After the image completion is done, we compared the evaluation metrics between the completed image and the ground truth to examine the performance of the model for different values of alpha 
In this table, we show the effect of the receptive filter size on our algorithm's capability to fill in bigger gap sizes The numbers in this table are showing the percentage of the time that DIP was successful to complete shapes with each gap size and corresponding receptive field size As predicted, the bigger the filter size, the more successful the algorithm is in filling in the gaps 

abstract

Humans can easily perceive illusory contours and complete missing forms in fragmented shapes  This work investigates whether such capability can arise in convolutional neural networks (CNNs) using deep structural priors computed directly from images  In this work, we present a framework that completes disconnected contours and connects fragmented lines and curves 
In our framework, we propose a model that does not even need to know which regions of the contour are eliminated  We introduce an iterative process that completes an incomplete image and we propose novel measures that guide this to find regions it needs to complete  Our model trains on a single image and fills in the contours with no additional training data 
Our work builds a robust framework to achieve contour completion using deep structural priors and extensively investigate how such a model could be implemented 

Introduction

The human visual system is used to seeing incomplete outlines  Our brains can effortlessly group visual elements and fragmented contours that seem to be connected to each other  This power enables us to make shapes, organize disconnected visual features, and even properties of 3D surfaces when projected on 2D planes 
demonstrated how early vision may quickly complete partially-occluded objects using monocular signals  This capability of perceptual grouping has been studied in vision science for decades   Although there has been some work on perceptual grouping in the past couple of years, it has been less studied in the past decade due to the enormous progress of deep neural networks and their success in dealing with the pixel-by-pixel inference of images 
Different types of lines and curves have been studied to maximize the connectivity of two broken ends in the planer contour completion problem   Different types of lines and curves have been studied to maximize the connectivity of two broken ends in the planer contour completion problem  Geometry-based constraints can be utilized to address some challenges of contour completion problems, such as smoothness and curvature consistency  
However, such approaches only work for simple, smooth contours and usually fail in more complex settings  On the other hand, we currently have deep models that could easily take an incomplete image and complete the missing regions using enough training data   The amazing capability of such models especially those that are trained on different modalities with millions or billions of training data raises the question of whether we need such a large amount of training to perceive all the visual cues that are present in an image, which underlies visual perception by humans 
In human vision, Gestalt psychology suggests that our brain is designed to perceive structures and patterns that are grouped by some known rules  In this work, we show that some perceptual structures can also be learned from the image itself directly using architectures that enable such learning  Earlier work has shown  This is an extraordinary capability of our human brain and in this paper, we tried to see whether convolutional neural networks can show such capabilities 
that some forms of perceptual grouping can be achieved using computational models, such as stochastic completion fields   This type of learning resonates with some of the Gestalt perceptual grouping principles including "proximity", "good continuation" and "similarity"  In scenarios where color and/or texture are present, the cue of "similarity" helps us group regions with consistent patterns  
When color and texture are present, they provide a collection of rich information for such cues  In the present article, we probe convolutional neural networks in a scenario where both are absent, and the neural network is dealing with just forms and shapes  Specifically, we explore whether the convolutional neural network architecture itself can give rise to some of these grouping cues when they are fed just contours and shapes alone 
For years, neural networks have been treated as black boxes that can generalize images very well to multiple classes when there are enough training exemplars  One of the reasons that neural networks are trained on many exemplars is to avoid the problem of overfitting  On the other hand, we know that CNNs that generalize well to large classes of exemplars can easily overfit when those class labels are randomly permuted  
Inspired by this observation, suggest that image priors can be learned to a large extent through a generator network architecture that is solely trained on a single image  This encouraged us to take a deeper look at what structural information can be learned from a single-shape image and whether we can reconstruct some of those perceptual grouping capabilities using a generator network 
Inspired by , in this work, we adopt a novel training regime to complete shapes and contours where we use a UNet architecture with random initial weights and try to complete the contours within a single image without any training data  In our model, we imagine that the input image (i e , the only image used to update the model's weights) is an image of fragmented contours 
In this work, instead of training the model on multiple images fetched from a big image dataset, we imagine a random fixed tensor noise image as input to this model  At each iteration, the random noise tensor is inferred through our generative network and the network produces an outcome image  We introduce a novel loss function that enables this network to complete contours 
This process repeats, and the weights of our network are updated gradually based on this loss function, which is an energy term defined based on the input image and the output of the network  The model will reconstruct the missing structures i e , group fragmented contours that perceptually seem to be connected, before it fully overfits to the incomplete input image 
Contributions of our work are summarized as follows: 1  In our pipeline, we propose a novel algorithm that enables us to complete contours that appear to be connected to each other in an illusory form  2  Our model is trained on just one single query image and does not need any training data  3  Our model does not need to know which regions of the image are masked or occluded, i e , we remove the dependency of the algorithm on the guiding mask (a guiding mask is a mask that informs the model on where the missing regions are located at) 
We also introduce two metrics to produce a stopping criterion to know when to stop training before the model fully overfits to the incomplete image, i e , we guide the model to stop when the completed image is produced 

