A simplified approach to merging partial plane images

Transcription

1 A siplified approach to erging partial plane iages Mária Kruláková 1 This paper introduces a ethod of iage recognition based on the gradual generating and analysis of data structure consisting of the 2D space using the difference fro the row and colun vectors sus as an application of the linear algebra The result of the data processing is a graphical interpretation of the easure of the siilarity of the generated results of overlapping of 2 iages Maxial easure of the siilarity is a easure for iage registration The study result is to create a list of the iages order, in which one follows the other, included in the non-registered set of iages that can be used for the final iage-stitching Keywords: iage-stitchining, iage recognition, easure of the siilarity Introduction The atheatical odel used for the study describes the 2D space and the odel application can be classified as iage recognition ethod with consequent registration Oldridge et al (2009), Szeliski and Coughlan (1997) In order to accoplish the final iage-stitching, we need to create a atheatical odel first Ch-YChen and Klette (1999) The atheatical expressions consist of the pixel values of the iage in the coordination syste Then the expressions of two iages are processed for overlapping Matheatical odels applied for the data frae were classified into patch-based and feature-based algoriths (Lowe (1999), Pazzi et al (2010), Chen and Willias (1993)) Applying the we can define the iage as graphical interpretation using siilarity of the patches and features In the linear algebra there is a part devoted to the atrix theory including the algoriths developed in order to find out the rate of siilarity based on the linear equations The process of the algoriths developent is based on the assuption that the values generated fro the iage data fraes into the row and colun vectors of sus can be applied using the in the foration of the square atrix The ain ai is to deterine the siilarities based on the estiation out of the differences of the atrixes In this study the algoriths based on the logical values coing fro the data frae of red coponent applied in the square binary atrices of the iage were used for the first stage of the pre-processing of the iages The RGB iage is stored as xnx3 data array that deterines red, green and blue coponent layer for each pixel separately The iage preprocessing is based on the transforation of the colour structure of the iage into the binary data atrix that consists of the red coponent layer only Binary iages, coonly called bi-level, have each pixel stored as a single bit (0 or 1) Pixel with 0 value is displayed in black that described the occurrence of the red coponent layer of the iage These values of the data frae of the red coponent layer of the iage are gradually sued up into the row and colun vectors of sus based on the rows and colun structures fro each data array of iage i Then the coponents of the vectors are generated and organized into the data atrix Rred xi of the row vector of sus and Cred ixn of the colun vector of the sus, where i is defined as nuber of processed of the data arrays of the iages The first absolute difference V row is a result of the 2 data arrays with arranged row vectors of sus into the 2 square atrixes V i Rred and V i+1 Rred The second absolute difference R v is a diffrence of the first absolute difference of the data atrix V row and the transponned atrix ade of the V row called Vrow T This will define the level of the siilarity of the coponents arranged in rows of the vector after the developent of the row vector of the sus Rrow j fro the second absolute difference of the square atrix R v The sae process is executed for generating of the colun vectors of the sus They will deterine the level of siilarity of the coponents arranged in the vector but the colun arrangeent of the vector s coponents after the row vector of sus Crow j fro the second absolute difference of the square atrix C u is created The arrangeent of the coponents of the row vectors of sus Rrow j a Crow j into square atrices and consequent calculation of the differences of the data arrays of the vectors lead to creating data array of the atrix F di f f fro which we can create definitive row vector of sus atch j This row vector of sus atchj consists of zero values in its coponents Based on these values the siilarity in two copared data arrays of the iages was deterined 1 Ing Mária Kruláková, Technical Univerzity of Košice, FBERG, Institute of Control and Inforatization of Production Processes, B Něcovej 3, Košice, ariakrulakova@tukesk 234

