A new look on the “Valid Detection Probability” of PIV Vectors

A new look on the “Valid Detection Probability” of PIV Vectors

Scharnowski, Sven; Sciacchitano, Andrea; Kähler, Christian J.

Publication date 2018

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Proceedings of the 19th international symposium on application of laser and imaging techniques to fluid mechanics

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Scharnowski, S., Sciacchitano, A., & Kähler, C. J. (2018). A new look on the “Valid Detection Probability” of PIV Vectors. In Proceedings of the 19th international symposium on application of laser and imaging techniques to fluid mechanics

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A new look on the “Valid D etection Probability” of PIV Vectors

Sven Scharnow ski1*, And rea Sciacchitano² , Christian J. Kähler1

1: Institu te of Flu id Mech anics and Aerod ynam ics,Bu nd esw ehr University Mu nich , 85577 N eu biberg, Germ any 2: Delft University of Technology, 2629H S Delft, The N etherland s

* Corresp ond ent au thor: sven.sch arnow ski@u nibw .d e

Keyw ords: PIV, cross-correlation fu nction, effective num ber of p article im ages

ABSTRACT

For the reliable estim ation of velocity vector field s by m eans of 2D and 3D p article im age velocim etry (PIV), the cross-correlation fu nctions calcu lated from the signal w ithin each interrogation w ind ow m u st featu re a d istinct p eak that rep resents the average shift of the p article im age ensem ble. A high valid d etection p robability (V DP) of the correct correlation p eak is necessary in ord er to com pu te valid and accu rate velocity field s. The follow ing analysis show s the sensitivity of VDPon flow param eters as w ell as on evalu ation param eters. The m ost im p ortant resu lt is that the so-called effective nu m ber of p article im ages N F F_{I I O} is not su ited to p red ict the VDPin the case of m od erate or strong ou t-of-p lane m otion. This can be exp lained by the fact that the VDPd ep end s not only on the nu m ber of particle im ages correctly p aired , bu t also on the nu m ber of p article im ages rem ain ing w ithou t p artner, w hich yield sp u riou s correlation p eaks. The find ings help to better u nd erstand the occu rrence of false vectors and enable the PIV u ser to im prove the m easu rem ent setu p as w ell as the PIV evalu ation in ord er to m inim ize sp u riou s vectors.

1. Introduction

In p article im age velocim etry (PIV) the velocity of a grou p of p articles is estim ated from the cross-correlation fu nction of tw o interrogation w ind ow s containing the corresp ond ing p article im ages at tw o consecu tive tim e instants. Und er id eal cond itions, the su b-p ixel location of the highest correlation valu e corresp ond s to the d isp lacem ent of the p article im age ensem ble w ithin the interrogation w ind ow . H ow ever, this only hold s for the case w here the flow w ithin the interrogation w ind ow is u niform as w ell as hom ogenou sly seed ed and a su fficient large nu m ber of p article im ages from tw o corresp ond ing im ages can be p aired . Keane and Ad rian (1992) show ed that the form ation of a w ell d etectable correlation p eak is very likely if the nu m ber of p article im ages w ithin the interrogation w ind ow is N_I t5. For N_I 5, the likelyhood that a rand om p eak in the correlation fu nction, that d oes not corresp ond to the correct p article

im age d isp lacem ent, is higher than the one corresp ond ing to the d isp lacem ent, increases strongly.

For realistic PIV im ages, the loss-of-correlation d u e to in-p lane m otion F_I, ou t-of-p lane m otion F_O, d isp lacem ent grad ients F_' or im age noise F_V lead to a d ecreased p robability for the d etection of valid d isp lacem ent vectors (Ad rian and Westerw eel, 2010, Raffel et al, 2018, Westerw eel, 2008, Scharnow ski and Kähler, 2016a/ 2016b). Keane and Ad rian conclu d ed that the loss of correlation can be com p ensated by increasing the nu m ber of p article im ages N_I su ch that:

I I O 5

N F F t (1)

The p rod u ct N F F_{I I O} is the so-called effective nu m ber of p article im ages accord ing to Keane and Ad rian (1992).

Fig. 1 Valid vector d etection p robability as a fu nction of the effective nu m ber of p article im ages

I I O

N F F for synthetic PIV im ages w ith zero in -p lane m otion (F_I 1) and varying ou t-of-p lane

m otion for d ifferent p article im age d ensities N_ppp and interrogation w ind ow sizes D_I.