Methods

Our eyes are trained to predict a missing region of an occluded object within a scene  We can easily perceive or make guesses about parts of objects or shapes that we do not necessarily see  Even when we are looking at an image, we might guess about the shape, property, or other attributes of an unknown object within a scene 
Such capability extends beyond just known objects or shapes  We can look at a disconnected set of contours and guess what the connected form may look like  This capability is rooted in our prior knowledge about the world  (see Figure )  In this work, we aim to achieve a similar capability using deep generative networks 
Most neural networks that we work with these days are trained with a massive amount of data and one might think that this is the only way that a neural network can obtain prior information  Authors of Deep Image Prior (DIP) suggest that the convolutional architecture can capture a fair amount of information about image distribution 
They show that the hourglass architectures like UNet can show some good performances in some inverse problems such as image denoising, super-resolution, and inpainting  In this work, we focus on completing fragmented contours end-to-end just by using a single image  To be able to address this problem, we first look at a similar problem in image editing, known as image inpainting 
Image inpainting is the task of completing an image where some regions of that image are covered or filtered by a mask  In image inpainting, the generative model receives a masked image with the mask that guides the algorithm to fill in those missing regions  Although in the problem of contour completion, we have a very similar goal, the additional challenge that we suffer from is that we do not necessarily have a mask that covers the regions of interest for us 
For example, when we look at Figure (left), we are not provided that which regions of the image are incomplete by a guiding mask  Our brain figures this out just by looking at the form and predicting those missing regions  Inspired by the image inpainting work of DIP , we propose a novel algorithm for the contour completion problem (see Figure ), where, unlike DIP, we do not have a guiding mask to know where to fill in the missing regions of our disconnected contours 
Let us assume that we are given a degraded image x I containing a fragmented contour  We propose an iterative process (see Figure ) that can connect those discontinuities and glue those fragmented pieces together as follows  We propose an hour-glass model structure (f ) that is initially set up with completely random parameters (θ 0 ) at first 
Through an iterative process, we start feeding our network with a fixed random tensor noise z signal and obtain the inferred output (f (z)) from that network  We then back-propagate the difference between the inferred output and the incomplete image to the network  We then repeat this process until the difference between the generated outcome of the network (f θ (z)) and the incomplete image (x I ) gets smaller and smaller and finally overfits the incomplete image (x I ) 
In this work, we propose a novel error metric to backpropagate in the model and update its weights  we set the metric in a way that enables us to complete the incomplete image before it overfits the incomplete image  This is where the magic of our algorithm happens  We also propose a stopping criterion, so that when the image is complete, we no longer overfit the outcome of the model and instead produce a plausible connected set of fragmented contour pieces 
As illustrated in Figure , this trajectory will pass through a complete version of the image in image space, which is close to the actual connected ground truth x gt , which we do not have access to directly 