2 Definitions and notation Red_I is a finite set of the Z 2 It is a atrix with rows and n coluns with the pixel values of 0 a 1 of binary iage as its eleents For Z there is row deterined as {Red_I(x,y) 2 : y = }, where is an index of particular row For n Z there is a colun n deterined as {Red_I(x,y) 2 : x = n}, where n is an index of particular colun Su of the row,rred is a nuber of the eleents of the atrix array Red_I generated fro the nuber of the eleents of rows that sus the values of the atrix coluns together into the row vector of sus Rred = n Z Red_I (x) Su of the colun, Cred n is a nuber of the eleents of the atrix array Red_I generated fro the nuber of the eleents of n coluns that sus the values of the atrix rows together into the colun vector of sus Cred n = Z Red_I (nxn) Then the Rred is the row vector of sus ( 1) and Cred n is the colun vector of the sus ( 2) generated fro the atrix Rred_I Rred = (rred 1,rred 2,rred 3,rred 4,,rred ) (1) Cred n = (cred 1,cred 2,cred 3,cred 4,,cred n ) (2) The result of the scipt createcoponentr is a atrix Rred xi The size of this atrix is [xi] The script createcoponentr has created the row vector of the sus Rred ( 1) as a su of the binary values of all 4 iages processed fro the data arrays of the atrices of each iage For exaple first eleent of resulting Rred x i is a su of eleents in the first colun of the Red_I Rred xi = rred 1,i i+1 rred 2,i i+1 rred 3,i i+1 rred,i i+1 rred 1,i i+2 rred 2,i i+2 rred 3,i i+2 rred,i i+2 rred 1,i i+3 rred 2,i i+3 rred 3,i i+3 rred,i i+3 rred 1,i i+i rred 2,i i+i rred 3,i i+i rred,i i+i The result of the scipt createcoponentc is a atrix Cred ixn The size of this atrix is [ixn] The script createcoponentc has created the colun vector of the sus Cred n ( 2) as a su of the binary values of all 4 iages processed fro the data arrays of the atrices of each iage Cred ixn = cred i i+1,1 cred i i+2,1 cred i i+3,1 cred i i+i,1 cred i i+1,2 cred i i+2,2 cred i i+3,2 cred i i+i,2 cred i i+1,3 cred i i+2,3 cred i i+3,3 cred i i+i,3 cred i i+1,n cred i i+2,n cred i i+3,n cred i i+i,n In case the input iage Red_I has the size [xn], and n >, then k = n and notation for for the sae output size of the Rred and Cred n k atrixes, if [xn k] Processing of the row vector of sus is created by script naed as organizerow This script processes the row vector of su in order to create a new square atrix as data array diplayed on ( 3) shows, where i = 1 is the first iage of out 4 Square atrix as data array diplayed on ( 4) represents the second iage of out 4, where i + 1 = 2 V i Rred = rred 1,i rred 1,i rred 1,i rred 1,i rred 2,i rred 2,i rred 2,i rred 2,i rred 3,i rred 3,i rred 3,i rred 3,i rred,i rred,i rred,i rred,i (3) 235

3 Mária Kruláková: A siplified approach to erging partial plane iages V i+1 Rred = rred 1,i+1 rred 1,i+1 rred 1,i+1 rred 1,i+1 rred 2,i+1 rred 2,i+1 rred 2,i+1 rred 2,i+1 rred 3,i+1 rred 3,i+1 rred 3,i+1 rred 3,i+1 rred,i+1 rred,i+1 rred,i+1 rred,i+1 (4) Fig 1 represents an algorith that describes the process of generating row and coluns vector sus into the atrixes Rred xi a Cred ixn Paraeter i deterines the nuber of the preprocessed iages Fig 1 Generation of the output atrixes Cred ix and Rred xi As the square atrixes V i Rred and V i+1 Rred are created out of the row vector of sus of the atrix Rred xi using ( 3), in the sae way the square atrixes V i Cred and V i+1 Cred are created out of the colun vector of sus of the Cred ixn atrix using ( 4) The arrangeent of the colun vector of sus is generated by script organizecol The process of how to find the differences between the atrixes includes following steps: 236

4 The first absolute difference V row is ade fro the 2 data arrays that are generated by the row vectors of the sus with result of the 2 square atrixes V i Rred a V i+1 Rred is ( 5): V row = V i Rred V i+1 Rred (5) The second absolute difference R v between the atrix V row and V T row is ( 6): R v = V row V T row (6) The row vector of sus Rrow j is generated fro the atrix R v ( 7): Rrow j = The colun vector of sus Rcol j is generated fro the atrix R v ( 8): R v(x) (7) Rcol j = R v(x) (8) Coparing the values of teh row vector of sus ( 7) and transposed colun vector of sus Rcol T j ( 8) it was found out that the vectors consists of the sae values of the copared eleents of both vectors ( 9): Rrow j = Rcol T j (9) The first absolute difference V col is ade fro the 2 data arrays that are generated by the row vectors of the sus with result of the 2 square atrixes V i Cred a V i+1 Cred is ( 10): V col = V i Cred V i+1 Cred (10) The second absolute difference C u between the atrix V col and Vcol T is ( 11): C u = V col V T col (11) The row vector of sus Crow j is generated fro the atrix C u ( 12): Crow j = C u(x) (12) The colun vector of sus Ccol j is generated fro the atrix C u ( 13): Ccol j = C u(x) (13) Coparing the values of teh row vector of sus ( 12) and transposed colun vector of sus Ccol T j ( 13) it was found out that the vectors consists of the sae values of the copared eleents of both vectors ( 14): Crow j = Ccol T j (14) 237