N ew investigations show that relation (1) is not su fficient if the nu m ber of u np aired p article im ages is of the sam e ord er as the nu m ber of p aired ones or even higher. This is how ever relevant for m any realistic exp erim ents w here the tim e betw een the tw o illu m inations

is op tim ized for low uncertainty and high sp atial resolu tion. To achieve a low u ncertainty and high sp atial resolu tion at the sam e tim e, the p article im age d isp lacem ent m u st be large com p ared to the rand om error of the evaluation m ethod . Ad d itionally, the light sheet m u st be thin and the interrogation w ind ow s m u st be sm all to achieve good ou t-of-p lane and in-p lane resolu tion , resp ectively. H ow ever, a large d isp lacem ent in a thin light sheet also increases the p robability of loss-of-p airs d u e to in-p lane and ou t-of-p lane m otion and thu s d ecreases F_I and

O

F . Figu re 1 illu strates how the valid d etection p robability changes w ith resp ect to the effective nu m ber of p article im ages over a w id e range of p article im age d ensities and interrogation w ind ow sizes. The valid d etection p robability is the p robability w it h that the highest correaltion p eak corresp ond s to the tru e m ean d isp lacem ent of the p article im age ensem ble. It w as evalu ated from O(1000) correlation fu nctions com p u ted from synthetic PIV im ages for each d ata p oint in the figu re.

It can be conclu d ed from Fig. 1 that the prod u ct N F F_{I I O} is not u sefu l in general to d eterm ine the effective nu m ber of p article im ages. Esp ecially if N_I is relatively large and F_O is relatively sm all the u np aired p article im ages affect the correlation fu nction significantly. For exam p le N_I 10 and F_O (red squares in Fig. 1) does not result in the same valid detection 1 p robability as N_I 100 and F_O 0.1 (black squ ares in Fig. 1). Althou gh the p rod u ct N F F_{I I O} is constant, the valid d etection p robability d ecreases from 100% to abou t 3% . Thu s, the ru le of thu m b that p red icts a valid d etection p robability of !95% for N F F_{I I O}t , does not apply for a 5 broad range of p aram eters.

In ord er to u nd erstand w hy the so-called effective nu m ber of p article im ages is not su fficient to pred ict the valid d etection probability, correlation fu nctions of synthetic PIV im ages w ere analyzed system atically. The follow ing section d iscu sses the heigh t d istribu tion of the d isp lacem ent p eak as w ell as the second ary correlation p eak and their relation to the valid d etection p robability. In Sec. 3 the effect of several im age p aram eters on the valid d etection p robability is d iscu ssed in d etail. Section 4 analyzes the p ossibility to optim ize the interrogation w ind ow size d ep end ing on the flow and im age p aram eters and conclu sions are d raw n in Sec. 5.

2. Correlation Peak Height D istribution & Valid D etection Probability

Figu re 2 show s an exam p le of a synthetic PIV im age p air w ith a relative ou t-of-p lane shift

of ' ' z/ z₀ 0.3 corresp ond ing to F_O 0.7. The in-p lane-shift w as set to zero, sim u lating the

final p ass of a m u lti-p ass evalu ation. On average 12.8 p article im ages are fou nd in each interrogation w ind ow size of D_I pixel for a particle image density of 16 N_ppp 0.05. H ow ever, as can be seen from the figu re, the tru e nu m ber of p article im ages w ithin each w ind ow v aries

significantly. For rand om ly chosen locations, the probability of find ing exactly NI p article im age centers w ithin a DI2 w ind ow is given by the binom inal d istribu tion:





I I I 2 I 2 I I ppp ppp ppp I ( , , ) N N 1 D N pdf N D N N N D  § ·_{˜ ˜ } ¨ ¸ © ¹ , (2)

w here D is exp ressed in p ixel. The d istribu tion in Fig. 3 clearly show s a strong variation of _I N_I

in agreem ent w ith the rand om ly d istribu ted p article im ages in Fig. 2. The p robability of find ing 12 p article im ages w ithin a w ind ow is abou t

11.4%

bu t extrem a valu es su ch as 6 and 20 p article im ages have still a significant p robability of abou t

1.5%

.

Fig. 3 Probability d ensity

d istribu tion of the nu m ber of p article im ages w ithin a 16 16u p ixel w ind ow for an average d ensity of N_ppp 0.05.