Energy Function

We can model this iterative process mathematically by maximizing a posterior distribution  Let us assume that the optimal image x * that we want to achieve is on a path that connects a random tensor noise z to the incomplete image x I   With this assumption, we can eventually overfit any random tensor noise to the incomplete image x I , and we can formulate the posterior distribution of our wanted optimal image x * as follows:
No Prior To better recapitulate what we want to achieve using our generative model, we solve an energy minimization problem on the parameter space of the model, rather than explicitly working with probability distributions and optimizing on x (image space)  Thus, we solve an energy minimization problem that incorporates the incomplete image (x I ) and model parameters (f θ (z)):
As shown in Figure , the pipeline starts from a random initialized set of parameters θ and updates those weights until it reaches a local minimum θ *   The only information provided for the network is the incomplete image x I   When we reach the optimal θ * , the completed image is obtained as x * = f θ * (z) where z is random tensor noise 
In this work, we use a U-Net architecture with skip connections as the generator model  " As we mentioned previously, in this work we were inspired by an earlier work known as Deep Image Prior (DIP)   In this work, the authors suggested a mean-squared-error loss term that enables the network to compare the output of the generator to the incomplete input image:
where x I is the incomplete image with missing pixels in correspondence of a binary mask m ∈ {0, 1} H×W and operator is for point-wise multiplication of two image matrices  In the inpainting tasks, the existence of a mask is essential as the algorithm needs to know where to fill in the missing area, whereas, in our work, we wanted to know whether the network can perform completion on its own without the need for the mask 
In other words, is it possible for the network to predict where to fill in at the same time that it is trying to reconstruct the incomplete image through the iterative process? To answer this question, we tried to solve a much harder problem in which the mask is not provided to the model and the model is agnostic to it 
To better understand how a solution could be hypothesized for this problem, we first imagine that we want to consider all the available regions in our image that could be potential places to fill in, i e , we set the mask in the previous formula 1 to be equal to the incomplete image x I   This is problematic as the model quickly tries to fill in all white space and quickly reconstructs the incomplete image by doing so 
On the other hand, we can take the inverse problem of the current problem, where the model tries to just fill in the regions that fragmented contour lives in  Taking these two at the same time, we came up with a novel loss term for energy minimization term that helps us remove the need for the mask in the case of the contour completion problem:
In this term, we introduce a linear combination of the two loss terms, where one focuses on reconstructing the missing regions in the foreground, and one focuses on avoiding inpainting regions in the background  The logic behind this is that, if we assume the original image to be representative of the mask, then the model tries to reconstruct in all white regions (the foreground), and in the inverse problem we just want to reconstruct the regions that are already part of the ground truth 

Stopping Criteria

As shown in Figure , knowing when to stop iterating to the over-fitted model is a key to obtaining a completed shape  Therefore, we equipped our model with a criterion that uses two individual novel terms to know when to stop and output the result of the network  These two metrics expand the capability of the generator network beyond what it does currently and achieve a full end-to-end contour completion model that trains and infers on a single image of divided contour fragments 
These new terms are: reconstruction_score (ρ) and overfit_score (ω) 

Reconstruction Score

The first score that this paper suggests is the reconstruction score, i  e , we have to make sure that the model is trained enough that it can reconstruct at least the entire set of fragmented contours within the image  This is a trivial score and to compute the reconstruction_score (ρ), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup to find the ratio of points in the original incomplete image (x 0 ) 
This score ranges from [0 − 100] 

Overfit Score

It is evident that the model overfits the fragmented contours  This is due to the fact that the error in our loss term is minimized as the x overfits to x I , i  e , replacing x with x I in the loss term would give us zero  As we hypothesize iterative process also produces the complete image before it overfits to the incomplete image, we can imagine that at some point the image is complete (x C ) and does not need to be fine-tuned any more to overfit to x I  
We suggest a new score called overfit_score  overfit_score determines how much of the reconstructed outcome is over the number of pixels that are already in the incomplete image (x I )  To compute the overfit_score (ω), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup of points in the outcome of the input image and see what portions of those points are novel and not already in the incomplete image (x I ) 
Similar to reconstruction_score, the overfit_score also ranges from [0 − 100]  Our goal is to maximize reconstruction_score and minimize the overfit_score, but we should consider that the minimization lower bound is data dependent and is not zero 

