CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE

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1 CHAPTER III IMAGE DEWARPING (CALIBRATION) PROCEDURE 3.1 Scheimpflug Configurtion nd Perspective Distortion Scheimpflug criterion were found out to be the best lyout configurtion for Stereoscopic PIV, becuse of its bility to ensure tht ll prticles, which re illuminted by the lser sheet in the object field, will be in good focus in the imge plne. In spite of the good estblishment of Scheimpflug rrngement to overcome some limittion tht rose in lte development er, it lso estblished some side effect tht could reduce the result of stereoscopic PIV processes. The most obvious one, which cn be seen in erly step of the processes, is the imge perspective distortion. Figure III.1 The Scheimpflug rrngement for Stereoscopic PIV As in clibrtion process, n imge with blck line rulings t 5 mm squre spcing is put inside n empty still-wter tnk. Becuse of the Scheimpflug Criterion, the grbbed imges from stereo cmers will shrink on one side producing trpezoidl form from the rel ones. This perspective distortion occurred becuse by using the Scheimpflug rrngement, the imge plne of the 6

2 cmer is not prllel to the object plne neither the lens principl plne. In this wy, the cmer nd the lens re seprtely rrnged with n ngle, so thtt it is possible to rotte the imge sensor into the plne of best focus without moving the cmer itself (only the cmer s lens). As result, the mgnifiction fctor of the grbbed imges is no longer constnt cross the field of view nd it produces perspective distortions (the trpezoidl form). Figure III. The 5mm blck-line spcing clibrtion grid Now tht the shrunk imges from stereo cmers stored s -D mtrices (.JPG-formtted pictures) in computer s hrd disk, the first thing tht must be hndled is the correction for the perspectivee distortions. Doing this improves the ccurcy for the next importnt step of PIV, the Imge Correltion. Since the principle property of the Imge Correltion is to find similrities between two imges nd the displcement shift between them, nerest-rel-perfect- symmetricl imge perspective is required (Ari, 006). With trpezoidl-shrunk imge, the complete set of velocity components (the three displcement component, i.e. Δx, Δy, nd Δz) tht hve been grbbed by the imge plne could not be extrcted perfectly to generte n (lmost) 3-D velocity field nor vorticity 7

3 clcultion s min objective of Stereoscopic PIV system. Therefore, clibrtion procedure to mesure nd correct the perspective distortion due to cmer-lensobject plne tilt nd other imge distortions in the opticl rrngement is necessry, in order to gurntee the next correltion step would run smoothly with less error. 3. The Dewrping Process As mentioned bove, the perspective distortions which mkes the grbbed imges shrunk in one side, re cused by the mgnifiction fctor tht re vrying cross the field of cmer s view. These cn implicte the ccurcy of the velocity mesurements through the cross correltion process. In order to reconstruct nd correct these distortions, the grbbed imges from those stereo cmers must be dewrped(ing) such tht the vrying mgnifiction fctors remin constnt over the entire imge. Figure III.3 Trpezoidl imge grbbed by stereo cmer using Scheimpflug rrngement The clibrtion method used in this thesis is written s n M-file in mtrices processing softwre, MATLAB R007B (v 7.5.0). Due to the perspective-viewing ngle of the cmers, the rel clibrtion grid will form trpezoidl insted of 8

4 rectngle. By finding the mpping coefficients using the liner nd nonliner lest squre methods (Willert, 1997), the trpezoidl imge cn be restored bck into the rectngle form nd the mgnifiction fctor cn be mde constnt. First, the M-file will identify the intersections of the grid lines in the clibrtion grid imge. By using the imge correltion process (-D mtrices convolution) provided in M-file syntx librry, the intersection of the clibrtion grid will be shown fter short time itertion. From there, the four reference points, which specify where the itertions for finding the new dewrped coordintes begin, cn be picked up. After the new coordintes re obtined, the next itertions to find the nerest possible displcement between the coordintes of the old trpezoidl imge nd the new guessed imge coordintes, cn be run by substituting the old nd the new coordintes into the lest squre methods. The result of the itertions will be the twenty four mpping coefficients tht will be used to dewrp the imge. All the process from finding the intersections, choosing the reference points, guessing the dewrp imge coordintes, until developing the dewrped imge will be briefly explined in the next sub-chpters Convolutionl Filtering The most importnt thing to be remembered before strting the dewrping process is tht the imges loded into the M-file must be n unsigned 8-bit integer imges(signed by uint8 in MATLAB ), i.e. the gryscle imges. If the grbbed imges re in ny other formt, the imges must be converted somewht using the imge processing softwre, such s Adobe Photoshop or other similr progrm. The Pulnix TM-1040 cmers grb nd sve those gryscle imges s 1008x104 mtrices (using the developed stereo cmers system) in computer memory. And for the first step, the intersection of the grid lines must be found so tht set of reference points cn be picked up to be used s the reference points to dewrp the shrunk imges. 9

5 Figure III.4 The comprison imges (blck nd white cross likes imge) This erly step is done thnkfully by the dvntge of digitl imge processing technology known s convolutionl filtering. The bsic principle of the convolutionl filtering is similr to the imge cross correltion techniques, but its gol is to filter out the intensity informtion in digitl imge, t which the luminous intensity vrying cross the imge, tht my be regrded s less relevnt to the intensity of the comprison imge. Therefore, the M-file identifies the intersections by serching for the highest locl mxim (of the intensity informtion) mongst the others. Figure III.5 The generl process of convolutionl filtering In spite of the imges tht been used in dewrping process re gryscle, hence the intensity (blck white) of the imges re simply recognized nd compred 30

