An Object Detection Method based on the Separability Measure using an Optimization Approach

Transcription

1 An Object Detecton Method based on the Separablt Measure usng an Optmzaton Approach Edward Y. H. Cho *, Wa Tak Hung 2 Hong Kong Poltechnc Unverst, Hung Hom, Kowloon, Hong Kong E-mal: mahcho@net.polu.edu.hk 2 Ke Unverst Research Centre n Health Technologes Unverst of Technolog, Sdne, Broadwa NSW 27, Australa E-mal: arthur_hung@hotmal.com * Correspondng author Receved: Aprl 4, 2 Accepted: June 24, 2 Publshed: Jul 3, 2 Abstract: The detecton of an object n an mage can be set up as a mamzaton problem usng separablt measure as the objectve functon. The parameters of ths objectve functon are the parameters used to defne two regons n a mask. Ths mask has the same dmenson as the mage beng consdered. The two regons n the mask correspond to two regons n the mage under nvestgaton. The pel value nformaton n these two regons n the mage would be used for the calculaton of the separablt measure. An optmzaton method would then be used to solve ths mamzaton problem. Ths stud demonstrated that the proposed method could detect the rectangle n the test mage successfull. It showed that object detecton and detaled segmentaton could be carred out at the same tme f the geometr of an object could be descrbed mathematcall. Kewords: Object detecton, Separablt measure, Optmzaton method. Introducton Threshold method s used to segment objects of nterest n an mage from the background. A common presentaton of segmentaton result s the use of a mask. A mask s a data fle specfng for each pel ether one or zero. Value one ndcates that the correspondng pel s part of the object beng segmented, whereas zero specfes a pel n the background. A new measure of class separablt has been demonstrated to be able to select the best mask (two regons representng respectve object and background) from a set of masks []. Each of them s the dfferent segmentaton result of the same mage. In ths case, the measure s used for evaluaton of segmentaton results. Pont, lne and edge detecton and object recognton are ke topcs n mage processng tetbook [2]. Also, computer aded detecton methods are frequentl used n dgtal medcal mages. For eample, breast cancer detecton n mammograms [3]. The use of the new measure of class separablt can be etended to detect an object n an mage usng an optmzaton approach. The objectve functon evaluates the new measure of separablt for a gven set of parameter values. Each set of parameter values refers to an enclosed regon n an mage that has a target object. The area outsde the enclosed regon of the mage s called the outer regon. The enclosed and outer regons represent the area of nterest and background respectvel. The calculaton of the new measure of separablt s based on pels value n these two defned regons accordng to a gven set of parameters. When the objectve functon s optmal, the locaton of the enclosed area defned b the correspondng set of parameters s epected to be near to the target object. Our objectve of ths stud s to eplore the applcaton the newl developed measure of separablt to the area of object detecton usng a smple test mage. 39

2 Materals and method Data The test mage has a small rectangle n whte colour as shown n Fgure. The dmenson of ths mage s 5 pels tmes 5 pels and ts background s black n colour. A pel s a square and has an area of unt square. Its top left corner and bottom rght corner are at (, ) and (,) respectvel n a 3-dmensonal co-ordnates sstem. The locaton of a pel n the test mage s smpl denoted b [ ]. The whte rectangle s defned b 4 parameters, namel, a and b. The centre of the whte rectangle s at pel, 2 a +,.e., a pel both to the left and to the rght [ ]. The wdth of ths rectangle s ( ) from the centre. Smlarl, the heght of the rectangle s ( 2 b +),.e., b pel both up and down from the centre. The rectangle n the test mage has the respectve values 22, 2, 2, 4 for, a and b. The pel values of the test mage are stored n a matr. For pel [ ], ts value s located at th row and th column of the matr. Pel values for black and whte colours are and 255 respectvel. Separablt Measure Fg. The test mage wth a whte rectangle and a background of black colour Measure of separablt The new measure of separablt s proposed b Cho and Hung []. It s based on the measure of separablt used b Otsu as an mage threshold method [4]. The new measure of separablt s smpl called separablt measure and s brefl descrbed. et and denote the sets of gra levels correspondng to the pel values n the background and n the object of an mage respectvel. In both and, the have gra levels,, 2, K,, where s the mamum gra level and + s the total number of possble gra levels. et n and n be the number of pels at level for and respectvel. Then the total number of pels s: N = ( n + n ). = 4