Combined Score

To be able to find the best possible set of completed contours, we combine the two and have a loop that tries to achieve close to full reconstruction and avoids over-fitting at the same time  This is what we call an "ideal" stopping point in the contour completion problem  In each run throughout all iterations, we pick an output of the network that minimizes a dissimilarity term:
where δ represents our dissimilarity score  The reconstruction_score and overfit_score are obtainable given network output and the incomplete image  Ideally, we want to achieve an output image that has a reconstruction_score equal to 100 and an overfit_score of γ which is a hyperparameter that is dependent on the image and complexity of the shape 
Empirically, we observed that this value is highly correlated with the distance between gaps that are observed in fragmented contours, i  e , the larger the gap results in a larger γ value  We will discuss this in more detail in the next section (see Section 3)  For one sample image, we computed the two metrics reconstruction_score and overfit_score and the combined value of dissimilarity (δ) and showed how these values change (see Figure ) 
Our initial observations show that the reconstruction_score will increase to 100 quickly for the incomplete image indicating that the already existing fragments of the contours have been reconstructed in the output  However, as mentioned previously, we cannot solely rely on this score since we also want to minimize the overfitting 
Remember that our goal is to produce an output that: a) preserves the original contours in the incomplete image and b) fills in the gaps between the fragmented contours  It is evident that overfit_score decreases throughout an iterative run of our process until it reaches zero  The dissimilarity will also decrease along with the overfit to a point, then it will increase, as the model tries to reproduce the incomplete image 
This is where an ideal γ value can be picked, i e , where to stop when the reconstruction is good but we have not done a full overfit to the incomplete image  Thus, one should pick the value of γ empirically in the scenario that the ground truth is not available, whereas, assuming that the ground truth is available, we can easily compute the best γ value 
In our experiments, we tried two datasets of images with different gap sizes  We observed that the best the γ for one set of samples is ∼ 5 (the set with shorter gaps) while it is ∼ 23 for samples from the other set, i  e, the set with longer gaps (see Figure for some completed examples) 

Experiments and Results

Performing unsupervised contour completion is a difficult task to benchmark as one can never know what fragments exactly are connected to each other in a real-world scenario  This makes the problem of contour completion a hard problem to solve  In this paper, we tried to create artificial shapes that are occluded by some masks and then tried to see if our model can regenerate the missing pieces and glue those divided contours together 
To demonstrate our model's behavior, we will conduct experiments on datasets created for this task and will report on them in this section  To compare network results in different settings, we will use pixel-wise Mean Squared Error (MSE) and Intersection over Union (IoU) between the produced result of the network and unmasked ground truth data and the reconstructed image on black pixels (where contours live) 

Data

We prepared two datasets, one labeled "Simple" and one "Complex", in accordance with the number of gaps in each shape  Both datasets contain nine different categories of shapes  In order to generate the Complex dataset, we used FlatShapeNet which is a dataset for the educational game Ariga  The dataset includes the following categories: Circle, Kite, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, Triangle and Overlap 
The "overlap" category contains images that are made as a mixture of two shapes that are overlapping from the previous categories  These are some standard shapes with a few gaps in simple dataset, while the complex dataset has some hand-drawn shapes with fragmented lines and more gaps that produce more variety in general 
For each instance, a ground truth image is available for comparison  Most of our experiments have been conducted using the complex dataset in order to evaluate the generalization of our approach  For the analysis of how γ values should be set for each shape, we used the simple dataset as a reference 

