Sampling the Disparity Space Image

Transcription

1 Accepte for publication, June To appear in IEEE Transactions on Pattern Analysis an Machine Intelligence (PAMI) Sampling the Disparity Space Image Richar Szeliski Daniel Scharstein Microsoft Research Milebury College Remon, WA Milebury, VT Abstract A central issue in stereo algorithm esign is the choice of matching cost. Many algorithms simply use square or absolute intensity ifferences base on integer isparity steps. In this paper we aress potential problems with such approaches. We begin with a careful analysis of the properties of the continuous isparity space image (DSI) an propose several new matching cost variants base on symmetrically matching interpolate image signals. Using stereo images with groun truth, we empirically evaluate the performance of the ifferent cost variants an show that proper sampling can yiel improve matching performance.

2 1 Introuction The last few years have seen a ramatic improvement in the quality of ense stereo matching algorithms [13]. A lot of this improvement can be attribute to better optimization algorithms an better smoothness constraints [6, 4, 17]. However, a remarkable amount of the improvement has also come from better matching metrics at the input [3]. In fact, Birchfiel an Tomasi s samplinginsensitive issimilarity measure is use by a number of toay s best performing algorithms [6, 4]. Using something better than just piel-sample intensity ifferences is not a new iea. For eample, Matthies et al. interpolate scanlines by a factor of 4 using a cubic interpolant before computing the SSD score [10]. Tian an Huhns wrote an even earlier survey paper comparing various algorithms for sub-piel registration [18]. In fact, some stereo an motion algorithms have always evaluate isplacements on a half-piel gri, but never mentione this fact eplicitly. The set of initial matching costs that are fe into a stereo matcher s optimization stage is often calle the isparity space image (DSI) [20, 5]. However, while the concept of stereo matching as fining an optimal surface through this space has been aroun for a while [20, 2, 5], relatively little attention has been pai to the proper sampling an treatment of the DSI itself. In this paper, we take a more careful look at the structure of the DSI, incluing its frequency characteristics an the effects of using ifferent interpolators in sub-piel registration. Among the questions we ask are: What oes the DSI look like? How finely o we nee to sample it? Does it matter what interpolator we use? We also propose a number of novel moifications to the matching cost that prouce a better set of initial high-quality matches, at least in teture, unocclue areas. It is our contention that filling in tetureless an occlue areas is best left to a later stage of processing [5, 6, 4, 17], which is why we o not consier global optimization techniques in this paper. The remainer of the paper is structure as follow. Section 2 presents our analysis of the DSI an iscusses minimal sampling requirements. Section 3 evelops several novel matching costs base on our analysis. The utility of these novel costs is valiate eperimentally in Section 4. We conclue with some ieas for future research. 2 Matching costs In this section, we look at how matching costs are formulate. In particular, we analyze the structure of the DSI an its sampling properties an propose some improvements to commonly use 1

3 matching costs. 2.1 The continuous isparity space image Given two input images, I L (, y) an I R (, y), we wish to fin a isparity map L (, y) such that the two images match as closely as possible I L (, y) I R ( L (, y),y). (1) In this paper we assume that the images have been rectifie to have a horizontal epipolar geometry [11, 8], i.e., that the images have been pre-warpe so that corresponing piels are on the same scanline. Stereo corresponence can uner such circumstances be reuce to one-imensional search. Note that our paper proposes interpolating images to a higher resolution after rectification. However, these two steps coul be combine into a single resampling operation to reuce aliasing artifacts. We efine the 3D signe ifference image (SDI) as the intensity (or color) ifference between the shifte left an right images, SDI L (, y, ) =I L (, y) I R (, y). (2) We also efine the raw isparity space image (DSI) as the square ifference (summe over all the color bans), DSI L (, y, ) = SDI L (, y, ) 2. (3) Alternate metrics such as absolute ifferences or robust functions are also possible. However, the quaratic case is easiest to analyze an also correspons to the case of Gaussian noise, as we will iscuss shortly. In the ieal (continuous, noise-free) case with no occlusions, we epect DSI L (, y, L (, y)) to be zero. Unfortunately, we o not actually have access to continuous, noise-free versions of I L (, y) an I R (, y). Instea, we have sample noisy versions, ÎL( i,y i ) an ÎR( i,y i ), Î L ( i,y i ) = [I L h]( i,y i )+n L ( i,y i ) (4) Î R ( i,y i ) = [I R h]( i,y i )+n R ( i,y i ), (5) where h(, y) is the combine point-sprea-function of the optics an sampling sensor (e.g., it incorporates the CCD fill factor [19]), an n L is the (integrate) imaging noise. Given that we usually only evaluate the DSI at the integral gri positions ( i,y i ),wehave to ask whether this sampling of the DSI is aequate, or whether there is severe aliasing in the 2