3 The respectve probablt dstrbuton for the occurrence of gra level for and are: n P = and N n P = N, =,, 2, K,. The zeroth- and frst-order cumulatve moments of the hstogram for are defned respectvel as: ω = P and = = Smlarl, µ. P = ω = P and µ = = P = 2 are the respectve moments for. The measure of class separablt, σ B, s defned as: 2 ( µ ) 2 T ω µ σ B = ω ω where µ T = µ + µ = ( n + n ) N = s the mean gra level of the orgnal matr (or mage). The followng four analses were desgned to detect and fnd the locaton of the whte rectangle n the test mage. Analss I A mask s used to determne two regons n the test mage. It has the same dmenson as the test mage. The rectangle n the mask s defned b four parameters descrbed above. The parameters a and b was assgned to the value of 2 and 4 respectvel. The values for and both range from to 5. When part of a rectangle s outsde the boundar of the mask, the would be truncated. For eample, when and are both, the top left corner and bottom rght corner of ths rectangle are at pels [, ] and [3, 4] respectvel. That means area above and to the left of the centre of rectangle s truncated. Totall, 25 dfferent rectangles are formed. Each of them dentfes two regons n the test mage,.e., pels wthn the rectangle and pels outsde the rectangle. Pels values wthn these two correspondng regons are used to calculate the separablt measure. The mamum separablt measure together wth ts rectangle centre locaton can be obtaned from these 25 solutons. Analss II Instead of evaluatng 25 separablt measures as shown n Analss I, random rectangle centres wth pel [ ] are generated n ths analss. These rectangles have a and b values equal to 2 and 4 respectvel. Ther separablt measures are then calculated. The mamum separablt measure together wth ts rectangle centre locaton can be obtaned from these solutons. 4

4 Analss III A mask that has the same dmenson of the test mage s used. It has a rectangle defned b four parameters (, a and b ) descrbed above. So, t has two regons,.e., area wthn the rectangle and area outsde the rectangle. These two regons n the mask correspond to two regons n the test mage. The separablt measure of the test mage can then be calculated based on the two regons dentfed b the mask. Hence, the detecton of the rectangle n the test mage can be set up as a mamzaton problem usng separablt measure as the objectve functon. The parameters of ths objectve functon are the four parameters that defne the rectangle n a mask. Ths objectve functon value vares as dfferent masks are generated from dfferent sets of rectangle parameters. An optmzaton method would then be used to solve ths mamzaton problem. It s epected the objectve functon s optmal when the respectve parameter values for the mask are 22, 2, 2 and 4. That means the rectangle n the mask corresponds eactl to the rectangle n the test mage. The functon called constroptm from R was used to perform the optmsaton [5]. B default constroptm performs mnmzaton. When the fnscale parameter was set to, t turns the problem nto a mamzaton problem. The Nelder-Mead method was selected for the method parameter [6]. It uses onl functon values and s robust but relatvel slow. The u and c parameters from the constroptm were used to set constrants for the parameter, a and b. These parameters were set to be greater than or equal to,, and respectvel. Snce the rectangle parameters were ntegers, the estmates returned from the constroptm are rounded before the were used n the separablt measure calculaton. The random set of parameters generated from Analss II to defne the rectangle n a mask are each used as the startng pont for the constroptm n ths analss. Analss IV Ths analss s smlar to Analss III ecept the mask s defned dfferentl. The mask has two rectangles. The frst one s defned b four parameters, a and b descrbed above. The centre of the second rectangle s the same as the frst one. The area of second rectangle s twce the area of the frst one. Therefore, the parameters for the second rectangle are, a + d and b + d, where 2 2 d = ( a + 6ab + b ( a + b) ) 4 and the smallest nteger not less than d should be taken to replace the orgnal d. The area wthn frst rectangle and the area between the two rectangles n the mask correspond to two regons n the test mage for the calculaton of the separablt measure. The random set of parameters generated from Analss II to defne the rectangle n a mask are each used as the startng pont for the constroptm n ths analss. Results The scatterplot3d functon from R was used to show the separablt measures, z of test mage for a gven mask or a mask determned b the optmzaton method. The vertcal lne was at the md-pont of the pel that represents the centre of rectangle wthn a mask. The heght of the lnes corresponds to the numerc value of the measure. The co-ordnate at the top of the lne s ( 5,. 5,. z ). In Analss I, each separablt measure value was shown n Fg. 2. Ths analss showed the value of the measure of separablt ncreased as the rectangle n the mask closed to the whte 42

5 rectangle n the test mage. The measure of separablt had a mamum when the mask was dentcal to the test mage. That meant the mask correctl dentfed the whte rectangle n the test mage as one regon and the background of the test mage as another regon. The separablt measure had a mamum value of 49.4 when the rectangle parameter values were 22, 2, 2 and 4. It was the optmal soluton. Separablt Measure Fg. 2 The dspla of separablt measure values calculated for each of the 25 masks In Analss II, the separablt measures assocated to the random masks were shown n Fg. 3. The separablt measure had a mamum value of 36.5 when the rectangle parameter values were 23, 5, 2 and 4. Ths was not the optmal soluton. In Analss III, the rectangle centre n these masks were randoml generated However, the parameters a and b were both set to 2 assumng the were unknown. Each of these sets of rectangle parameters was used as the startng values for the constroptm functon to solve for the separablt measure. Totall 7 solutons converged but onl 4 of them were optmal. Ther separablt measures were shown n Fg. 4. The separablt measure had the optmal value of 49.4 (the longest vertcal lne) from four sets of startng ponts. Ther respectve rectangle centres were at pels [44, ], [3, ], [27, ] and, [23, 8]. The optmal soluton has the parameter values 22, 2, 2 and 4. Twent-fve solutons ndcated the 5 teratons lmt for the Nelder-Mead method had been reached. The remanng fve solutons ndcated degenerac of the Nelder-Mead smple. In Analss IV, the rectangle centres of the masks used here were same as the ones used n Analses III. However, two rectangles were constructed n each mask. Each of these sets of rectangle parameters was used as the startng values for the constroptm functon to solve for the separablt measure. Totall 99 solutons converged but onl of them were optmal. Ther separablt measures were shown n Fg. 5. When the startng pont had rectangle centre at pel [23, 5], the separablt measure had the optmal value of (the longest vertcal lne n Fg. 5). Ths optmal value was dfferent from the one obtaned n Analss III because the masks were defned dfferentl. The optmal soluton had the parameter values 22, 2, 2 and 4. The remanng one soluton ndcated the 5 teratons lmt for the Nelder-Mead method had been reached. 43