Evaluation

In this section, we compare our model to the original Deep Image Prior (DIP) inpainting model  DIP's inpainting module accepts a degraded image and a binary mask corresponding to that image  In order to make a fair comparison, instead of providing a binary mask, we used the incomplete images both as input and as a mask in order to see whether it can produce a result similar to ours 
For DIP, we run the iterative process for a maximum number of 2500 iterations with the U-net backbone  We used the exact same architecture and setting in our model for a fair comparison  Using our ground truth dataset images, we calculate the MSE loss between the network output and ground truth during each iteration instead of relying on our stopping mechanism described in the previous section 
We then store the output with minimal loss throughout all the iterations  Finally, we select the best output among all iterations, report the MSE and IoU with the ground truth, and save the iteration number which resulted in the lowest MSE  Table compares the results that are obtained using the DIP method, the DSP method (ours), and the difference between raw images and the ground truth 
We have presented the average MSE-loss, average IoU, and the average number of iterations for the best output for different methods  As can be seen from the table, our model improves both MSE and IoU between the incomplete image and ground truth in fewer iterations  The DIP method can neither generate a better result than the raw image nor provide stopping criteria to prevent overfitting 
We provide a more detailed analysis of this result in Figure   As results show, our algorithm not only provides a much faster convergence but also consistently provides a better-completed image (consistently less MSE loss and better IoU), whereas it is challenging for the DIP method to accomplish better results without a guiding mask 
We compare MSE loss between the degraded raw images that these algorithms started with (shown in blue) (a) Mean Squared Error Loss: we clearly see that for almost all images, DSP (green) achieves a lower MSE than the incomplete images (blue) whereas, the DIP completed images either do not improve the MSE or even worsen that for the incomplete images 
Note that, the MSE is computed to an available ground truth image hidden from our methods (the lower is better)  (b) Intersection Over Union: here, we are looking at the IoU metric that specifies the amount of intersection over the union between the obtained images and the ground truth data  Again, we see that DSP produces images that are much closer to the ground truth (in most cases) whereas the DIP can not achieve a similar result 
While we see few DIP completed images produce a better IoU than the degraded images (in terms of IoU), most of them are worse than the starting image (the higher is better)  (c) The number of iterations that are needed for each algorithm to obtain its best results  Here, we see that DSP can quickly produce the best outcome with the least MSE loss whereas the DIP algorithm's best results are when we run the iterative process for more iterations (the lower is better) 

Correlation of γ with the Gap Size

To better understand the γ parameter of our combined score, we conducted the following experiment  A total of 12000 samples were obtained by merging all of our images from the two datasets, simple and complex  As we have access to the ground truth for each degraded image in the combined dataset, we can easily calculate reconstruction_score and overfit_score for each degraded-ground truth pair 
As expected, we obtain a reconstruction_score of 100 for all samples, but the overfit_score varies among them  Intuitively, we hypothesized that an optimal value of overfit score should be intertwined with the total area of gaps  To test this hypothesis, we did the following experiment  We first define a function φ(x) which takes a binary, black and white image x and returns the number of black pixels in it 
Then we define a gap term as follows: where x I is the incomplete image and x gt is the ground truth  In this case, gap indicates the total area of the gap with respect to the whole shape  We found out that this term and the best result have a correlation of 97 43%  This indicates that the value of γ is highly correlated with the gap size, that is something expected in a way 

Effect of α

We conducted additional experiments concerning how α affects the quality of reconstruction  In the previous section, we defined Equation 2 as the loss term that guides our iterative process  The term α specifies the amount of emphasis the model should place on reconstructing missing regions, rather than filling in fragmented contours 
A lower α indicates a better grouping quality, as shown in Equation   However, we will not achieve completion if we remove the first term completely from the loss by setting α = 0  Therefore, the first term should be kept, but its weight should be very low in order to achieve a good completion  On the other hand, if we set α = 1 and omit the second term, we lose the contour completion regularization term and obtain the same output as a vanilla deep image prior, which does not complete shapes 

Effect of Receptive Field Size

To better understand the effect of receptive field size on our algorithm, we test the following hypothesis: can models with bigger receptive field size complete shapes with bigger gaps? In Table , we report showing the results of this experiment  As we can see, the bigger the receptive field size, the more complete shapes we can reconstruct using DSP 

Implementation Details

As shown in Figure , we use a model with 5 layers and 128 channels for downsampling and upsampling convolutions, and 64 channels for skip convolutions  The upsampling and downsampling modules use 3 × 3 filters, while the skip module uses 1 × 1 filters  In the upsample part of the network, the nearest neighbor algorithm is used 
We used 256 × 256 images with three channels in all of our experiments  In training, we use the MSE loss between the degraded image and the output of the network, and we optimize the loss using the ADAM optimizer and a learning rate equal to 0 01   In our experiments, we also used α = 0 15 as an optimal proportion coefficient for reconstruction loss 

