SCIENCE CHINA Physics, Mechanics & Astronomy
. Article .
October 2015 Vol. 58 No. 10: 104704 doi: 10.1007/s11433-015-5719-y
Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application PAN Chong1 , XUE Dong1 , XU Yang1 , WANG JinJun1* & WEI RunJie2 1 Fluid
Mechanics Key Laboratory of Ministry of Education, Institute of Fluid Mechanics, Beihang University, Beijing 100191, China; 2 Beijing MicroVec Incorporated Company, Beijing 100083, China Received May 29, 2015; accepted August 3, 2015
Lucas-Kanade (LK) algorithm, usually used in optical flow filed, has recently received increasing attention from PIV community due to its advanced calculation efficiency by GPU acceleration. Although applications of this algorithm are continuously emerging, a systematic performance evaluation is still lacking. This forms the primary aim of the present work. Three warping schemes in the family of LK algorithm: forward/inverse/symmetric warping, are evaluated in a prototype flow of a hierarchy of multiple two-dimensional vortices. Second-order Newton descent is also considered here. The accuracy & efficiency of all these LK variants are investigated under a large domain of various influential parameters. It is found that the constant displacement constraint, which is a necessary building block for GPU acceleration, is the most critical issue in affecting LK algorithm’s accuracy, which can be somehow ameliorated by using second-order Newton descent. Moreover, symmetric warping outbids the other two warping schemes in accuracy level, robustness to noise, convergence speed and tolerance to displacement gradient, and might be the first choice when applying LK algorithm to PIV measurement. PIV, Lucas-Kanade (LK) algorithm, accuracy performance, FFT-PIV algorithm PACS number(s): 47.80.Cb, 47.80.Jk, 42.30.Tz Citation:
Pan C, Xue D, Xu Y, et al. Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application. Sci China-Phys Mech Astron, 2015, 58: 104704, doi: 10.1007/s11433-015-5719-y
1 Introduction Particle Image Velocemetry (PIV) has become a standard tool in the arsenal of fluid dynamic experimentalists. Since the early work which laid the theoretical foundation of PIV technique [1,2], the community has seen a vast boost in its hardware capacity. Nowadays, powerful laser and fast camera enables to capture a flow-field dynamics with high temporal resolution [3–6], and the usage of multi-camera and massive sampling credits turbulence measurement covering a broad scale spectrum [7]. For the latter scenario, thousands of snapshots are essentially needed for statistical convergence [8–12]. On considering the time of resolving velocity vector
field from PIV images, i.e. O(100 )–O(102) secs for a standard FFT-based cross-correlation (CC) PIV scheme with multipass iteration (abbreviated as FFT-PIV in the following), the computational efficiency becomes a critical issue in practical applications. To deal with this issue, Champagnat et al. [13] proposed a fast post-processing algorithm, FOLKI (French acronym for Iterative Lucas-Kanade Optical Flow), for two-dimensional two-component (2D2C) PIV. This algorithm is adopted from the widely-used Lucas-Kanade (LK) algorithm in the field of computer vision [14,15], with additional modification to accommodate special requirement in PIV scenario. To avoid losing generality, we denote this algorithm as LK instead of FOLKI in the following. In contrasted to the state-of-the-
*Corresponding author (email:
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c Science China Press and Springer-Verlag Berlin Heidelberg 2015
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art FFT-PIV method, the LK algorithm is based on the Sum of Square Difference (SSD) optimization. Its major feature, as summarized in Table 1, is the “matrixwise” operation, i.e. the velocity vector field is resolved by manipulating the grayscale intensity matrix as a whole. This credits the ability of massive parallelization, making it suitable for Graphic Processing Units (GPU) calculation. For example, a GPU speedup factor of 50 was achieved by Champagnat et al. [13], much more prominent than those efforts of enhancing the calculation efficiency of “pointwise” FFT-PIV [16,17]. A byproduct of matrixwise operation is the dense field output up to pixellevel; however, the spatial resolution is still determined by the size of interrogation window, same to FFT-PIV. Note that this is different from other optical flow solvers [18–20], which don’t involve window integration, so that the spatial resolution can be enhanced to pixel level theoretically. In addition to the above two major features, advanced interrogation techniques [21,22], including iterative shifting, multi-layer pass and window deformation, are implicitly imbedded in LK algorithm. Owing to the enhanced computation efficiency, LK algorithm shows great potential in situation where fast calculation is essential. For instance, Gautier and Aider applied it to a feed-forward control of a perturbed backward-facing step flow where image-based real-time velocity field monitoring is needed [23]. Recent efforts have been made to extend it to stereo/volumetric PIV [24,25]. Moreover, the fast calculation feature makes it well suited in the scenario of an incrementally updated Dynamic Mode Decomposition (DMD) scheme for a quick online reduced-order-model diagnose [26]. Note that LK algorithm is originally designed for optical flow, in which spatial variation of image’s brightness intensity is smooth and continuous. It is critical to know whether this algorithm is suitable for discrete particle images, i.e. will the gray-scale discontinuity deteriorate its feasibility in PIV scenario? By both synthetic and experimental tests, Champagnat et al. [13] have shown that LK algorithm is comparable to that of FFT-PIV in terms of both spatial frequency response and accuracy. However, without a detailed exploration covering a broad range of particle image parameters, such conclusion is, to our opinion, insufficient or at least incomplete. In this sense, the present study is devoted to inTable 1