Fig. 2 Exam p le PIV d ou ble im age w ith a p article im age d iam eter of

D

3

p ixel, a p article im age

d ensity of N_ppp 0.05 and an ou t-of-p lane m otion of ' ' z/ z₀ 0.3 (left and m id d le). Right: corresp ond ing norm alized correlation fu nctions com p u ted from an interrogation w ind ow size of D_I pixel .The true displacement peak and the secondary peak are marked by circles and 16 squ ares, resp ectively. Green and red color of the circles and squ ares ind icates the highest and second highest p eak, resp ectively.

On the right sid e of Fig. 2 the norm alized cross-correlation function of the tw o PIV im ages is show n qu alitatively. The tru e d isp lacem ent p eak and the second ary p eak are m arked by circles and squ ares, resp ectively. Green color of the circle and squ ares ind icates the highest p eak w hile the second highest p eak is colored in red . It can be seen from the figu re that the second ary p eak is som etim es higher than the d isp lacem ent p eak (refer to second and third row from top in the m ost right colu m n). This is m ainly cau sed by the loss -of-correlation d u e to ou t-of-p lane m otion. In the case of valid m easu rem ents, the d isp lacem ent p eak is higher than the second ary p eak, and a valid vector is d etected . Instead , w hen the second ary p eak is larger than the d isp lacem ent p eak, an erroneou s d isp lacem ent vector (ou tlier or sp u riou s vector) is com p u ted .

Figu re 4 illu strates the d istribu tion of the height of the d isp lacem ent p eak and the second ary correlation p eak for the case w ith significant ou t -of-p lane shift ' ' z/ z₀ 0.3. The correlation p eak heights w ere analyzed by m eans of synthetic PIV im ages w ith zero in -p lane m otion as show n in Fig. 2. The p article im age d iam eter, the particle im age d ensity and the interrogation w ind ow size w ere set to D 3p ixel, N_ppp 0.05 and D_I pixel, respectively. 16 Figu re 4 show s that for the m ajority of correlation fu nctions the d isp lacem ent p eak is higher than the second ary p eak. H ow ever, for som e cases the d isp lacem ent p eak becom es sm aller than the second ary one, lead ing to an erroneou s d isp lacem ent vector. This is in agreem ent w ith the correlation fu nctions p resented in Fig. 2.

Fig. 4 Probability d ensity d istribu tion of the norm alized correlation height for the d isp lacem ent

The actu al valu e of the valid d etection p robability VDP d ep end s on the p robability d ensity fu nction of the d isp lacem ent p eak pdf₁ and of the second ary p eak pdf₂ as follow s:

 

 

2 2 2 · 1 1 d 1d 2 c VDP pdf c pdf c c c f f f

³

³

(3) Where c is the norm alized height of the d isp lacem ent p eak and ₁ c₂ the norm alized height of the second ary p eak. For the case show n in Fig. 2 a valid d etection probability of

95.2%

is com p u ted from the height d istribu tions. Thu s, for

4.8%

of the cases the second ary p eak becom es larger than the d isp lacem ent p eak.

3. Effect of Image and Flow Parameters on Correlation Statistic

Figu re 5 show s the d istribu tions of the d isp lacem ent p eak and the second ary correlation p eak for a variation of the in-p lane-m otion (top left), the ou t-of-p lane m otion (m id d le left), the in-p lane grad ients (bottom left), the p article im age d ensity (top right), the interrogation w ind ow size (m id d le right) and the im age noise level (bottom right).

Generally, it can be conclu d ed from the figu re that all p aram eters tested affect the valid d etection p robability of PIV vectors. It is im p ortant to note that the three p aram eters on the left hand sid e, F_I, F_O, and F_', d o not alter the probability of the second ary p eak. This is d u e to the fact that the locations of the u np aired p article im ages are rand om . H ow ever, the w id th of the height d istribu tion of the d isp lacem ent p eak increases w ith d ecreasing F_I, F_O, and F_', ind icating that a broad er range of d isp lacem ent p eak heights becom es p ossible. If the d isp lacem ent p eak is alw ays higher than the second ary p eak the valid d etection p robability is u nity. This is only the case if the d istribu tions of both p eak heights are w ell sep arated , as show n for F_I!0.75, F_O !0.8,

0.45

F_'! .

On the other hand , the p aram eters on the right sid e of Fig. 5, N_ppp, D , and SN R, clearly _I

influ ence the d istribu tion of both p eaks. Figu re 5 show s in the top right corner the effect of d ifferent p article im age d ensities N_ppp on the correlation height d istribu tion. The interrogation w ind ow size D w as ad ju sted to keep the nu m ber of p article im ages _I N_I constant. The d isp lacem ent correlation p eak is not affected for N_ppp 0.1 becau se it is still com p osed of the sam e nu m ber p article im ages. For N_ppp!0.1 the p article im ages start to overlap m assively lead ing to a slightly increasing w id th of pdf₁. The second ary p eak how ever, strongly d ep end s on

ppp

N over the fu ll range: the w id th and the m ean valu e of the d istribu tion increase w ith increasing p article im age d ensity.