4 resulting signal. We cannot, of course, reconstruct the true DSI since we have alreay banlimite, corrupte, an sample the original images. However, we can (in principle) reconstruct continuous signals from the noisy samples, an then compute their continuous DSI. The reconstructe signal can be written as I L (, y) = i Î L ( i,y i )g( i,y y i ) (6) = ĨL(, y)+ñ L (, y), (7) where g(, y) is a reconstruction filter, ĨL(, y) is the sample an reconstructe version of the clean (original) signal, an ñ L (, y) is an interpolate version of the noise. This latter signal is a ban-limite version of continuous Gaussian noise (assuming that the iscrete noise is i.i.. Gaussian). We can then write the interpolate SDI an DSI as SDI L (, y, ) = I L (, y) I R (, y) an (8) DSI L (, y, ) = SDI(, y, ) 2. (9) What can we say about the structure of these signals? 2.2 Frequency analysis an aequate sampling The answer can be foun by taking a Fourier transform of the SDI. Let us fi y for now an just look at a single scanline, F{SDI} = F { I L () I R ( ) } = H L (f ) H R (f )e j2π(f f), (10) where H L an H R are the Fourier transforms of I L an I R, an f an f are the an frequencies. Figure 1 shows the SDIs an DSIs an their Fourier transforms for two scanlines taken from the 38th an 148th row of a test image pair containg newsprint on a slante surface. The first term in (10) correspons to the horizontal line in the SDI s Fourier transform (secon column of Figure 1), while the secon term, which involves the isparity, is the slante line. Squaring the SDI to obtain the DSI (thir column in Figure 1) is equivalent to convolving the Fourier transform with itself (fourth column in Figure 1). The resulting signal has twice the banwith in an as the original SDI (which has the same banwith as the interpolate signal). It is also interesting to look at the structure of the DSI itself. The thin iagonal stripes are spurious 3

5 L38 Left image L38, left L38, right I I L SDI F(SDI) DSI F(DSI) f f L38 sinc f f f f L38 linear f f f f L148 sinc f f Figure 1: Sample SDIs an DSIs an their Fourier transforms. Top row: Original image with two selecte scanlines an intensity profiles of the first selecte scanline (L38); notice how the sinc-interpolate signals (soli) are more similar than the linearly interpolate ones (ashe). Bottom rows: Signe Difference Image (SDI) an its transform, an Disparity Space Image (DSI) an its transform; first for L38 using perfect (sinc) interpolation, then for L38 using piecewise linear interpolation, then for L148 using perfect interpolation. The correct isparity in the DSI images is marke with an arrow. 4

6 ba matches (ark-light transitions matching light-ark transitions), while the horizontal stripes are goo matching regions (the straighter an arker the better). What can we infer from this analysis? First, the continuous DSI has significant frequency content above the frequencies present in the original intensity signal. Secon, the amount of aitional content epens on the quality of the interpolator applie to the signal. Thus, when perfect banlimite reconstruction (a sinc filter) is use, the resulting DSI signal only has twice the frequency of the image. It is therefore aequate (in theory) to sample the DSI at 1/2 piel intervals in an. When a poorer interpolant such as piecewise linear interpolation is use, the sampling may have to be much higher. The same is true when a ifferent non-linearity is use to go from the SDI to the DSI, e.g., when absolute ifferences or robust measures are use. This is one of the reasons we prefer to use square ifference measures. Other reasons inclue the statistical optimality of the DSI as the log likelihoo measure uner Gaussian noise, an the ability to fit quaratics to the locally linearize epansion of the DSI (see Section 3.3). We can summarize these observations in the following Lemma: Lemma 1: To properly reconstruct a Disparity Space Image (DSI), it must be sample at at least twice the horizontal an isparity frequency as the original image (i.e., we must use at least 1/2 piel samples an isparity steps). It is interesting to note that if a piecewise linear interpolant is applie between image samples before ifferencing an squaring, the resulting DSI is piecewise quaratic. Therefore, it suffices in principle to simply compute one aitional square ifference between piels, an to then fit a piecewise quaratic moel. While this oes reconstruct a continuous DSI, there is no guarantee that this DSI will have the same behavior near true matches as a more properly reconstructe DSI. Also, the resulting minima will be sensitive to the original placement of samples, i.e., a significant bias towars integral isparities will eist [14]. For eample, if the original signal is a fairly high-frequency chirp (Figure 2a), applying a piecewise linear interpolant will fail to correctly match the signal with a fractionally shifte version. Figure 2b an c show the results of aggregating the original raw DSIs with a 7-piel filter (see Section 3). Clearly, using the linear interpolant will result in the wrong isparity minimum being selecte in the central portion (where the central horizontal line is weak). One might ask whether such high-frequency signals really eist in practice, but it shoul be clear from Figure 1 that they o. 5

