CONTACT: JBARRAL@STANFORD.EDU

Lucas-Kanade Algorithm for Displacement Extraction from NavigatorsJ.K. Barral1, M. Lustig1,2, and D.G. Nishimura1

1Electrical Engineering, Stanford University, Stanford, CA2Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley, CA

INTRODUCTIONSeveral algorithms have been proposed to extract displacement information from Cartesian nav-igators. The Ahn-Cho algorithm (AC) is extremely fast but lacks accuracy [1]. The Cross-Correlation (XC) [2] and Least-Squares (LS) algorithms are equivalent when the projection profilesare not truncated and all data points are used. XC is often preferred, as it avoids the exhaustivegrid search necessary with LS and is therefore much faster. However, when projection profiles aretruncated, LS is more robust against noise and profile deformation [3, 4].

The Lucas-Kanade algorithm (LK) is a nonlinear registration algorithm that has been widely usedfor optical flow estimation and stereo vision [5]. Like LS, it minimizes the dissimilarity betweentwo signals, but it is much faster as it avoids any exhaustive search.

This work proposes the use of LK as an efficient way to extract displacements from navigators.The performance of LK as compared to AC, LS, and XC is demonstrated in simulations. Thealgorithm is then applied to in vivo larynx data.

THEORYLet us call S a given profile, N its number of points, and Sref the reference profile. The projectionprofile S[x] (in image-space) is obtained from the k-space navigator profile S[k] through a centered1D-FFT.

LK minimizes the dissimilarity ∑N/2−1x=−N/2 (Sref [x]− S[x + d])2 over d [5]. At each iteration, S[x+d]

is linearized as S[x] + d∂S∂x[x], the derivative is estimated using a centered difference scheme, and

dLK is obtained as: dLK = (∂S∂x)′(Sref − S)/‖∂S

∂x‖22, where ′ denotes the complex conjugate trans-

pose. Sref is then translated by dLK (using phase modulation in k-space), dLK is updated, and theprocedure is repeated for r iterations.

The linearization of S[x + d] assumes small displacements. To extend the range of displacementsaccurately estimated with LK, we further propose a hybrid algorithm, where LK is initialized withAC. The initialization also speeds up the convergence of LK.

LK can be generalized to handle multi-coil data by minimizing the total dissimilarity over d. Al-ternatively, the projection profiles can be combined before processing.

METHODSIMPLEMENTATION AC, LS, XC, and LK were implemented using Matlab. In this work,the projection profiles are not truncated, and no cropping is necessary. We also assume that thenavigator acquisition has been optimized as to provide navigators with adequate signal-to-noiseratio (SNR), and no smoothing of the profiles is performed.

SIMULATIONS The different algorithms were compared using Monte-Carlo simulations.An in vivo projection profile with 256 points was taken as reference. Displaced profiles werecomputed using phase modulation in k-space. Circularly Gaussian-distributed, zero-mean, whitecomplex noise was added independently to the reference profile and the displaced profiles. Thereference profile was scaled to peak amplitude 1 and contrast-to-noise ratio (CNR) was defined asthe inverse of the noise standard deviation [6]. For LS, a precision of 0.1 pixel was chosen, with agrid spanning 30 pixels. For LK, r = 20 iterations was used, unless mentioned otherwise.

IN VIVO EXPERIMENTS A patient with a chondrosarcoma of the larynx was scanned ona 1.5 T GE system. A FLASE sequence [7] was used with the following parameters: TR = 80 ms,TE = 10 ms, flip angle = 140◦, BW = ±32 kHz, FOV = 12×6 cm2 , slice thickness = 2 mm, matrixsize = 256×128×32, frequency direction = Left/Right. Cartesian navigators were acquired in allthree directions, in an interleaved manner. ±32-kHz bandwidth and 256 sampling points wereused, with a FOV of 12 cm. Scan time was 5 min 34. The patient was instructed to remain asstill as possible. A dedicated three-coil array was used [8] and projection profiles were combinedusing sum-of-squares before processing. For LS and XC, a precision of 0.1 pixels was chosen,with a grid spanning 10 pixels. For LK, r = 5 iterations was used. The middle profile was chosenas reference.

RESULTS AND DISCUSSION

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Fig. 4: Comparison of LS, XC, and LK on Patient Data.Note a global shift to which is superimposed the respiratorycomponent (TR = 80 ms).

SIMULATIONS Figure 1 displaysthe root-mean-square error (RMSE) vs.CNR for the different algorithms when arelatively large displacement of 7.31 pixelsis used. For each CNR level (1000 simula-tions), AC took 0.046 s, LS 94 s, XC 1.6 s,and LK 5.4 s. At low CNR levels, when 20iterations are used, LK lacks accuracy un-less initialized with AC (hybrid algorithm).

Figure 2 presents the distributions of dis-placement estimates vs. true displacementsfor a CNR level of 5. As can be seen, theaccuracy of LK degrades as the displace-ment gets larger, and the algorithm even-tually fails. LK initialized with AC avoidsthis problem.

Figure 3 shows that, as expected, LK con-verges faster when CNR increases (dis-placement = 7.31 pixels). At all CNR lev-els, initializing LK with AC speeds up con-vergence. Equivalently, for a given numberof iterations, the hybrid algorithm providesbetter accuracy than LK.

IN VIVO EXPERIMENTSFigure 4 shows that LS, XC, and LK givevery similar displacement estimates on pa-tient data. LS took 37 s, XC 1.9 s, and LK1.5 s (1024 profiles).

Figure 5 compares image quality withoutcorrection and with correction when LS orLK are used to extract displacement infor-mation from navigators. Severe blurring and ghosting artifacts can be seen in the uncorrectedimage. Blurring artifacts are removed upon corrrection, and ghosting artifacts are attenuated. Im-age quality improves significantly. Non-rigid motion is not accounted for and remaining artifactsare expected. As anticipated from Fig. 4, LS and LK lead to images without visible differences.

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Fig. 5: FLASE Larynx Images (resolution = 0.5 × 0.5 × 2 mm3): no correction (a) and correction when comput-ing displacements in all three directions with LS (b) and LK (c). The sharpness of subcutaneous fat (F) and thyroidcartilage (C) is restored upon correction.

In summary, the Lucas-Kanade algorithm is a simple, fast, and accurate method to extract displace-ments from Cartesian navigators. With truncated profiles, LK is expected to perform similarly toLS and therefore outperform XC [3, 4]. This remains to be validated for specific applications.[1] C. B. Ahn and Z. H. Cho. IEEE Trans Med Imaging, 6(1):32–36, 1987.[2] R. L. Ehman and J. Felmlee. Magn Reson Imaging, 173:255–263, 1989.[3] Y. Wang, R. C. Grimm, J. P. Felmlee, S. J. Riederer, and R. L. Ehman. Magn Reson Med, 36(1):117–123, 1996.[4] E. Trucco and A. Verri. Introductory techniques for 3-D computer vision. Prentice Hall, Upper Saddle River, NJ, 1998.[5] B. D. Lucas and T. Kanade. In Proc 7th Intl Joint Conf on Artificial Intelligence, pages 676–679, Vancouver, BC, 1981.[6] T. D. Nguyen, Y. Wang, R. Watts, and I. Mitchell. Magn Reson Med, 46(5):1037–1040, 2001.[7] J. Ma, F. W. Wehrli, and H. K. Song. Magn Reson Med, 35(6):903–910, 1996.[8] J. K. Barral, H. H. Wu, E. J. Damrose, N. J. Fischbein, and D. G. Nishimura. In Proceedings of the 17th Annual Meeting of ISMRM, page 1318, Honolulu, HI, 2009.