Conclusion

In this work, we introduced a novel framework for contour completion using deep structure priors (DSP)  This work offers a novel notion of a maskless grouping of fragmented contours  In our proposed framework, we introduced a novel loss metric that does not require a strict definition of the mask  Instead, it lets the model learn the perceivable illusory contours and connects those fragmented pieces using a generator network that is solely trained on just the single incomplete input image 
Our model does not require any pre-training which demonstrates that the convolutional architecture of the hour-glass model is able to connect disconnected contours  We present an extended set of experiments that show the capability of our algorithm  We investigate the effect of each parameter introduced in our algorithm separately and show how one could possibly achieve the best result for their problem using this model 
In future work, we plan to extend this model and try to see how it performs with real images  In particular, we want to determine whether we can inpaint real-world photographs while retaining perceptually aware scene structures  The importance of shape in perception by deep neural networks has been highlighted in many adversarial examples to appearance-based networks  
The outcome of this work has strong potential to impact the designing and implementation of models that are robust to such perturbations 



ground_truth: 3-D printing and software development.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What was the population of McPherson County according to the 2020 census?

Read the following text and answer briefly.

Context 2: McPherson County (standard abbreviation: MP) is a county located in the U S  state of Kansas   As of the 2020 census, the county population was 30,223  The largest city and county seat is McPherson  The county is named for Civil War General James B  McPherson 

History

Early history

For many millennia, the Great Plains of North America was inhabited by nomadic Native Americans  From the 16th century to 18th century, the Kingdom of France claimed ownership of large parts of North America  In 1762, after the French and Indian War, France secretly ceded New France to Spain, per the Treaty of Fontainebleau  In 1802, Spain returned most of the land to France, but keeping title to about 7,500 square miles 

In 1803, most of the land for modern day Kansas was acquired by the United States from France as part of the 828,000 square mile Louisiana Purchase for 2 83 cents per acre  In 1848, after the Mexican–American War, the Treaty of Guadalupe Hidalgo with Spain brought into the United States all or part of land for ten future states, including southwest Kansas  In 1854, the Kansas Territory was organized, then in 1861 Kansas became the 34th U S  state 

19th century

From the 1820s to 1870s, the Santa Fe Trail passed through, what is now McPherson County  The trail entered the county, east of Canton, then south of Galva, then north of Inman, and west towards Lyons  In 1855, Charles O  Fuller established a ranch adjacent to the Running Turkey Creek Crossing about two miles south and one mile east of Galva  Fuller's Ranch provided accommodations for travelers on the Santa Fe Trail and was probably the first white settlement in McPherson County 

Peketon County was established in 1860, by the passage of a bill by S  N  Wood:  An act to establish Peketon County  Section 1  - That all that territory west of the sixth principal meridian and south of Township 16, in Kansas Territory, be and the same is hereby erected into a county, to be known by the name of Peketon County  On February 17, 1865, Peketon County was abolished, and McPherson County was made a part of Marion County, which extended from the west line of Chase County to the present western boundary of Kansas 

In 1868, Solomon Stephens and L  N  Holmberg were appointed Justices of the Peace—the first officers in what is now McPherson County  The next year (1869) occurred the first election for the township, now the county of McPherson  McPherson was regularly organized as a county in the spring of 1870, a mass meeting being held at Sweadal  Sweadal, the county seat thus selected, was located about one mile and a half southwest of the present site of Lindsborg  In September, however, the County Commissioners resolved to meet at the latter place, McPherson which had already been located some two years 

In April, 1873, a petition was filed for the county seat re-location  It was signed by 483 voters, and a special election was accordingly ordered for June 10  Upon that day, McPherson received 605 votes, New Gottland 325, King City 3 and Lindsborg 1; McPherson's majority over all, 276  In May the McPherson Town Company had offered, as an inducement for the location of the county seat at this point, the free use of rooms for ten years, and the donation of two squares of land on the town site  The offer was accepted the next month, the County Commissioners selecting blocks 56 and 65  Thus the county seat was established at McPherson and has remained since 