7 I (a) (b) (c) () Figure 2: Chirp signal matching: (a) a continuous signal an its shifte an iscretely sample versions; (b) Disparity Space Image (DSI) for linear interpolation; (c) horizontally aggregate DSI for sinc interpolation, showing the correct minimum; () horizontally aggregate DSI for linear interpolation, with incorrect minima near the center. The correct isparity in the DSI images is marke with an arrow. 3 Improve matching costs Given the above analysis, how can we esign a better initial matching cost? Birchfiel an Tomasi [3] an Shimizu an Okutomi [14] have both observe problems with integral DSI sampling an have propose ifferent methos to overcome this problem. Birchfiel an Tomasi s sampling-insensitive issimilarity measure compares each piel in the reference image against the linearly interpolate signal in the matching image, an takes the minimum square error as the matching cost. It then reverses the role of the reference an matching images, an takes the minimum of the resulting two cost measures. In terms of our continuous DSI analysis, this is equivalent to sampling the DSI at integral locations, an computing the minimum value vertically an iagonally aroun each integral value, base on a piecewise linear reconstruction of the DSI from integral samples. Shimizu an Okutomi [14] compute cancellation costs using half-piel interpolate signals, an a these to the original cost measure to reuce the bias towars integral estimates (which they call piel locking ). 6

8 >0 =0 (a) (b) (c) Figure 3: Interval analysis: (a b) two signals with their corresponing half-sample intervals; (c) three intervals being compare (ifference). The ifference between the first two intervals is =0because their ranges overlap. The ifference between the secon two intervals is >0, i.e., the ifference between the nearest two points in the two intervals. In this paper, we efine a family of improve matching costs incluing generalizations of Birchfiel an Tomasi s matching measure. 3.1 Symmetric matching of interpolate signals First, we interpolate both signals up by a factor s using an arbitrary interpolation filter. In this paper, we stuy linear (o =1) an cubic (o =3) interpolants. (The cubic interpolant is a compact approimation to a sinc filter [15] an is often use as the stanar high-quality interpolant in many image-processing applications.) We then compute the square ifferences between all of the interpolate an shifte samples, as oppose to just between the original left (reference) image piels an the interpolate an shifte right (matching) image samples. This ifference signal is then reuce back to the original horizontal image sampling rate (i.e., to a single value per original piel) using a symmetric bo (moving average) filter of with s an then ownsampling. A higherorer filter coul potentially be use, but we wish to keep iscontinuities in epth sharp in the DSI, so we prefer a simple bo filter. 3.2 Interval matching If we wish to apply the iea of a sampling-insensitive issimilarity measure [3], we can still o this on the interpolate signals before ownsampling. However, rather than treating the reference an matching images asymmetrically an then reversing the roles of reference an matching (as in [3]), we have evelope the following relate variant that is base on interval analysis. Figure 3 shows two signals that have been interpolate to yiel the set of iscrete intensity samples shown as vertical lines. (A piecewise linear interpolant is use here since we epect the original interpolation stage to take care of aliasing.) The original Birchfiel-Tomasi measure com- 7