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vestigating the accuracy of LK algorithm affected by various particle image quantity, so that a practical guideline for PIV application can be drawn. Secondly, recalling that a wide variety of variants of LK algorithm have been proposed in history. They differ from each other in either update form, warping direction or gradient descent method [15]. It is thus interesting to make a comparison of different LK variants in the circumstance of PIV scenario, in a hope of choosing a scheme with “optimal” accuracy performance. This forms the secondary aim of the present study. To accomplish the above two purposes, synthetic test by means of artificial PIV images with pre-defined velocity fields is carried out. In sect. 3, we cover a broad parameter subspace to discuss the effect of particle image quantity on LK algorithm. FFT-PIV is used as a baseline, based on which different warping methods of LK algorithm are compared in detail. In sect. 4, the gradient descent is updated from a first-order Gauss-Newton scheme to a second-order Newton scheme, the performance improvement in combination with the computation penalty is discussed there. Before these two sections, the deduction of LK algorithm, its variants and limitation are briefly reviewed in sect. 2.
2 LK algorithm To provide necessary background, we briefly review the basic construction of LK algorithm and its variants in this section. The covered topics include minimization core, image warping choice, incremental formulation v.s. direct formulation, and gradient descent scheme. Readers should refer to Champagnat et al. [13] and Baker and Matthews [15] for more detailed descriptions. 2.1 Minimization core LK algorithm deals with pattern registration from a template to a target. For PIV scenario, a pair of particle images I(x, 0) and I(x, t) straddling over a short time t, where I is the image’s brightness intensity at pixel coordinate x=[x, y]T, are regarded as template and target, respectively. Using the brightness consistency assumption [27], the template is mapped to
Comparison of major features between LK algorithm and FFT-PIV algorithm Feature Matching criteria Interrogation window Matching method Calculation mode GPU acceleration Dense field output Iterative window shifting Multi-layer pass Window deformation Sub-pixel interpolation
LK algorithm minimize SSD score needed gradient descent optimization parallel matrixwise easy yes implicitly imbedded implicitly imbedded implicitly imbedded at each pass
FFT-PIV algorithm maximize CC score needed direct searching sequential pointwise difficult no optional optional optional at final pass
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the target by a displacement vector u=[u, v]T at each interrogation window (IW): I(x, 0) → I(x + u, t), ∀x ∈ IW. This implies a constant u over IW, namely, the optical pattern inside IW is assumed to translate as a whole during t. A score of Sum of Square Difference (SSD) integrated over IW is used to measure the correctness (or similarity) of the mapping [I (x, 0) − I (x + u(xc ), t)]2 . (1) min u
IW
In eq. (1) the displacement vector u of IW is assigned to the local displacement at xc , the centroid pixel of IW. And the window averaging characteristic determines that the spatial resolution of LK algorithm and its spatial frequency response should be exactly the same to that of FFT-PIV. 2.2 Forward additive algorithm with multi-layer pyramid Eq. (1) describes a non-linear optimization problem which can be handled by Gaussian-Newton (GN) descent approximation properly. Let u0 be a guess of the displacement u, and Δu an increment update of the guess: u0 +Δu0 →u. I(x+u,t) can thus be approximated by I(x+u0 +Δu,t). Expanding it around x+u0 to the first-order, and rewrite the SSD score as: 2 (2) I 0 (x) − I t (x + u0 ) − ∇I t (x + u0 )T Δu , IW
in which ∇I(·)=[∂I/∂x, ∂I/∂y]T is the spatial gradient of I in the shifted coordinate (·). For simplicity, the time stamp is moved to the superscript, and u(xc ) is denoted by u. According to Baker and Matthews [15], the algorithm based on eq. (2) is named as forward additive method. “Forward” describes that the increment Δu is applied to the target image I t , i.e. along forward time direction. “Additive” indicates that the increment is directly applied to the displacement, rather than to the “compositional” warping function [15]. Note that for linear translation in optical flow, both two update methods (additive and compositional) are identically the same. To minimize eq. (2), we set its partial derivative with respect to Δu to be zero ∇I t (x + u0 ) · ∇I t (x + u0 )T Δu IW
− ∇I t (x + u0 ) · I 0 (x) − I t (x + u0 ) = 0.