As early as 1875, city leaders of Marion held a meeting to consider a branch railroad from Florence  In 1878, Atchison, Topeka and Santa Fe Railway and parties from Marion County and McPherson County chartered the Marion and McPherson Railway Company   In 1879, a branch line was built from Florence to McPherson, in 1880 it was extended to Lyons, in 1881 it was extended to Ellinwood  The line was leased and operated by the Atchison, Topeka and Santa Fe Railway  The line from Florence to Marion, was abandoned in 1968  In 1992, the line from Marion to McPherson was sold to Central Kansas Railway  In 1993, after heavy flood damage, the line from Marion to McPherson was abandoned  The original branch line connected Florence, Marion, Canada, Hillsboro, Lehigh, Canton, Galva, McPherson, Conway, Windom, Little River, Mitchell, Lyons, Chase, then connected with the original AT&SF main line at Ellinwood 

In 1887, the Chicago, Kansas and Nebraska Railway extended its main line from Herington to Pratt   This main line connected Herington, Ramona, Tampa, Durham, Waldeck, Canton, Galva, McPherson, Groveland, Inman, Medora, Hutchinson, Whiteside, Partridge, Arlington,  Langdon, Turon, Preston, Natrona, Pratt   In 1888, this main line was extended to Liberal   Later, this line was extended to Tucumcari, New Mexico and Santa Rosa, New Mexico, where it made a connection with the Southern Pacific from El Paso, Texas   The Chicago, Kansas and Nebraska Railway was absorbed by the Chicago, Rock Island and Pacific Railway  This line is also called the "Golden State Route" 

20th century
The National Old Trails Road, also known as the Ocean-to-Ocean Highway, was established in 1912, and was routed through Windom, Conway, McPherson 

Geography

According to the U S  Census Bureau, the county has a total area of , of which  is land and  (0 3%) is water 

Adjacent counties
 Saline County (north)
 Dickinson County (northeast)
 Marion County (east)
 Harvey County (southeast)
 Reno County (southwest)
 Rice County (west)
 Ellsworth County (northwest)

Major highways
  Interstate 135
  U S  Route 56
  U S  Route 81
  K-4
  K-61
  K-153

Demographics

The McPherson Micropolitan Statistical Area includes all of McPherson County 

2000 census
As of the census of 2000, there were 29,554 people, 11,205 households, and 7,966 families residing in the county   The population density was 33 people per square mile (13/km2)   There were 11,830 housing units at an average density of 13 per square mile (5/km2)   The racial makeup of the county was 96 53% White, 0 81% Black or African American, 0 34% Native American, 0 32% Asian, 0 06% Pacific Islander, 0 79% from other races, and 1 16% from two or more races   1 94% of the population were Hispanic or Latino of any race  37 1% were of German, 12 9% Swedish, 12 1% American, 6 7% English and 6 3% Irish ancestry according to Census 2000 

There were 11,205 households, out of which 33 00% had children under the age of 18 living with them, 62 50% were married couples living together, 6 00% had a female householder with no husband present, and 28 90% were non-families  25 50% of all households were made up of individuals, and 11 80% had someone living alone who was 65 years of age or older   The average household size was 2 49 and the average family size was 2 99 

In the county, the population was spread out, with 25 40% under the age of 18, 10 30% from 18 to 24, 25 20% from 25 to 44, 21 80% from 45 to 64, and 17 30% who were 65 years of age or older   The median age was 38 years  For every 100 females there were 95 90 males   For every 100 females age 18 and over, there were 92 90 males 

The median income for a household in the county was $41,138, and the median income for a family was $48,243  Males had a median income of $33,530 versus $21,175 for females  The per capita income for the county was $18,921   About 4 20% of families and 6 60% of the population were below the poverty line, including 5 20% of those under age 18 and 8 10% of those age 65 or over 

Government

Presidential elections
McPherson county is often carried by Republican candidates  The last time a Democratic candidate has carried this county was in 1964 by Lyndon B  Johnson 

Laws
Following amendment to the Kansas Constitution in 1986, the county remained a prohibition, or "dry", county until 1996, when voters approved the sale of alcoholic liquor by the individual drink with a 30 percent food sales requirement 

Education

Colleges
 McPherson College in McPherson
 Bethany College in Lindsborg
 Central Christian College in McPherson