9 pares a piel in the reference image with the interval in the matching image efine by the center piel an its two 1/2-sample interpolate values (rectangular boes in Figure 3a b). (This ifference is 0 if the piel falls within the intervals, else it is the smaller of the ifferences from the two enpoints.) It then performs this same computation switching the reference an matching images, an takes the minimum of the resulting two costs. Our version of the algorithm simply compares the two intervals, one from the left scanline, the other from the right, rather than comparing values against intervals. The unsigne ifference between two intervals is trivial to compute: it is 0 if the intervals overlap (Figure 3c), else it is the gap between the two intervals. A signe ifference coul also be obtaine by keeping track of which interval is higher, but in our case this is unnecessary since we square the ifferences after computing them. When working with color images, we currently apply this interval analysis to each color ban separately. In principle, the same sub-piel offset shoul be use for all three channels, but the problem then becomes a more complicate quaratic minimization problem instea of simple interval analysis. 3.3 Local minimum fining (quaratic fit) An alternative to oing such interval analysis is to irectly compute the square ifferences, an to then fit a parabola to the resulting sample DSI. This is a classic approach for obtaining subpiel isparity estimates [18, 1, 10], although applying it irectly to integer-value isplacements (isparities) can lea to severe biases [14]. When the DSI has been aequately sample, however, this is a useful alternative for estimating the analytic minimum from the (fractionally) sample DSI. Note that we use the parabola fit here not to obtain sub-piel isparities, but rather to reconstruct the minimum DSI value, i.e., the actual smallest matching cost in the vicinity of the sample value. In orer to reuce the noise in the DSI before fitting, we apply spatial aggregation (averaging with neighbors) first. In this paper, we use fie uniformly weighte square winows (i.e., bo filters), which perform well in teture areas, as long as the winow oes not strale a epth bounary. While the use of shiftable winows (winows offset from the center piel) [13] can improve the performance of matching near epth iscontinuities, it makes the analysis more ifficult, an is not the main focus of our paper. 8

10 3.4 Collapsing the DSI Finally, once the local minima in the DSI at all piels have been aequately moele, we can collapse the DSI back to an integral sampling of isparities. This step is often not necessary, as many stereo matchers o their optimization at sub-piel isparities. It oes, however, have several potential avantages: For optimization algorithms like graph cuts [6] where the computation compleity is proportional to the square of the number of isparity levels, this can lea to significant performance improvements. Certain symmetric matching algorithm (e.g., ynamic programming) require an integral sampling of isparity to establish two-way optima. To collapse the DSI, we fin the lowest matching score within a 1 isparity from each integral 2 isparity, using the results of the parabolic fitting, if it was use. We also store the relative offset of this minimum from the integral isparity for future processing an for outputting a final highaccuracy isparity map, as well as the local certainty in the match, which can be etermine from the parabolic fit [1, 10]. Alternately, sub-piel estimates coul be recompute at the en aroun each winning isparity using one of the techniques escribe in [18], e.g., using a Lucas-Kanae graient-base fit [9] to nearby piels at the same isparity. 4 Eperimental evaluation of matching costs Since there are so many alternatives possible for computing the DSI, how o we choose among them? From theoretical arguments, we know that it is better to sample the DSI at fractional isparities an to interpolate the resulting surface when looking for local minima. However, real images have noise an other artifacts such as aliasing an epth iscontinuities. We therefore evaluate our new techniques using the Sawtooth, Tsukuba, an Venus stereo test sequences with groun truth from [13], which are available at Two of these sequences are shown in Figure 4a. We shoul note that the Sawtooth an Venus ata sets have high-quality sub-piel accurate groun-truth estimates (compute using piecewise planar surface fitting), while the Tsukuba groun truth has only integer isparities. We are not using the Map ata set, which can be solve almost perfectly, making it ill-suite for comparing ifferent matching costs. In this paper, we focus on the accuracy of these techniques in unocclue teture areas. The effect of ifferent matching costs in tetureless areas is harer to evaluate, since the results epen 9

11 (a): Tsukuba Venus (b): groun truth isparity maps an teture maps (c): traitional integer-isparity SSD isparities an errors (s=1, i=0, SD) (): Birchfiel-Tomasi isparities an errors (s=1, i=0, BT) (e): half-piel symmetric interval-ifference results (s=2, i=1, ID) Figure 4: Test images an selecte results: (a) input images; (b) true isparity maps an teture nonocclue regions (shown in black) in which errors are being evaluate; (c) traitional SSD results (isparity maps an error maps); () Birchfiel-Tomasi results; (e) fractional isparities with symmetric matching an interval ifference. The error maps in (c e) show in black the ba matches in teture, nonocclue regions, i.e., piels whose floating point isparity iffers from the groun truth by more than 1. 10