Fig. 5 Effect of F_I (top left), p article im age d ensity N_ppp (top right), FO (m id d le left), interrogation w ind ow size D (m id d le right), F_{I '} (bottom left) and signal-to-noise ratio SN R (bottom right) on the height d istribu tion of the d isp lacem ent p eak and the second ary p eak of the norm alized cross-correlation fu nction. If not varied , the sim u lation p aram eters are: D_I 16 p ixel, N_ppp 0.05, F_O 0.7,

SNR

10

. The d otted lines and the d ashed lines ind icate the highest p robability and the m ean height, resp ectively. The shad ed areas represent the

90%

coverage of the pdf _{and the red solid line is the valid d etection p robability VDP from Eq. (3).}

Besid es the p article im age d ensity also the nu m ber of p article im ages w ithin the interrogation w ind ow N_I influ ences the d istribu tion of the second ary p eak pdf₂. This is illu strated in Fig. 5 in the m id d le row on the right sid e for the evalu ation of the sam e synthetic im ages w ith d ifferent interrogation w ind ow size ranging from D_I 8 p ixel to 64 p ixel. Ad d itionally, N_I also affects the d istribu tion of the d isp lacem ent p eak pdf₁: While its m ean height is rather constant, the w id th and the height w ith the highest p robability (d otted line) increase w ith d ecreasing w ind ow size. As a resu lt, the valid d etection p robability d ecreases for sm aller interrogation w ind ow sizes, as exp ected . The best sp atial resolu tion is reached as soon as both pdf are sep arated , w hich is arou nd D_I|20 pixelfor the sp ecific cases tested here.

Im age noise red u ces the norm alized correlation height of the d isp lacem ent p eak and the second ary p eak, as show n in Fig. 5 in the bottom right corner. In the relevant region

V V

_A/ _n ! 1 the height of the d isp lacem ent p eak d ecreases m u ch faster than the second ary one. H ere

V

_A is the intensity stand ard d eviation of the noise-free im age and V_n is the im age noise level (see Scharnow ski 2016b).

In su m m ary it can be stated that the pdf of the d isp lacem ent p eak and the second ary correlation p eak is qu ite sensitive on flow p aram eters, im age p aram eters and evalu ation p aram eters. As a resu lt, the estim ation of the valid d etection p robability becom es rather com p lex.

4. Optimized Interrogation Window Size

The strongest ad vantage of PIV over classical p oint-w ise m easu rem ent techniqu es is its ability to p rovid e flow field s from w hich the organization of tu rbu lent stru ctu res of variou s length scales can be d etected . In ord er to m axim ize the flow inform ation acqu ired w ith PIV m easu rem ents, it is im p ortant to cap tu re a large field of view and to resolve sm all d etails at the sam e tim e (Ad rian, 1997; Kähler et al., 2012). To achieve this, cam era sensors w ith a large nu m ber of p ixel or m u ltip le cam era ap p roaches (Cu vier 2017) can be com bined w ith sop histicated im age evalu ation techniqu es that iteratively d ecrease the interrogation w ind ow size (Scarano, 2001; Sciacchitano et al. 2012).

The op tim u m interrogation w ind ow size d ep end s on several factors: For the first evalu ation step , the in-p lane-m otion as w ell as the in -p lane grad ients strongly affect the VDP . To accou nt for this the shift of the p article im age ensem ble shou ld not exceed one qu arter of the interrogation w ind ow size for the first iteration (Keane and Ad rian, 1990). For the follow ing iterations the in -p lane m otion is com p ensated by w ind ow shifting and im age d eform ation techniqu es so that the interrogation w ind ow s can be red u ced in size from iteration to iteration .

The sm allest su ited w ind ow size for a reliable d etection of the m ean shift of the p article im age ensem ble w ithin the interrogation w ind ow is reached if only N_I|5 p article im ages are fou nd in each w ind ow on average (Keane and Ad rian, 1992). Thu s, the constraint of the p article im age d ensity lim its the sp atial resolu tion of PIV. If an ou t-of-p lane m otion is p resent the nu mber of p article im ages w ithin an interrogation w ind ow m u st be larger than 5 as illu strated in Fig. 1 to keep the valid d etection p robability on the sam e level. Ad d itionally, for strong ou t-of-p lane m otions it is not su fficient to keep the p rod u ct N F_{I O} !5, becau se the VDP also d ep end s on the second ary correlation p eak, w hich is form ed from all p article im ages (see Fig. 4 and Eq. 3). Fu rtherm ore, as show n in Fig. 5 in the top right corner, the p article im age d ensity N_ppp

influ ences the second ary correlation p eak and thu s the VDP .