Unified school districts
 Smoky Valley USD 400
 McPherson USD 418
 Canton-Galva USD 419
 Moundridge USD 423
 Inman USD 448

School district office in neighboring county
 Goessel USD 411
 Little River-Windom USD 444

Museums
 Birger Sandzén Memorial Gallery in Lindsborg
 McCormick-Deering Days Museum in Inman
 McPherson Museum in McPherson
 Lindsborg Old Mill & Swedish Heritage Museum in Lindsborg
 Kansas Motorcycle Museum in Marquette

Communities

Cities

 Canton
 Galva
 Inman
 Lindsborg
 Marquette
 McPherson (county seat) 
 Moundridge
 Windom

Unincorporated communities
† means a Census-Designated Place (CDP) by the United States Census Bureau 
 Conway
 Elyria†
 Groveland
 Johnstown
 New Gottland
 Roxbury†

Ghost towns
 Alta Mills
 Battle Hill
 Christian
 Doles Park
 Elivon
 King City
 Sweadal

Townships
McPherson County is divided into twenty-five townships   The cities of Lindsborg and McPherson are considered governmentally independent and are excluded from the census figures for the townships   In the following table, the population center is the largest city (or cities) included in that township's population total, if it is of a significant size 

See also
 List of people from McPherson County, Kansas
 National Register of Historic Places listings in McPherson County, Kansas
 McPherson Valley Wetlands
 Maxwell Wildlife Refuge

References

Notes

Further reading

 Wheeler, Wayne Leland  "An Analysis of Social Change in a Swedish-Immigrant Community: The Case of Lindsborg, Kansas " (PhD dissertation, University of Missouri-Columbia; ProQuest Dissertations Publishing, 1959  5905657) 

County
 Through the Years: A Pictorial History of McPherson County; McPherson Sentinel' Heritage House Publishing Co; 1992 
 McPherson County First Courthouse Built About 1869 or 1870; Lindsborg News-Record; March 30, 1959 
 Pioneer Life and Lore of McPherson County, Kansas; Edna Nyquist; Democratic-Opinion Press; 1932 
 A History of the Church of the Brethren in Kansas (includes McPherson College history); Elmer LeRoy Craik; McPherson Daily; Republican Press; 397 pages; 1922 
 Portrait and Biographical Record of Dickinson, Saline, McPherson, and Marion Counties, Kansas; Chapman Bros; 614 pages; 1893 
 Standard Atlas of McPherson County, Kansas; Geo  A  Ogle & Co; 82 pages; 1921 
 Plat Book of McPherson County, Kansas; North West Publishing Co; 50 pages; 1903 
 Edwards' Atlas of McPherson County, Kansas; John P  Edwards; 51 pages; 1884 

Trails
 The Story of the Marking of the Santa Fe Trail by the Daughters of the American Revolution in Kansas and the State of Kansas; Almira Cordry; Crane Co; 164 pages; 1915  (Download 4MB PDF eBook)
 The National Old Trails Road To Southern California, Part 1 (LA to KC); Automobile Club Of Southern California; 64 pages; 1916  (Download 6 8MB PDF eBook)

Mennonite Settlements
 Impact of Mennonite settlement on the cultural landscape of Kansas; Brenda Martin; Kansas State University; 1985/1988  
 Mennonite settlement : the relationship between the physical and cultural environment; Susan Movle; University of Utah; 1975/1886 
 Status of Mennonite women in Kansas in their church and home relationships; Eva Harshbarger; Bluffton College; 1925/1945 

External links

County
 
 McPherson County - Directory of Public Officials
Historical
 , from Hatteberg's People'' on KAKE TV news
Maps
 McPherson County Maps: Current, Historic, KDOT
 Kansas Highway Maps: Current, Historic, KDOT
 Kansas Railroad Maps: Current, 1996, 1915, KDOT and Kansas Historical Society

 
Kansas counties
1867 establishments in Kansas
Populated places established in 1867



ground_truth: 30,223.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the receptive field size affect the completion of shapes?

Read the following text and answer briefly.

Context 4:  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}




ground_truth: Bigger receptive field size leads to more successful shape completion.