12 strongly on the aggregation or global optimization algorithm. We therefore restrict our analysis to teture areas an use a simple winow-base corresponence algorithm. Unteture areas can be hanle in many ways; for eample, after establishing certain matches in teture areas, unteture areas can be fille in using iffusion [12], aggregation with successively larger winows [16], or global optimization methos [6]. We also eclue piels near epth iscontinuities, which present problems for winow-base methos. For our analysis, we select teture piels by computing the square horizontal graient at each piel (averaging the left an right values to remain symmetrical). These values are then average in a 3 3 neighborhoo an threshole, using a threshol of 6 gray levels square. Occlue piels are foun by forwar-warping the true isparity maps, an epth iscontinuity regions are selecte by ilating the locations of strong isparity jumps an occlusion [13]. The resulting teture unocclue piels are shown as black piels in Figure 4b. The parameters that we vary in our eperiments are as follows: s =1, 2, 4: interpolation rate (inverse of fractional isparity); o =1, 3: interpolation orer linear or cubic; i =1, 0: symmetric matching of interpolate scanlines (Section 3.1) on or off; = SD, ID, BT: issimilarity metric (Section 3.2) square ifferences, interval ifferences, or Birchfiel-Tomasi measure; f =1, 0: parabola fit for minimum cost estimation (Section 3.3) on or off. We have also varie other parameters, incluing winow size (which is 7 7 in all eperiments reporte here), an using absolute (rather than square) ifferences. The effect of changing these parameters is iscusse below. In orer to be able to evaluate subpiel isparity performance, we o not collapse the DSI to an integer sampling in this stuy. The statistics we gather for each eperiment are the RMS isparity errors an the percentage of ba matches, i.e., piels whose floating point isparity iffers from the groun truth by more than 1. Table 1 shows the numerical results of some of our eperiments. The top three rows list the ifferent values of parameters s, i, an ; the other parameters are hel constant at o =3(cubic interpolation), f =0(no cost fitting), an a winow size of 7 7. Each of the remaining five rows compares the matching performance uner the ifferent parameter settings in the teture, unocclue regions of a given ata set. The lowest score for each ata set is highlighte in bolface. While there is no single setting that consistently outperforms the others, our new cost variants generally o better than the original costs. We now evaluate the effect of the ifferent parameters, 11

13 Interpolation rate s : s =1 s =2 s =4 Symmetric matching i : Dissimilarity metric : SD ID BT SD ID SD ID SD ID SD ID Ba Sawtooth piel Tsukuba % Venus RMS Venus error Venus / subpi Table 1: Matching performance as a function of parameters s, i, an. (SD=square ifferences, ID=interval ifferences, BT=Birchfiel-Tomasi.) Parameters o an f are hel constant at o =3(cubic interpolation) an f =0(no cost fitting). The mile three rows show the percentages of ba matching piels for the three ata sets teste; the last two rows show the RMS isparity errors for the Venus ata set without an with a final subpiel fitting step. The lowest number in each row is highlighte in bolface. The unerline numbers correspon to the results shown in Figure 4. focusing first on the ba piel percentages, which give a goo inication of the overall performance of the ifferent cost variants. SD vs. ID Interval ifferences outperform square ifferences on the Tsukuba an Venus ata sets accross other parameter variations. On the Sawtooth images, however, they result in ecrease performance. Careful analysis of the images an the error maps reveals that there is a small vertical misregistration present in the original images, an that the errors occur in areas with near-horizontal lines. This suggest that interval ifferences are a goo iea, but may amplify aliasing problems cause by misaligne images. For comparison, we have inclue the results for the Birchfiel-Tomasi measure (BT, thir column). Given integer sampling (s =1), BT an ID yiel similar results; using interpolation (iscusse net), however, we can clearly improve upon BT s performance. Interpolation rate s Interpolating the images (s =2an s =4) yiels an obvious improvement over integer-base costs (s=1), verifying our theoretical results from Sections 2 an 3. Quarter-piel steps (s =4) perform similar to half-piel steps (s =2); the numbers are slightly better on Sawtooth an Venus images, but worse on Tsukuba (reasons for this anomaly are iscusse below). This suggests that for most practical applications half-piel steps are sufficient, as long as a goo interpolant (e.g., cubic) is use. Symmetry (i =1vs. i =0) Symmetric matching (i =1) yiels slightly better results in most cases. 12