(3)
IW
A closed-form solution of Δu can be obtained as: ∇I t (x + u0 ) I 0 (x) − I t (x + u0 ) , Δu = H−1 IW
(4)
u0 + Δu → u. In eq. (4), H is the 2×2 Gauss-Newton approximation to the Hessian [15] H= ∇I t (x + u0 ) · ∇I t (x + u0 )T . (5) IW
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Eq. (4) is the incremental formulation. Once Δu is calculated, u0 is updated by u0 +Δu→u0 for the next iteration. If some convergence condition is satisfied, the iteration can be stopped, and the final update u0 +Δu→u gives an estimation of the translation displacement of the corresponding IW between I 0 and I t . We would like to give four remarks about the above procedure. First of all, due to the first-order Taylor expansion in eq. (2), the increment Δu should be small enough; meanwhile, the initial guess u0 is usually set to be zero for convenience. This imposes a critical constraint on the maximum displacement this method can handle, i.e. usually smaller than 1–2 pix. The technique of multi-layer image pyramid [13,18] offers a practical solution for such constraint. The principle is to downsample the image pair (I 0 , I t ) successively to generate a pair of image pyramids (I 0j , I tj ) with j=1,· · ·,N, in which the size of the jth layer is a quarter of that of its lower (j– 1)th neighborhood, and the displacement u j is thus 1/2 times smaller. This makes uN at the top layer to be reduced by a factor of 2N−1 , i.e. uN =u1 /2N−1 . In practice, gradient descent begins from the top Nth layer where uN,0 =0 can be a reasonable initial guess once N is large enough. The output uN is then upsampled to serve as an initial guess uN−1,0 for the lower (N–1)th layer. This looping continues till the bottom j=1 layer, where the output u1 provides the final estimation. Note that similar coarse-to-fine multi-resolution scheme [21] is an optional feature in FFT-PIV; however, it severs as a “building block” here, with the pyramid depth N determining the up limit of the displacement. Secondly, eq. (4) yields real-valued u. This means that the image shifting operation imbedded in I(x + u) implicitly takes sub-pixel interpolation at each iteration. In contrast, the correlation peak searching in multi-pass FFT-PIV outputs integer value, and the sub-pixel fitting is only conducted at the final pass. The third remark is that to enable matrixwise calculation, the image shifting operation in I(x + u(xc )) is replaced by a image warping I(x + u(x)). Namely, IW is warped by a coordinate-dependent displacement, instead of being uniformly translated by the displacement at xc . As a conse quence, the local integration over IW in eq. (2), [·], upIW
dates to a global convolution w ∗ [·] for the whole image, in which w is a uniform weighting function with the same size of IW. All the IWs can then be handled simultaneously, making GPU acceleration possible. A byproduct is to output the displacement in a dense field style, i.e. one velocity vector at one pixel. Details on the GPU implementation can be referred to Champagnat et al. [13]. Finally, in order to get eq. (4), Δu is moved outside the IW integration (or global convolution) in eq. (3). This requires a constant Δu (so that a constant u) over IW. Recalling that in the third remark, each pixel in IW is shifted over an independent displacement to formulate an image warping scheme, while here Δu is constraint to be constant over IW so
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that maxtrixwise operation can be fulfilled. As will be shown later, such confliction is the primary issue to affect the accuracy performance of the current LK algorithm. To reconcile it, u(x) inside IW should be close to uniform, which imposes a constraint on either the size of IW or its displacement gradient. In addition to the incremental formulation of eq. (4), the following direct formulation can be obtained by substituting u–u0 for Δu in eq. (3) u =H−1 ∇I t (x + u0 ) I 0 (x) − I t (x + u0 ) IW
+ ∇I t (x + u0 )T u0 ,
(6)
with H defined in eq. (5). Although eqs. (4) and (6) are mathematically identical, our experience shows that the latter has better numerical stability, because the above constant displacement contradiction becomes less critical for largevalued u than a small increment Δu. Therefore, we will always use the direct formulation in the following. 2.3 Inverse/symmetric additive algorithm In eq. (5), the spatial gradient ∇I t (x + u0 ) and the Hessian H depends on u0 , thus needs to be re-calculated at each pass. To save computation time, an inverse additive scheme was developed [15,28]. The key is to apply Δu to the template image I 0 , i.e. along the inverse temporal direction. The minimization problem becomes 2 I 0 (x − Δu) − I t (x + u0 ) , u0 + Δu → u. (7) min u
IW
Following the GN descent in sect. 2.2, the solution of eq. (7) takes the direction formulation as: u = H−1 ∇I 0 (x) I 0 (x) − I t (x + u0 ) + ∇I 0 (x)T u0 , IW
∇I t (x) · ∇I t (x)T . H=
(8)
IW
In eq. (8), H is independent to image warp, and can be precomputed and re-used for each pass, so dose ∇I 0 (x). Alternatively, the image warping can be evenly applied to I 0 and I t , this forms the symmetric warping scheme [29]. The SSD minimization becomes 2 min I 0 (x − u0 /2 − Δu/2) − I t (x + u0 /2 + Δu/2) , u (9) IW u0 + Δu → u. The direct formulation of u with GN descent is u = H−1 ∇I¯ I 0 (x − u0 /2) − I t (x + u0 /2) + ∇I¯T u0 , IW
∇I¯· ∇I¯T . H= IW
(10)
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In eq. (10), ∇I is the spatial gradient averaged between the template and the target ∇I¯ =
1 0 ∇I (x − u0 /2) + ∇I t (x + u0 /2) . 2
(11)
Note that ∇I¯ should be re-calculated at each pass; moreover, eq. (10) enforces second-order accuracy via central difference [30].
