Image Processing for Bubble Detection in Microfluidics

Transcription

1 Image Processng for Bubble Detecton n Mcrofludcs Introducton Chen Fang Mechancal Engneerng Department Stanford Unverst Startng from recentl ears, mcrofludcs devces have been wdel used to buld the bomedcal devces and mcro fuel cells. In the research on mcrofludcs, hgh speed camera s wdel used to record the moton of lqud bubbles n the mcrochannel, based on whch the dnamcs of bubbles can be nterpreted. However, the mechancal vbraton of the sstem causes the channel to devate from ts orgnal poston durng the mage acquston process. Fgure (a) shows a tpcal mage of a slcon mcrochannel wth sdewall water njecton, n whch the njected water bubble and the skewness of the channel are evdent. Hence, t s necessar to perform the skewness and translaton correcton on the raw mages acqured b the hgh speed camera before the correct nformaton of the bubble trajector can be obtaned from the mages. Also, to extract the moton nformaton from a huge number of mages, we need to solate the front bubble object from the background mage, whch wll facltate the automatc calculaton of bubble veloct. he spatal devaton of the mage can be decomposed nto angular and translatonal components. In the current project, we wll frst develop an algorthm to effcentl calculate the angular devaton of the mage based on solvng a mnmzaton problem usng Least Square SVM. Also, the translaton correcton of mage can be accomplshed n a straghtforward wa. After the realgnment of the mage s fnshed, the movng bubble can be separated from the background mage b usng Gaussan mxture model and EM algorthm to classf each pxel n the mage as ether the foreground or background. he test result shows that the present skewness correcton algorthm works ver well for the tlt angle less than 0 degree. Also, the background separaton algorthm can successfull dstngush the movng object from the background mage wth stll droplets n t for varous llumnaton condtons. he major advantage of the current approach s that no teraton s requred and computaton s ver cheap. Skewness Correcton o correct for the angular devaton, we frst convert the orgnal gra-scale mage (Fg.(a)-4(a)) nto bnar mage(fg.(b)-4(b)). Suppose the mage tlt angle should be corrected b α, then the whte pxel at ( x, ) s moved to ( x ', ' ), where we have: x ' ' x ', where ' cosα snα snα cosα Snce the bggest feature n the bnar mages s a par of straght black lnes representng the channel border, to mplement the skewness correcton, we onl need to calculate the, such that the projecton of the black pxels onto the axs after the correcton assumes the mnmal value,.e.: ' ' mn ( ) or mn ( [ x, ] [ x, ] ) () Here, we construct a tranng data set{ X, Y } correspondng to all the black pxels n the mage, such that X R ndcates the x, coordnates of the pxel, and regress { X, Y }, then the regresson error s : Y R can be set as a constant. If we use f ( X ) X + b,( 0) to ()

2 ε f ( X ) Y X + b Y (3) It s nterestng to recognze that the skewness correcton objectve () can be expressed as: mn ( [ x, ] [ x, ] ) mn ( ε b Y ε b Y ) mn ( ε ) In dervng (4), we recognzed that Y R s set as a constant n our applcaton. B comparng (3) and (4), we fnd that to determne, t s equvalent to regress the tranng set { X, Y } usng the lnear equaton f ( X ) X + b,( 0). Here, we use the least square SVM to mplement the regresson. In partcular, we consder the mnmzaton problem: mn + ζ s. t. Y X b e 0 n e (5) B constructng the Lagrangan functon, takng dervatve wth respect to, b, e and Lagrangan multpler a, we have: (4) 0 Ω γ I b 0 α Y (6) wth Y [, K, ], [, K,], α [ α, K, α ], I s an dentt matrx, and the kernel matrx Ω R wth Ω x x. Snce the ntercept b does not nfluence the, (6) can be smplfed as: j j γ Ω a Ia (7) hen non-zero vector a correspond to the frst egenvector of Fnall, the correcton angular dsplacement a X Ω R. After fnshng the skewness correcton, the next step s movng the mage n X and Y drecton to fnall fnsh the mage algnment. In the present case, such algnment can be acheved based on the poston of the water njecton pont. We wll not talk about ths part n detal. Movng Bubble Detecton o estmate the bubble trajector n an automatc manner, we need movng bubble detecton algorthm to separate the movng bubble from the channel and other stll objects n the background for each frame n the vdeo sequence. he easest wa to detect movng object n an mage ncludes obtanng the background mage frst, and then subtract the background mage from the orgnal mage, leavng the foreground movng objects. However, ths technque s not applcable to the present stud, snce the background mage vares wth the fluctuaton of the llumnaton condton and the exstence of some other stll droplets durng the mage capture process, makng t ver dffcult to fnd a general background mage applcable to all frames n the vdeo. Here, we consder a sngle pxel and the dstrbuton of ts values over tme. At a specfc moment, a partcular pxel ma be ether n the background state or n the foreground