5 Mária Kruláková: A siplified approach to erging partial plane iages The final absolute difference F di f f (15) is a result of the rearrangeent of the values fro the row vector of sus Rrow j ( 7) generated fro the atrix R v into atrix Row and row vector of sus Crow j ( 12), that is generated fro atrix C u into atrix Colun ( 15): F di f f = (Row Colun (15) The row vector of the sus atch j was generated fro the atrix F di f f ( 16): atch j = (F di f f ) x (16) The easure of siilarity was deterined according to the ost frequent occurrence of the zero values within the vector eleents (see Fig6) All used scripts that are used in the study (createcoponentr, createcoponentc, organizerow, organizecol, di f f erencesu) can be found in the apendix at the end of the paper Graphical coparison of the vector values Fig 2 shows the list of iages that was preprocessed and analysed The content of the list includes 4 preprocessed binary atrixes corresponding with the 4 bianry iages (A, B, C, D) copared in pairs Fig 3 and Fig 4 show the Fig 2 Generated binary iages utual coparison of the vector values between overlapped iages A&B and A&C, Fig 3 and B&C and C&D Fig 4 Fig 3 Coparison of the iages A&B and A&C The calculation of F di f f ( 15) results in the row vector of su atch j ( 16) This vector includes also zero values that occur when there is the sae value in the row and in the colun The frequency of zero values in the row vector of su atch j is final result of the overlapping of all copared iages Fig 5 It provides the inforation about the easure of siilarity (higher the nuber of zero values out of pairwise coparison, greater the siilarity) Tab 1 includes all possible cobination of the utual coparison of overlapping the iages (based on pairwise coparison) For each coparison (A&B, A&C, A&D, B&C, B&D, C&D) there is nuber of zero values found to easure the siilarity 238

6 Fig 4 Coparison of the iages B&C and C&D Fig 5 Result of pairwise coparison (siilarity of iages) Tab 1 Iage registration Cobination of the pairwise coparison Nuber of the zero values The sequence of iages A&B 8 A B A&C 1 A&D 2 B&C 36 B C B&D 12 C&D 18 C D Conclusion The ain goal of the study was to use data processing in graphical interpretation of the easure of the siilarity by coparing and assessing how 2 iages overlap In order to fulfill the target we have created algoriths of linear algebra The result of the iage data processing can be seen in Tab 1 It infors about the easure of the siilarity and a list of the iages order It presents iage registration as an outcoe of the pairwise coparison of 4 iages They were indexed as following: A B C D This type of indexing that is iportant for iage registration It can be applied in case of ore extended list of iages with siilar data frae for aking the final iage-stitching faster The study help us to find algoriths for iage processing and the way of creating the sequence of the iages, in which one follows the other, that can be practically used for non-registered set of iages and for aking the final iage-stitching faster 239

7 Mária Kruláková: A siplified approach to erging partial plane iages Acknowledgeent This work was supported by the Slovak Research and Developent Agency with contract No APVV References Ch-YChen, Klette, R, 1999 Iage stitching-coparisons and new techniques, in: Coputer Analysis of Iages and Patterns Springer Berlin / Heidelberg volue 1689 of Lecture Notes in Coputer Science, pp Chen, S, Willias, L, 1993 View interpolation for iage synthesis, in: Coputer Graphics (SIGGRAPH 93), pp Lowe, DG, 1999 Object recognition fro local scale-invariant features, in: International Conference on Coputer Vision, Corfu, Greece pp Oldridge, S, Miller, G, Fels, S, 2009 Autoatic classification of iage registration probles, in: LNCS 5815 Springer-Verlag Berlin Heidelberg, pp Pazzi, RW, Broukerche, A, Feng, J, Huang, Y, 2010 A novel iage osaicking technique for enlarging the field of view of iages transitted over wireless iage sensor networks Mobile Networks and Applications 15, Szeliski, R, Coughlan, J, 1997 Spline-based iage registration International Journal of Coputer Vision 22,