2.4 Second-order Newton descent In sects. 2.2 and 2.3, GN descent is used for the SSD minimization. Other alternatives have been proposed in the past, including Newton descent, steepest descent, diagonal approximation and Levenberg-Marquardt algorithm [15]. Among them, Newton descent uses second-order Taylor expansion and is well balanced between algorithm complexity and convergence rate. Baker and Matthews [15] have shown that it outbids steepest descent and diagonal approximation in terms of accuracy, but is still worse than the first-order GN descent. Note that this observation was based on an empirical test on the affine warp of continuous images. No performance evaluation on discrete particle images has been taken before. Our deduction of Newton descent and the final formulation is slightly different from the original version in Baker and Matthews [15]. Taking the forward additive warp as an example, Newton descent stars from expanding I t (x+u0 +Δu) in the SSD term around x+u0 with second-order accuracy, setting the partial derivative ∂SSD/∂u0 to zero, expanding the left-hand side and only keeping the leading-order terms, the direct formulation can then be obtained as: u =H−1
I t · ∇ It I0 −
IW
∂2 T It 0 t t t
+ ∇ I ∇ I − I − I · 2 · u0 , ∂x
(12)
and the Hessian becomes H=
⎞ ⎛⎜⎜ ∂2 T I t ⎟⎟ ⎜⎜⎝∇ It − I0 − I t · 2 ⎟⎟⎠ . I t ∇ ∂x IW
(13)
In eqs. (12) and (13), I(x,0) and I(x + u0 , t) are simplified as I 0 and I˜t , respectively, and ∂2 (·)/∂x2 is the 2×2 second-order spatial derivative. If ∂2 I˜t /∂x2 = 0, eqs. (12) and (13) degrade to eqs. (6) and (5), respectively. In this sense, (I 0 – I t )· ∂2 It in eqs. (12) and (13) can be regarded as a second-order ∂x2 “add-on” term. Following the above procedure, the direct formulation for inverse/symmetric additive scheme with Newton descent can
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displacement gradient range. In sects. 3 and 4, when evaluating the effect of one of these parameters, the others take the following default value if no more specification is given: ΔIW=32 pix, dp =3 pix, rp =0.2, ρp =22.8, SNR=∞, N=2 and M=2 in FFT-PIV, N=4 and M=6 in LK algorithms with GN descent, and N=3 and M=3 with Newton descent. The reason for selecting these default values will be given in proper place.
be derived as eqs. (14) and (15), respectively −1 I0 − I t · ∇I 0 u =H IW
∂2 I 0
t
+ ∇I ∇I + I − I · u0 , ∂x2 ∂2 I 0 0 0T 0 t
H= ∇I ∇I + I − I . ∂x2 IW
u =H−1
0
0T
0
(14)
I t · ∇I¯ I 0 −
3.1 Synthetic method We use a lab-made image generator based on the framework of Lecordier and Westerweel [31] to create synthetic particle image pairs with pre-given displacement field. In this image generator, the brightness intensity of randomly distributed seeding particles follows an ideal Gaussian model. The particle image diameter is defined at the edge where local brightness reaches 1/e2 of one particle’s maximum intensity. Discretization and quantization effect has been considered by projecting this Gaussian model onto the CCD units with specified fill ratio [31]. Particles’ non-uniformity is modeled by a Gaussian distribution of the diameter whose standard deviation σ(dp ) is fixed as 20% of the mean value dp . The illuminating laser sheet has a Gaussian-type intensity profile with thickness zl =1 pix and standard deviation of 0.5 pix. Particles are dispersed uniformly within the laser sheet. Gaussian-type out-of-plane displacement is added to each particle, with the standard deviation of 20%zl . Finally, Gaussian-type white noise with zero-mean is added to each pixel. SNR is used to measure the inverse of the relative noise strength. Neither the illumination inhomogeneity nor lens aberration is simulated; the particle deformation due to long exposure and the light reflection due to high seeding density are not considered, too. Figures 1(c) and (d) give an example of one image pair. More details about this image generator can be referred to Shen et al. [32]. Unlike other studies using elementary benchmark flows [13,33], a more complicate flow field containing a series of 2D vortices is used here. As shown in Figure 1(a), the size of 2D vortices decreases from the left-upper corner to the right-
IW
∂2 I 0 ∂2 I t
¯ I¯T + 1 I 0 − It − · u + ∇I∇ 0 , 4 ∂x2 ∂x2 ⎞⎤ ⎛ 0 ⎡⎢⎢ I I t ⎟⎟⎟⎥⎥⎥ 1 0 t ⎜⎜⎜ ∂2 ∂2 T ⎢ ¯ ¯ ⎢⎣∇I∇I + I − I ⎜⎝ 2 − 2 ⎟⎠⎥⎦ . H= 4 ∂x ∂x IW
(15)
I t ; while In eq. (14), I(x, 0) is denoted by I 0 and I(x+u0 , t) by
0 t I =I(x + u0 /2, t), and ∇I¯ = ineq. (15), I =I(x − u0 /2, 0), 1
0 t
I . Note that the second-order “add-on” term 2 ∇ I + ∇ 2 0 2 t 1 0 ∂ I t