6 even by the humn eyes. So, by dding the comprison imge, in this cse, the blck nd white (+) cross likes convolutionl msk with smller mtrix size (depend on the size of rw imges). The filtering process cn be done by using the complex -D mtrices convolution lgorithm provided in MATLAB syntx librry. The process consumes insignificnt mounts of time, while the result cn be seen s the red nodes in every intersection in the rw (shrunk) imges. It mens tht, the dewrping lgorithm hs succeeded identifying ech intersection in the imge. Also from the result, the perspective distortion for ech unprllel grid lines cn be seen obviously. Figure III.6 The result: intersection nodes of the grid imge 3.. Gussin Interpoltion To increse the ccurcy of the convolution filtering result, it is necessry to locte the highest locl mxim more ccurtely within sub-pixel ccurcy in horizontl nd verticl directions. Therefore, n interpoltion between the locl mxim vlues must be dded to the lgorithm. This sub-pixel ccurcy process conducted using three points of interpoltion: the locl pek mxim point nd the 31

7 other two locl mxim points next to the locl pek mxim point. According to Ari (006), the best sub-pixel interpoltion method is the Gussin interpoltion. Hence, this convolution filtering process uses the Gussin interpoltion s its subpixel interpoltion. The Gussin interpoltion cn be written s: m n G G ' = m + ln R i ' = n + ln R i ln R( i 1, j) ln R( i + 1, j) ( + 1, j) ln R( i, j) + ln R( i + 1, j) ln R( i, j 1) ln R( i, j + 1) (, j + 1) ln R( i, j) + ln R( i, j 1) (3.1) where m G ' nd n G ' is the sub-pixel displcement given by Gussin interpoltion Guessing the coordintes of new dewrped imges Since the intersections from the rw imge hve been identified ccurtely within the sub-pixel ccurcy, the x nd y-coordinte of ech intersection nodes were sved s (n x ) mtrix in MATLAB virtul memory. Therefore, four reference nodes cn be chosen to form reference rectngle, which will be ct s reference grid for determining the new guessed mpping coordintes. considered: As when picking the four reference nodes, there is strict rule tht hs to be 1. Strt by choosing the best symmetricl grid from tht similr list of grids in the imge;. After tht, pick the upper left node of the chosen grid s the first origin node, where the itertion process of finding the new dewrped imge coordintes begun; 3. Followed by picking the upper right node; 4. Then going down to the bottom right node; 5. And ended by picking the lst node of the grid, the bottom left. This simple rule must be well thought-out to void the errors, which cn be occurred long the process. 3

8 Figure III.7 The rule for picking the reference grid (four reference nodes) Principlly, the lgorithm of this guessing process is simply by replcing ll x or y-coordintes of the intersection nodes, which re ligned with the ctive reference node (the first ctive reference node is the upper left node of the chosen grid, which hs been picked up for the first time), with the x or y-coordintes of the ctive reference node. Therefore, the replcement of the coordintes depends on the ctive xis of the itertion. As exmple, the X-xis is the ctive xis nd the ctive reference node is the origin node (the upper left node of the chosen grid, which is locted t the center of the imge). Hence, the x-coordintes of ll intersection nodes which re ligned verticlly (bove nd below the origin node) or in other word, hve the minimum X-distnces ( 0) from the x-coordinte of the origin node, re replced by the x-coordintes of the ctive reference node (the origin node). After tht, the ctive reference node moved to the next node on the right side of the origin node nd the sme itertion repplied until reched the lst node of the right hnd side of the imge. The sme guessing itertions re lso 33

9 clculted for the left hnd side of the imge (by tking ll verticlly ligned nodes t the center of the imge s the reference strting line of itertions). When ll itertions in X-xis re done, ll intersection nodes in the imge will be ligned verticlly in ech verticl column. However, the lignments in Y- direction still need to be considered. Therefore, the sme itertion method is reused in Y-xis. As finl result, ll nodes re verticlly nd horizontlly dewrped in respect with the grid spcing distnces (in x nd y-direction) of the reference grid. Moreover, the guessed coordintes results re plotted s the cyn cross (x) in M-file figure. Figure III.8 The finl result of the guessing process 3..4 The twenty four mpping coefficients From the Introduction, we hve known tht the grbbed imges shrunk on one side of the imge. In order to dewrp the imge, the mpping coefficients between the new-dewrped imges (x, y) nd the old-trpezoidl imge (X, Y) hs 34

10 to be determined. The determintion of these coefficients in solver equtions by mens of liner lest squre method is not s simple s for those second orders wrping pproch (Eq..13), becuse the equtions no longer constitute liner polynomils but rther re rtios of two polynomils of the sme order. Using one of the nonliner squre methods such s Levenberg-Mrqurt method, the twentyfour mpping coefficients results re relible to find the best mtch to solve this geometric perspective problem. After the new guessed coordintes obtined in subchpter 3..3 bove, the X nd Y-coordintes of the old trpezoidl imge nd the new x nd y-coordintes of the guessed dewrped imge, re used together in the equtions of lest squre method. First nd foremost, the liner lest squre method is implemented by solving for the six unknowns in the first order projection equtions of Eq..15 (rewritten below): x = y = X X X 1 X 3 3 Y Y Y Y By setting 31 nd 3 equl to zero nd 33 equls to 1, the perspective trnsformtion eqution reduces to the more frequently used eqution. In ddition, by solving two equtions bove, the six coefficients ( ) could be used s initil estimtes in the Levenberg-Mrqurt method for the solution of the higher order equtions (referred to Eq..16): x = y = X X X X Y Y Y Y X X X X Y Y Y Y XY XY XY XY The result of these itertions will be the twenty-four mpping coefficients tht hve been mentioned in the previous explntion. And by mens of those twenty-four coefficients, the trpezoidl imge cn be restored into the good-symmetricl clibrtion grid imge. 35