ECE Digital Image Processing and Introduction to Computer Vision

Transcription

1 ECE59064 Dgtal Image Processg ad Itroducto to Computer Vso Depart. of ECE NC State Uverst Istructor: Tafu Matt Wu Sprg 07 Outle Recap Le Segmet Detecto Fttg Least square Total square Robust estmator Hough trasform RANSAC

2 4/4/7 Le Segmet Detecto Le Fttg ad Vashg Pot Detecto Vertcal vashg pot at ft Vashg le Vashg pot Vashg pot Recap LSD: a Le Segmet Detector Goal: detectg locall straght cotours.e. le segmets o gra mages Rafael Grompoe vo Go et al paper ad code

3 Recap LSD: a Le Segmet Detector Algorthm Recap ELSD: Ellpse ad Le Segmet Detector V. Pătrăucea P. Gurdjos ad Rafael Grompoe vo Go ECCV0 code: 3

4 Recap ELSD: Ellpse ad Le Segmet Detector Fttg Fgure credt: Lazebk 4

5 RecallBasc Relatoshps betwee Pels Neghbors of a pel are defed w.r.t. Coordates ad Dstace measures 4 / D / 8eghbors artfacts due to samplg.e. dgtalzed coordates Adjacec s defed w.r.t. The tpe of eghbors specfed ad A set of pel values specfed How to geeralze ths settg? 4 / D / 8 / madjacec Path: a sequece of pels the whole lattce b default wth successve pels beg adjacet 4 / D / 8 / mpath Coectvt s defed w.r.t. A subset of pels specfed whch could be the whole lattce ad A path s etrel cotaed the subset. Coected compoets the subset The subset s a coected set / rego f ol oe coected compoet ests Now we have taught a computer what a rego s. Fttg Choose a parametrc model to represet a set of data / features smple model: les smple model: crcles complcated model: car Source: K. Grauma 5

6 Fttg Desg challeges Desg a sutable goodess of ft measure Smlart should reflect applcato goals Ecode robustess to outlers ad ose Desg a optmzato method Avod local optma Fd best parameters quckl Fttg: Overvew If we kow whch pots belog to the le how do we fd the optmal le parameters? Least squares What f there are outlers? Robust fttg RANSAC What f there are ma les? Votg methods: RANSAC Hough trasform 6

7 Least squares le fttg Data: Le equato: m + b Fd m b to mmze E å m b é ù E Y XB where Y! ë û E de db X T Y XB T T X XB X Y 0 XB X T T Y XB Y XB Y Y é X! ë ù! û T m+b émù B ëb û T T Y XB Y + XB XB Matlab: B X \ ; Normal equatos: least squares soluto to XBY Problem wth vertcal least squares Not rotatovarat Fals completel for vertcal les 7

8 Total least squares a+bd Ut ormal: Dstace betwee pot ad le a+bd a +b E : a + b d å a + b d Na b Total least squares Dstace betwee pot ad le a+bd a +b : a + b d Fd a b d to mmze the sum of squared perpedcular dstaces E å a + b d a+bd Ut ormal: Na b E å a + b d 8

9 9 Total least squares Dstace betwee pot ad le a+bd a +b : a + b d Fd a b d to mmze the sum of squared perpedcular dstaces å + d b a E a+bd å + d b a E Ut ormal: Na b 0 + å d b a d E b a b a d + + å å UN UN b a b a E T û ù ë é û ù ë é + å!! 0 N U U dn de T Soluto to U T UN 0 subject to N : egevector of U T U assocated wth the smallest egevalue least squares soluto to homogeeous lear sstem UN 0 Total least squares û ù ë é U!! û ù ë é å å å å T U U secod momet matr

10 0 Total least squares û ù ë é U!! û ù ë é å å å å T U U N a b secod momet matr F&P d ed. sec.. Least squares as lkelhood mamzato Geeratve model: le pots are sampled depedetl ad corrupted b Gaussa ose the drecto perpedcular to the le ø ö ç è æ + ø ö ç è æ ø ö ç è æ b a v u e a+bd u v ε pot o the le ose: sampled from zeromea Gaussa wth std. dev. σ ormal drecto