− ∂∂xI2 in eq. (15) approaches to zero more 4 I −I ∂x2 quickly than its correspondences in eqs. (12) and (14). This is reasonable since the symmetric warping already enforces second-order accuracy by a symmetric expansion.
3 Comparison of different warping schemes The accuracy performance of different variants of LK algorithm are synthetically evaluated in a large domain of various calculation/particle image parameters, as listed in Table 2. The effect of warping direction (with GN descent) is studied here, while the benefit of adapting Newton descent is discussed later in sect. 4. The convergence speed with respect to pyramid depth N and iteration number M, and the accuracy dependency on the interrogation window size ΔIW are both tested. The particle image’s mean diameter dp , its seeding concentration rp , the corresponding particle density ρp and the signal-to-noise ratio SNR are all taken into consideration. Moreover, all the above parameters are discussed in separate Table 2
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Investigated algorithm variants and influence parameters a) Variants of LK algorithm
Forward additive + Gaussian-Newton descent (LK-FA-GN) Inverse additive + Gaussian-Newton descent (LK-IA-GN) Symmetric additive + Gaussian-Newton descent (LK-SA-GN) Forward additive + Newton descent (LK-FA-Newton) Inverse additive + Newton descent (LK-IA-Newton) Symmetric additive + Newton descent (LK-SA-Newton)
Calculation parameters
Particle image parameters
N, M √
ΔIW √
dp √
rp √
ρp √
SNR √
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
×
×
×
×
×
×
a) N: total number of image pyramid layer; M: total number of iteration pass at each pyramid layer; ΔIW: width (height) of the interrogation window; dp : mean diameter of particle image, unit: pix; rp : seeding concentration, ratio of the area covered by particle images to the whole image area; ρp : particle density, number of particles in an interrogation window with size of 32 pix×32 pix; SNR: signal-to-noise ratio, the ratio between maximum brightness to standard deviation of the noise intensity.
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(b)
V
Inside ROSB Outside ROSB ×10
(c)
(d)
Figure 1 (Color online) Illustration of the prototype multi-vortex flow. (a) Spatial distribution of the displacement vector and the vector length; (b) histogram of the displacement vector length; (c) and (d) example of a pair of synthetic particle images with dp =5. ROSB is the region defined by the dashed black box in (a).
bottom, while their swirling strength increases simultaneously. Meanwhile, the displacement magnitude also increases. The reason for choosing such a fabricated flow field is that it offers a hierarchy of displacement gradient whose strength is inversely related with the structure’s scale, resembling the case in many complex flows. The accuracy & efficiency test was taken in Matlabc environment on a DELL workstation with CPU of Intel Xeon X5650. Jacketc is used to adapt Matlab code to GPU calculation on a NVidia GTX-760 VGA card. As for the baseline FFT-PIV, we use the open source code MatPIV, which is also based on Matlabc and poses the feature of multiresolution and iterative pre-offset. Note that in MatPIV, the classical local median method is used to remove outliers during each pass, while LK algorithm does not involve any postprocessing. Moreover, local zero normalization [13] is used for image pre-processing in both cases. Figure 2(a) illustrates the spatial distribution of the displacement error ΔV yielded by FFT-PIV. ΔV is defined as the difference between the length of the output displacement vector length and that of the pre-defined input. Neither the component error nor the directional error is discussed for simplicity. ΔV in the right-bottom region (the dashed box in both Figure 1(a) and Figure 2(a)) is relatively large, i.e. beyond ±2 pix. Referring to Figure 1(b), the displacement dynamic range of this prototype flow is V=[0 20] pix, with large displacement mainly concentrating inside this rightbottom corner. On considering that the uncertainty inside this right-bottom corner is directly related with the large local displacement, we denote it as the region of systematic bias (ROSB). LK algorithms with forward/inverse/symmetric warping present similar error distribution; however, ΔV inside ROSB is biased mainly towards negative value. The former two (Figures 2(c) and (d)) have deteriorated bias there if compared to FFT-PIV (Figure 2(b)), while the last (Figure 2(e)) seems to ameliorate the bias to some extent.