3 state. hus, the ntenst value x, of a pxel (x, ) can be treated as the weghted sum of two Gaussan dstrbutons: φ g + ( φ ) g (8) x, background, x, background, x, background, x, foreground, x, where : g g ( µ, ) background, x, background, x, background, x, ( µ, ) foreground, x, foreground, x, foreground, x, he model for pxel (x,) s parameterzed b the parameter { φ, µ, {, }} (9) x, Σ l foreground background Let be the pxel ntenst, L be a random varable ndcatng the label of the pxel n the mage, our model defnes the probablt dstrbuton: φ P( L l, I( x,, t) ) exp( ( µ ) Σ ( µ )) (0) ( π ) n Σ Knowng the parameter of the above probablt dstrbuton of each pxel, we can classf each pxel as ether background or foreground based on the hghest posteror probablt P( L l I( x,, t)). Here, our objectve s to fnd parameters arg max t P( L lt, I( x,, t) ), b takng dervatve, t can be found that: φ l, x, µ l, x, l, x, l, x, l, x, l, x, l, x, l, x, l, x, l, x, w here { L l} l, x, x,, t t M ({ L l} I ( x,, t)) Z l, x, x,, t t l, x, M Z µ µ t ({ L l} I ( x,, t ) I ( x,, t) ) x,, t Snce we do not have the labels L x,, t for the tranng data, the above equatons can not be calculated drectl. Instead, we calculate a sequence of parameter settngs, n whch each settng s found b usng the prevous one to classf the data. In partcular, suppose at the current step we have some dstrbuton parameterzed b,, values of dfferent labels accordng to as an estmate of ther true value. akng,, l x l x, we can use the expect as an example, we have: E[ ] P( L l I( x,, t), ) () t t o classf each new frame, ths approach requres calculatng the summaton over all the prevous frames, whch s qute expensve. As an alternatve wa, whenever we classf a new frame, we add ts contrbuton to the current

4 statstcs, whch means that we are ncreasng our tranng set at each step, wthout reprocessng the prevous frames n the tranng set. Hence, we have followng algorthm: Intalze for each pxel n the frst mage n the vdeo. For each new frame { For each pxel n the new frame {. Update the parameter for mxture model: } } : ( α) + α P( L l I( x,, t), ) t M : ( a) M + α P( L l I ( x,, t), ) I ( x,, t) t Z : ( a) Z + α P( L l I( x,, t), ) I ( x,, t) I( x,, t) () t f (, M, Z ). Classf the pxel as background or foreground, based on the current mxture model Snce the llumnaton condton ma change over the tme, each frame s contrbuton to the background mage need to be weghted accordng how far t s awa from the current frame. herefore, we ntroduce the relaxaton factor α n (). α les between 0 and ; he nfluence of the prevous mage on the current mage decas exponentall wth the dstance. In the present stud, we choose α 0.5. Result Fgure through Fgure 4 shows the mage processng result for four frames usng the present algorthm for skew correcton and movng bubble extracton. It s shown that the skewness correcton method works ver well for the mages n varous angles. In partcular, the algorthm s not susceptble to the nose (scattered black pxels) n the bnar mage converted from the source mage. Also, the movng bubble s successfull separated from the background channel. In partcular, the algorthm correctl classfes the stll bubble n the channel as background and excludes t from the resulted movng bubble mage. B recognzng that such stll object ma exst at an place n the channel and ma not be correctl removed b drectl subtractng a predetermned background mage from the source mage, the advantage of our method s evdent. Summar In the present stud, we correct a sequence of skew mages b solvng a mnmzaton problem based on LS SVM. he movng bubble object s extracted b classfng each pxel n the mage usng Gaussan mxture model. he result s qute satsfactor for the present applcaton. he major advantage of the current methods s that no teraton s nvolved n both steps. In partcular, the correcton angle can be drectl obtaned b calculatng the egenvector of the resulted matrx, and the classfcaton model for the pxels n the current frame can be updated b addng the contrbuton of the present mage to the parameters of the prevous mage. herefore, the computaton s ver cheap and an mplementaton n real tme s possble.

5 x Water njecton pont Movng bubble Stll bubble Fg. mage processng result for frame ((a) orgnal mage, (b) bnar mage, (c) corrected mage, (d) extracted movng bubble, note our algorthm can successfull remove the stll bubble from the extracted mage) Fg. mage processng result for frame Fg.3 mage processng result for frame 3