6 Separablt Measure Fg. 3 The dspla of separablt measure calculated for each of random masks Separablt Measure Fg. 4 The dspla of separablt measures obtaned from 7 solutons that converged Separablt Measure Fg. 5 The dspla of separablt measures obtaned from 99 solutons that converged 44

7 Dscusson The results from Analss III and IV demonstrated that the proposed method could detect the rectangle n the test mage successfull. Also, the mask used to dentf two regons n the test mage could have man defntons. The Nelder-Mead method was used to solve the mamzaton problem n ths stud. Other optmzaton technque such as partcle swarm optmzaton could be consdered [7]. Ths research potentall opens up a new methodolog of object detecton and segmentaton usng the separablt measure n the framework of an optmzaton technque. The proposed method s generc. Other possble applcatons are to detect a mssng boat n the sea or to detect an arcraft n the ar. Object detecton and detaled segmentaton could be carred out at the same tme f the geometr of an object could be descrbed mathematcall. Further development of ths method s requred when t s used to detect object embedded n a background of low contrast. Acknowledgement Ths research s partall supported b the Hong Kong Poltechnc Unverst (Department of Appled Mathematcs) Internal Research rant. We thank Mss Cand am for her assstance n the computer programmng. References. Cho Y. H., W. T. Hung, R. Mtchell, X. Zhou (27). An Automatc Statstcal Method to Detect the Breast Border n a Mammogram, Int. Journal of Boautomaton, 6, onzalez R. C., R. E. Woods, S.. Eddns (29). Dgtal Image Processng usng MATAB, Second Edton, atesmark Publsherng. 3. Olver A., J. Freenet, J. Martí, E. Pérez, J. Pont, R. E. Erka, E. R. E. Denton, R. Zwggelaar (2). A Revew of Automatc Mass Detecton and Segmentaton n Mammographc Images, Medcal Image Analss, 4, Otsu N. (979). A Threshold Selecton Method from ra-evel Hstograms, IEEE Transactons on Sstems, Man, and Cbernetcs, SMC, 9(), R Development Core Team (29). R: A anguage and Envronment for Statstcal Computng, R Foundaton for Statstcal Computng, Venna, Austra, ISBN , UR 6. Nelder J. A., R. Mead (965). A Smple Algorthm for Functon Mnmzaton. Computer Journal, 7, Kenned J., R. C. Eberhart (995). Partcle Swarm Optmzaton, In: Proceedngs of IEEE Internatonal Conference on Neural Networks, IEEE Press, Pscatawa,

8 Assoc. Prof. Edward Y. H. Cho, Ph.D. E-mal: Dr. Edward Y. H. Cho s an Assstant Professor of the Department of Appled Mathematcs, Hong Kong Poltechnc Unverst. He has taught varous subjects n Mathematcs and Statstcs to dfferent levels of students, rangng from Assocated degree up to Master levels. He holds the degrees of M.Sc. and Ph.D. and s also a Chartered Statstcan of the Roal Statstcal Socet of Unted Kngdom. Hs research nterests are n Medcal Statstcs, Epdemolog, partcularl through the applcaton of Meta-analss. He s the prncpal nvestgator of several projects funded b the research grants of the Department. He has also provded varous knds of consultanc commssoned b the government or publc bodes. He enjos readng, hkng, theatre, watchng soccer matches, and hstor. Dr. Wa Tak Hung, Ph.D. E-mal: arthur_hung@hotmal.com Dr. Wa Tak Hung attaned hs Ph.D. n Statstcs from Macquare Unverst n 989. He also have been awarded (twce) wth Post Doctoral Fellowshps n appled mathematcs and statstcs, the most recent beng wth the CRC for Cardac Technolog jontl wth the Unverst of Technolog Sdne (UTS). He s a member of the Statstcal Socet of Australa and s an Assocate Fellow of Insttute of Mathematcs and ts Applcatons (Unted Kngdom). He s the Senor Analst at Cancer Insttute NSW. Hs research nterests are Statstcs n Medcne and Computer Aded Dagnoss or Detecton. 46