Root mean square (RMS) of the displacement error σΔV is used as the chief measure of the accuracy performance. σΔV inside ROSB and outside ROSB is calculated separately. To compensate the randomness rooted in the particle image generator, multi-repetition evaluation is taken. An ensemble of 300 repetitions is found to be enough for a statistical convergence of σΔV even at the extreme case of SNR=2, so that this ensemble size is kept everywhere. 3.2 Effect of interrogation window size Figure 3 shows the correlation between V and ΔV with two IW sizes ΔIW=32 pix and 16 pix (IW is always square in the following). Besides the scatter plot, both the isolines of joint probability density function (JPDF) and the most probable ridge lines ΔVmost (V) are given. For FFT-PIV, ΔV inside/outside ROSB both present quasi-symmetric distribution about ΔV=0. Jittering of the most probable ridge is observed, which can be attributed to the sub-pixel fitting that attracts the output around integer number. As ΔIW decreases to 16 pix×16 pix, negative ΔV appears in the large V side, indicating that reducing the IW size will deteriorate the accuracy of FFT-PIV algorithm where the local displacement is large. As for all three LK algorithms, ΔV inside ROSB present negative lobs directing towards large V when ΔIW=32 pix (Figure 3(a)), according well with the observation that systematic error mainly concentrates inside ROSB (Figure 2(c– e)). On the other hand, the vertical extend of the JPDFs of the whole flow field is somehow compressed, the number of points outside ROSB with JPDFforward warping>inverse warping. It holds both inside and outside ROSB, and validates for all IW sizes except the left extreme of ΔIW=8 pix where effective particle number is too small. This hierarchy is reasonable, since symmetric warping ρp 10
1
1.4
5.7
12.8
22.8
35.6
51.2
69.7
91.0 115.2
σ ∆V (pix)
Inside ROSB
10
10
0
FFT−PIV LK−FA−GN LK−IA−GN LK−SA−GN
Outside ROSB
−1
8
16
24
32
40 48 ∆ IW (pix)
56
64
72
Figure 4 (Color online) σΔV as a function of ΔIW in different algorithms. Hollow symbols for outside ROSB, solid symbol for inside ROSB. The meaning for the abbreviation of LK-FA-GN, LK-IA-GN and LK-SAGN is listed in Table 2, the same in the following figures.
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enforces second-order accuracy, while inverse warping sacrifices accuracy by assuming a constant Hessian (eq. (8)) to save calculation time. More importantly, all three LK algorithms are superior to FFT-PIV in terms of RMS error in small IW size. This can be partly attributed to the fact that LK algorithms are less sensitive to the drop of effective particle number, which will be discussed in sect. 3.3. 3.3 Effect of particle size and particle concentration In this subsection, the effect of particle size dp and seeding concentration rp is compared among different algorithms, with IW size fixed at the “optimal” ΔIW=32 in FFT-PIV. Figures 6(a) and (b) show the contour plot of σΔV in FFT-PIV as a function of both dp and rp outside/inside ROSB, respectively. For the former (Figure 6(a)), minimum of σΔV appears in the region with high seeding concentration and small particle size; while for the latter (Figure 6(b)), large particle size will somehow ameliorate the systematic bias once the seeding concentration is high enough. Moreover, σΔV always decreases with the increase of rp at given dp . Figures 6(c) and (d) show ΔσΔV , the relative difference of σΔV in LK algorithms to that in FFT-PIV, negative sign for accuracy improvement with respect to FFT-PIV and positive sign for deterioration. Outside ROSB (Figure 6(c)), all three LK algorithms lead to accuracy improvement in a large (dp , rp ) domain, with the hierarchy of symmetric warping>forward warping>inverse warping observed in sect. 3.2. A maximum profit of 40% is obtained by symmetric warping at dp =3–4 pix. Referring to Figure 7(a), increasing dp from 3 pix at fixed rp leads to a continuous reduction of the accuracy improvement in general. As for the case inside ROSB, forward/inverse warping scheme both deteriorates the accuracy if compared to FFT-PIV, with the maximum σΔV increment close to 100%. However, symmetric warping scheme still poses a small net benefit. Figure 