11 Least squares as lkelhood mamzato Õ Õ ø ö ç ç è æ + µ d b a d b a P d b a P ep s! Lkelhood of pots gve le parameters a b d: å + d b a d b a L s! Loglkelhood: a+bd u v ε Geeratve model: le pots are sampled depedetl ad corrupted b Gaussa ose the drecto perpedcular to the le ø ö ç è æ + ø ö ç è æ ø ö ç è æ b a v u e Least squares: Robustess to ose Least squares ft to the red pots:

12 Least squares: Robustess to ose Least squares ft wth a outler: Problem: squared error heavl pealzes outlers Robust estmators Geeral approach: fd model parameters θ that mmze å r q s r ; r θ resdual of th pot w.r.t. model parameters θ ρ robust fucto wth scale parameter σ The robust fucto ρ behaves lke squared dstace for small values of the resdual u but saturates for larger values of u

13 Choosg the scale: Just rght The effect of the outler s mmzed Choosg the scale: Too small The error value s almost the same for ever pot ad the ft s ver poor 3

14 Choosg the scale: Too large Behaves much the same as least squares Robust Estmator. Italze: e.g. choose θ b least squares ft ad s.5 meda error q data. Choose params to mmze: å s + error q data E.g. umercal optmzato 3. Compute ew error s.5 meda error 4. Repeat ad 3 utl covergece 4

15 Robust estmato: Detals Robust fttg s a olear optmzato problem that must be solved teratvel Least squares soluto ca be used for talzato Scale of robust fucto should be chose adaptvel based o meda resdual Outlers are ot the ol ssue Whe multple structures appear Source: Colls 5

16 Other was to search for parameters for whe o closed form soluto ests Le search. For each parameter step through values ad choose value that gves best ft. Repeat utl o parameter chages Grd search. Propose several sets of parameters evel sampled the jot set. Choose best or top few ad sample jot parameters aroud the curret best; repeat Gradet descet. Provde tal posto e.g. radom. Locall search for better parameters b followg gradet Hpothesze ad test. Propose parameters Tr all possble Each pot votes for all cosstet parameters Repeatedl sample eough pots to solve for parameters. Score the gve parameters Number of cosstet pots possbl weghted b dstace 3. Choose from amog the set of parameters Global or local mamum of scores 4. Possbl refe parameters usg lers 6

17 Hough Trasform: Outle. Create a grd of parameter values. Each pot votes for a set of parameters cremetg those values grd 3. Fd mamum or local mama grd Hough trasform P.V.C. Hough Mache Aalss of Bubble Chamber Pctures Proc. It. Cof. Hgh Eerg Accelerators ad Istrumetato 959 Gve a set of pots fd the curve or le that eplas the data pots best m m + b b Hough space Slde from S. Savarese 7

18 Hough trasform m Slde from S. Savarese m b b Hough trasform P.V.C. Hough Mache Aalss of Bubble Chamber Pctures Proc. It. Cof. Hgh Eerg Accelerators ad Istrumetato 959 Issue : parameter space [mb] s ubouded Use a polar represetato for the parameter space r r q cosq + sq r Hough space q Slde from S. Savarese 8

19 Hough trasform features votes Slde from S. Savarese Hough trasform Nos data feature s Need to adjust grd sze or smooth votes Slde from S. Savarese 9

20 Hough trasform featu res votes Issue: spurous peaks due to uform ose Slde from S. Savarese E. Image à Ca 0

21 Ca à Hough votes Hough votes à Edges Fd peaks ad postprocess

22 Fdg les usg Hough trasform Usg mb parameterzato Usg r theta parameterzato Usg oreted gradets Practcal cosderatos B sze Smoothg Fdg multple les Fdg le segmets RANSAC Robust fttg ca deal wth a few outlers what f we have ver ma? Radom sample cosesus RANSAC: Ver geeral framework for model fttg the presece of outlers Outle Choose a small subset of pots uforml at radom Ft a model to that subset Fd all remag pots that are close to the model ad reject the rest as outlers Do ths ma tmes ad choose the best model M. A. Fschler R. C. Bolles. RANdom SAmple Cosesus: A Paradgm for Model Fttg wth Applcatos to Image Aalss ad Automated Cartograph. Comm. of the ACM Vol 4 pp