The tw o p aram eters F_O and N_ppp are the d riving p aram eters for d eterm ing the VDP . Figu re 6 show s an exam p le of the valid d etection p robability VDP as a fu nction of F_O and N_ppp

for an interrogation w ind ow of 16 16u p ixel. It is clear from the figu re, that a high valid d etection p robability (e.g.

95%

) requ ires an increasing p article im age d ensity for sm aller valu es of F_O in ord er to keep the sp atial resolu tion constant. In other w ord s: Dep end ing on the ou t-of-p lane m otion and the p article im age d ensity, the interrogation w ind ow size m u st be selected to ensu re a su fficiently high VDP .

Fig. 6 Valid d etection p robability VDP as a fu nction of the p article im age d ensity N_pppand the loss-of-correlation d u e to out-of-p lane m otion F_O for an interrogation w ind ow size of D_I 16 p ixel.

Figu re 7 illu strates iso-contou rs w ith VDP 0.95 for interrogation w ind ow s betw een 12² and 64² p ixel. The solid lines in the figu re show for each w ind ow size the accep table valu e of F_O

that resu lts in VDP 0.95 as a fu nction of N_ppp. For sm aller w ind ow sizes the VDP d ecreases and for larger ones the VDPincreases. The d ashed lines in Fig. 7 ind icate the average nu m ber of p article im ages p er interrogation w ind ow . The figu re clearly show s that the cond ition N F_{I O}!5, as p rop osed by Keane and Ad rian (1992), is not su fficient to achieve a high valid d etection p robability: For N_I and 20 N_ppp 0.02 the loss-of-correlation d u e to out-of-p lane m otion of

O 0.5

F | resu lts in VDP 0.95 althou gh N F_{I O} | , for example. The required effective number of 10 p article im ages becom es even larger for cases w ith stronger out -of-p lane m otion and higher p article im age d en sity: For N_I 100 and N_ppp 0.1 the loss-of-correlation d u e to ou t-of-p lane m otion of F_O |0.4 resu lts in VDP 0.95 althou gh N F_{I O} |40. Fu rtherm ore, for N F_{I O} 5 (blu e d otted line in Fig. 7) the loss-of-correlation d u e to ou t-of-p lane m otion m u st be F_O!0.8 for the tested p article im age d ensity in ord er to achieve VDP!0.95.

Fig. 7 Iso-lines w ith VDP 0.95 show ing the level of accep table F_O w ith resp ect to the p article im age d ensity N_ppp for d ifferent interrogation w ind ow sizes (red solid lines). The d ashed black lines ind icate the requ ired nu m ber of p article im ages w ithin the interrogation w ind ow .

5. Summary & Conclusions

The analysis show s that the effective nu m ber of p article im ages N F F_{I I O} is not su fficien t to p red ict the valid d etection p robability of a PIV vector field s as p rop osed by Keane and Ad rian (1992). This is becau se not only the nu m ber of p article im ages that can be p aired m u st be consid ered bu t also the nu m ber of those that cannot be p aired is im p ortant. The form er d eterm ine the hight of the d isp lacem ent p eak and the latter contribu te to the second ary correlation p eak. Only if the nu m ber of p aired p article im ages is large enou gh com p ared to the u np aired ones the correct correlation p eak is the highest one.

From the variation of several p aram eters in Sec. 3 it can be conclu d ed that the highest valid d etection p robability is reached if no m otion is present at all and the p article im age d ensity is very low . H ow ever, this resu lts in velocity m easu rem ents w ith high u ncertainty and p oor resolu tion. To achieve accu rate m easu rem ent resu lts som e ou tlieres m u st be accep ted (Scharnow ski and Kähler, 2016a). An op tim ization regard ing the sp atial resolu tion and / or the u ncertainty m u st be p erform ed d u ring d ata acqu isition (tim e sep aration betw een d ou ble im ages, op tical m agnification, ap ertu re, light sheet w id th and e nergy, p article concentration, ...) as w ell as d u ring d ata evalu ation (interrogation w ind ow size, im age d eform ation ap p roach, vector p ost -p rocessing, ...).

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