7(b) further shows that as dp increases, σΔV in forward/inverse warping scheme gradually restore to the level of FFT-PIV, while σΔV in symmetric warping scheme even descents a bit lower, again evidencing its ability of inhibiting the systematic error in large displacement gradient. Given a combination of dp and rp , the particle density ρp , which describes how many effective particles are contained inside one IW with ΔIW=32 pix, can be exclusively determined. Figure 8 plots σΔV as a function of ρp . Outside ROSB, all dots with varied dp and rp collapse onto one single curve. Inside ROSB, slight scattering appears in the middle range of ρp =[5 50]; however, an asymptotic state is gradually achieved in large ρp . This observation is valid for both FFT-PIV and LK algorithms. The existence of a universal σΔV –ρp curve suggests that the particle density is the dominant particle parameter, which unifies the effect of dp and rp , to characterize the accuracy performance. This observation is consistent with both the
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analytical prediction for FFT-PIV [2] and the empirical test on single pixel ensemble correlation [32]. Comparing the universal σΔV –ρp curve outside ROSB in FFT-PIV to those
in LK algorithms, all present quick decay with the increase of ρp , the curve in FFT-PIV achieves a convergence state a bit earlier (around ρp =20), but the converged magnitude
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3.4 Effect of noise level When talking about the noise effect, the influence of particle density should be considered in together. This is because high particle concentration might suppress the influence of noise which is evenly added to each pixel. Figures 9(a) and (b) plot σΔV outside/inside ROSB in FFT-PIV as a function of both ρp and SNR. Figures 9(c) and (d) show the relative improvement of σΔV by using LK algorithms. And Figure 10 illustrates the variation of σΔV with respect to SNR at ρp =10 and 100. In general, noise plays a critical role when SRN 10, beyond which its effect becomes attenuated. This statement comes from the fact that the contour of σΔV (ρp , SNR) outside/inside ROSB in FFT-PIV (Figures 9(a) and (b)) can be divided into two parts: the negatively inclined isolines below SNR=10 suggest an interchangeable effect of SNR and ρp ; while the quasi-vertical isolines beyond SNR=10 indicate weakened SNR dependency there. Such division is also
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valid for LK algorithms. Figure 10 further shows that after SNR>10, σΔV of all four algorithms quickly converge to the level without noise, but increase rapidly towards the direction of small SNR. LK algorithms improve σΔV outside ROSB (Figure 9(c)) in a large (ρp , SNR) domain, with the maximum profit up to 60%; however, σΔV inside ROSB (Figure 9(d)) is elevated in forward/inverse warping if SNR>10. Finally, the noise suppression effect by high particle density is clearly depicted in Figure 10: increasing ρp from 10 to 100 will significantly reduce σΔV outside ROSB at small SNR in LK algorithms, similar to the case without noise (Figure 8). This means that in practice, the tolerance on SNR can be relaxed by deliberately increasing particle density. Note that the benefit is more remarkable in symmetric warping, which even shows robustness in ameliorating the systematic bias inside ROSB in the circumstance of large noise level.
is higher than that in forward/symmetrical warping scheme. Note that no convergence state is observed in inverse warping scheme; instead, σΔV presents a re-growth at large ρp , thus exceeds the converged value of FFT-PIV finally. This phenomenon can be still attributed to the constant Hessian assumption invoked in inverse warping: high concentration of particles causes severe change of the spatial gradient of brightness intensity; therefore, the Hessian H is far from constant over iteration. As for the curve of σΔV –ρp inside ROSB, only symmetric warping scheme is equivalent to FFTPIV in terms of converged σΔV , other two perform comparably worse in large ρp . The reason is that inside ROSB, it is the constant displacement constraint that dominantly affecting the accuracy performance of LK algorithms.