14 Figure 4c e shows the results corresponing to the unerline numbers in the first (s=1, SD), thir (s =1, BT), an seventh (s =2, i =1, ID) column of Table 1. Note that the seventh column consistently outperforms columns 1 3. Given the small ifferences in the numerical scores, the question arises whether the results are statistically significant. Careful eamination of the error maps for the ifferent parameter settings (incluing those in Figure 4c e) shows that our new costs o inee result in a significant reuction of errors in high-frequency image regions, as preicte by our theoretical analysis. This is most apparent for the Venus images, which contain many such regions. Errors are also reuce in other areas affecte by aliasing, such as strong intensity iscontinuities or near-horizontal eges. Other errors, however, are not a irect result of the matching cost, an can obscure the numerical results. The Tsukuba images in particular contain fewer high-requency regions, but several areas with repetitive patterns an fine isparity variations that are challenging for a winow-base metho, an thus result in spurious errors that are not irectly a function of the matching cost use. Although not shown in Table 1, we have also analyze the effect of changing other parameters: Using linear interpolation (o =1) gives clearly inferior results than cubic interpolation, again valiating our observations from Section 3. Refining the cost values by local fitting (f =1) results in minor ifferences, an oes not yiel a clear improvement. Using absolute ifferences rather than square ifferences yiels comparable results, an in some cases even small improvements. Decreasing the winow size increases the errors overall, but oes not significantly change the relative performance of the ifferent matching cost variants. An interesting question is to what etent the new cost variants improve the quality of subpiel isparity estimation. The last two rows of Table 1 show that the RMS isparity errors on the Venus ata set ecrease slightly when subpiel (floating-point) isparity estimation is turne on (using a stanar parabola fit aroun the winning cost values). Note that the RMS numbers are contaminate by gross errors; visual inspection of the isparity maps shows an obvious improvement over the typical stair-casing effect ehibite by our iscrete matching algorithm (which is noticeable even at quarter-piel steps). In summary, it can be seen that symmetric interpolate matching (i =1an s =2or s = 4) usually outperforms traitional, integer-base matching, in particular in high-frequency image 13

15 regions. Cubic interpolation shoul always be use. Interval ifferences help as well, but seem less tolerant to calibration errors. The benefit of interval matching also epens on the winner selection strategy (for eample, it can cause problems for algorithms that analyze cost istributions, because goo matches often yiel a matching cost of 0). Cost refinement by parabola fitting oes not seem to increase matching performance. However, using the same fitting technique to refine the winning (half or quarter-piel) isparities into true floating-point isparities generally further reuces the remaining isparity errors an results in smoother isparity maps. 5 Conclusion In this paper we have presente novel matching costs base on interpolate image signals. The nee for such costs was motivate by a frequency analysis of the continuous isparity space image (DSI). We have eplore several symmetric cost variants, incluing a generalize version of Birchfiel an Tomasi s matching criterion [3]. While there is no clear winner among the ifferent variants, we have emonstrate that our new matching costs result in improve performance, particularly in high-frequency image regions, an that they also yiel improve subpiel isparity estimates. An interesting generalization of our approach is to use a smaller interval from each image, e.g., to only interpolate ± 1 piel away (or in general ɛ away). This coul be use to compensate for 4 small unmoele shifts in the images, e.g., resiual vertical paralla. We call this ilation of a piel value to an interval etermine by its neighbors values a partial shuffle, since it is relate to Kutulakos shuffle transform [7]. Another major irection for future work is to etermine which piels can be matche with high certainty (negligible error), an to use these matches as a set of anchors points for resolving the remaining ambiguous matching regions [5, 16, 17]. It is our hope that this approach coul be use to prouce high-quality corresponence maps without the higher computational requirements of global optimization methos. In general, we believe that paying close attention to the quality of local evience (matching costs) will play a significant role in computing high-quality stereo reconstructions. References [1] P. Ananan. A computational framework an an algorithm for the measurement of visual motion. International Journal of Computer Vision, 2(3): , January

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