23 RANSAC for le fttg eample Source: R. Raguram RANSAC for le fttg eample Leastsquares ft Source: R. Raguram 3

24 RANSAC for le fttg eample. Radoml select mmal subset of pots Source: R. Raguram RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model Source: R. Raguram 4

25 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto Source: R. Raguram RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model Source: R. Raguram 5

26 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpotheszeadverf loop Source: R. Raguram RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpotheszeadverf loop Source: R. Raguram 5 6

27 RANSAC for le fttg eample Ucotamated sample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpotheszeadverf loop Source: R. Raguram 53 RANSAC for le fttg eample. Radoml select mmal subset of pots. Hpothesze a model 3. Compute error fucto 4. Select pots cosstet wth model 5. Repeat hpotheszeadverf loop Source: R. Raguram 7

28 RANSAC for le fttg Repeat N tmes: Draw s pots uforml at radom Ft le to these s pots Fd lers to ths le amog the remag pots.e. pots whose dstace from the le s less tha t If there are d or more lers accept the le ad reft usg all lers Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p e.g Zeromea Gaussa ose wth std. dev. σ: t 3.84σ Number of samples N Choose N so that wth probablt p at least oe radom sample s free from outlers e.g. p0.99 outler rato: e Source: M. Pollefes 8

29 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p e.g Zeromea Gaussa ose wth std. dev. σ: t 3.84σ Number of samples N Choose N so that wth probablt p at least oe radom sample s free from outlers e.g. p0.99 outler rato: e s N e p s p / log e N log proporto of outlers e s 5% 0% 0% 5% 30% 40% 50% Source: M. Pollefes Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p e.g Zeromea Gaussa ose wth std. dev. σ: t 3.84σ Number of samples N Choose N so that wth probablt p at least oe radom sample s free from outlers e.g. p0.99 outler rato: e s N e p s p / log e N log Source: M. Pollefes 9

30 Choosg the parameters Ital umber of pots s Tpcall mmum umber eeded to ft the model Dstace threshold t Choose t so probablt for ler s p e.g Zeromea Gaussa ose wth std. dev. σ: t 3.84σ Number of samples N Choose N so that wth probablt p at least oe radom sample s free from outlers e.g. p0.99 outler rato: e Cosesus set sze d Should match epected ler rato Source: M. Pollefes Adaptvel determg the umber of samples Outler rato e s ofte ukow a pror so pck worst case e.g. 50% ad adapt f more lers are foud e.g. 80% would eld e0. Adaptve procedure: N sample_cout 0 Whle N >sample_cout Choose a sample ad cout the umber of lers If ler rato s hghest of a foud so far set e umber of lers/total umber of pots Recompute N from e: Icremet the sample_cout b s p / log e N log Source: M. Pollefes 30

31 RANSAC Sog Whe ou have outlers ou ma face much frustrato; f ou clude them a model fttg operato. But f our model's ft to a sample set of mmal sze the probablt of the set beg outlerfree wll rse. Brute force tests of all sets wll cause computatoal costpato. N radom samples wll provde a eample of a ftted model uflueced b outlers. No eed to test all combatos! Each radom tral should have ts ow uque sample set ad make sure that the sets ou choose are ot degeerate. N the umber of sets to choose s based o the probablt of a pot beg a outler ad of fdg a set that's outler free. Updatg N as ou go wll mmse the tme spet. So f ou gamble that N samples are ample to ft a model to our set of pots t's lkel that ou wll w the bet. Select the set that boasts that ts umber of lers s the most ou're almost there. Ft a ew model just to those lers ad dscard the rest a estmated model for our data s ow possessed! Ths marks the ed pot of our model fttg quest. Pros RANSAC pros ad cos Smple ad geeral Applcable to ma dfferet problems Ofte works well practce Cos Lots of parameters to tue Does t work well for low ler ratos too ma teratos or ca fal completel Ca t alwas get a good talzato of the model based o the mmum umber of samples c/ 3

32 Summar Fttg If we kow whch pots belog to the le how do we fd the optmal le parameters? Least squares What f there are outlers? Robust fttg RANSAC What f there are ma les? Votg methods: RANSAC Hough trasform 3