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4.1 Improvement in convergence speed Since the minimization problem of eqs. (1), (7) or (9) relies on gradient descent regression, both the pyramid depth N and the iteration number M are critical parameters affecting the accuracy. Champagnat et al. [13] have demonstrated that N determines the maximum displacement that GN descent can reliably handle, i.e. Vmax –2N pix. This statement is supported by Figure 11(a), in which N=4 leads to a convergence of σΔV outside/inside ROSB in all three warping schemes with GN descent. Recalling that the displacement dynamic range of the prototype flow is V=[0 20] pix. As for the choice of M, the formal way is to stop the iteration at each pyramid layer when certain convergence criterion is reached [13,34]. As shown in Figure 12(a) where GN descent is evaluated, the convergence of σΔV outside ROSB is fast if the requirement of Vmax –2N is satisfied, i.e. only 6 iteration is needed for forward/inverse warping, while 2 iteration
is enough for symmetric warping. To reduce the algorithm complexity, M in the present study is fixed among pyramid layers. Note that the convergence speed of σΔV inside ROSB with respect to M is comparably slow. This is still related with the systematic bias caused by large displacement gradient there. In contrast to LK algorithms, the baseline FFT-PIV presents a weak dependence on multi-layer iteration: N=2 and M=2 is enough for a good convergence of σΔV outside/inside ROSB. This fact should be noticed when making calculation efficiency comparison. Updating to Newton descent, the convergence behavior of all three warping schemes is improved remarkably. Figure 13 compares σΔV outside/inside ROSB obtained by either GN descent or Newton descent in (M, N) plane. The isolines of the latter are generally shifted towards the left-bottom direction. Referring to Figure 11(b), Newton descent will
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advance the convergence of σΔV outside/inside ROSB from N=4 to N=3 in symmetrical warping scheme, and σΔV in the other two warping schemes at N=3 is also reduced significantly. Meanwhile, Figure 12(b) shows that the critical M in forward/inverse warping scheme, measured by σΔV outside ROSB, is advanced from M=5 to M=3, while a convergence of σΔV inside ROSB at M=6 can be even achieved in symmetric warping scheme. 4.2 Improvement in accuracy performance Besides the convergence speed, the improvement in accuracy performance is a more practical issue. Figures 12 and 13 already indicate that Newton descent can somehow lower σΔV inside ROSB, implying its superiority in dealing with large displacement gradient. Figure 14 further shows ΔσΔV , the relative difference between σΔV obtained by Newton descent and that by GN descent, at various ΔIW. It is clear that updating to Newton descent positively improves ΔσΔV outside ROSB once ΔIW>32 pix: a persistent reduction up to 20% is seen in inverse/symmetric warping scheme, while the improvement in forward warping scheme is less prominent. The same error reduction hierarchy exists inside ROSB; however, the maxi-
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mum occurs around ΔIW=28–32 pix, and quickly diminishes with the increase of ΔIW. This implies that Newton descent is still incapable of relaxing the constant displacement constraint in the region where the displacement difference is relatively large. Turning to the effect of other particle parameters, the positive role of Newton descent becomes more evident. Figure 15 shows the variation of ΔσΔV in (ρp , SNR) plane with ΔIW=32 pix. Figure 16 presents ΔσΔV as a function of SNR at ρp =10 and 100. Combing with Figure 15 and Figure 16, it can be concluded that if ρp is not too large, i.e. ρp forward warping>symmetric warping is observed, which is exactly the inverse of the accuracy hierarchy shown in sect. 3. More interestingly, updating to Newton descent will result in slight efficiency improvement in inverse warping scheme, while the calculation overload in forward/symmetric warping is also small. This is because the
minimum pyramid depth and iteration number required for an error convergence is saved by Newton descent.
5 Conclusion In summary, the feasibility of applying LK algorithm in the circumstance of PIV application has been evidenced by synthetic test. Systematic accuracy comparison to classical FFTPIV method has been conducted in a large domain of influential parameters. LK algorithm is shown to be comparably accurate, stable and reliable. On considering its potential in GPU acceleration, it should be regarded as a promising
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choice with full confidence. Moreover, its dense calculation feature makes it suitable for the scenario where dense velocity field is needed. The most critical problem in the current version of LK implementation is the constant displacement constraint, which is rooted in the algorithm construction, so that is unavoidable. This constraint restricts the local displacement gradient inside the interrogation window. In FFT-PIV, it is not a determining factor. Previous studies [13,25] haven’t stressed its importance in affecting accuracy performance of LK algorithm. Empirical experience from sect. 3.1 suggests that the limitation for local averaged displacement gradient is on the order of 1 pix/pix. Once this constraint is violated, the estimation error degrades to systematic bias being trapped in local minima, thus resulting in a significant deterioration of the accuracy performance. Symmetric warping scheme can somehow ameliorate this constant displacement constraint, since it enforces secondorder accuracy by evenly warping both template and target image. Figure 18 gives a performance radar chart of forward/inverse/symmetric warping schemes. The superiority of symmetric warping lies in accuracy level, robustness to noise, convergence speed and tolerance to displacement gradient. On considering all these benefits, the slight decrease of the calculation efficiency seems to be an affordable price. Finally, using second-order Newton descent to replace the first-order GN descent will yield positive accuracy improvement for inverse/symmetric warping schemes. This is in distinct contrast to the previous observation in optical flow
scenario [15]. The reason might lay in the fact that the second-order spatial derivatives used here are only treated as “add-on” correction terms via sufficient smoothing. Moreover, due to the discrete & sparse feature of particle images, these second-order terms are less affected by random noise or sub-pixel interpolation error, if compared to the case in continuous optical flow images. Interestingly, inverse warping scheme, which ranks in the bottom of the accuracy hierarchy, will get the most significant improvement. As for the symmetric warping scheme, its tolerance to displacement gradient will be also enhanced remarkably. Meanwhile, since the convergence speed is improved, no remarkable calculation penalty will be introduced. In this sense, it is valuable to further explore the potential margin of Newton descent in the future, and make full use of it in PIV application.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11372001 and 11490552).
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