1 Introduction
Optical flow estimation has been widely applied to computer vision applications, e.g. segmentation, image deblurring and stabilization, etc. In many cases, optical flow is often estimated on the videos captured by a shaking camera. Those footages may contain a significant amount of camera blur that bring additional difficulties into the traditional variational optical flow framework. It is because such blur scenes often lead to a fact that a pixel may match multiple pixels between image pair. It further violates the basic assumption – intensity constancy – of the optical flow framework.
In this paper, we investigate the issue of how to precisely estimate optical flow from a blurry video footage. We observe that the blur kernel between neighboring frames may be near linear, which can be parameterized using linear elements of the camera motion. In this case, the camera trajectory can be informatic to enhance the image deblurring within a variational optical flow framework. Based on this observation, our major contribution in this paper is to utilise an RGBMotion Imaging System – an RGB sensor combined with a 3D pose&position tracker – in order to propose: (A) an iterative enhancement process for camera shake blur estimation which encompasses the tracked camera motion (Sec. 3) and a Directional Highpass Filter (Sec. 4 and Sec. 7.2); (B) a BlurRobust Optical Flow Energy formulation (Sec. 6); and (C) a hybrid coarsetofine framework (Sec. 7) for computing optical flow in blur scenes by interleaving an iterative blind deconvolution process and a warping based minimisation scheme. In the evaluation section, we compare our method to three existing stateoftheart optical flow approaches on our proposed ground truth sequences (Fig. 1, blur and baseline blurfree equivalents) and also illustrate the practical benefit of our algorithm given realworld cases.
2 Related Work
Camera shake blur often occurs during fast camera movement in lowlight conditions due to the requirement of adopting a longer exposure. Recovering both the blur kernel and the latent image from a single blurred image is known as Blind Deconvolution which is an inherently illposed problem. Cho and Lee FMD propose a fast deblurring process within a coarsetofine framework (Cho&Lee) using a predicted edge map as a prior. To reduce the noise effect in this framework, Zhong et al. Zhong introduce a prefiltering process which reduces the noise along a specific direction and preserves the image information in other directions. Their improved framework provides high quality kernel estimation with a low runtime but shows difficulties given combined object and camera shake blur.
To obtain higher performance, a handful of combined hardware and softwarebased approaches have also been proposed for image deblurring. Tai et al. tai08 introduce a hybrid imaging system that is able to capture both video at high frame rate and a blurry image. The optical flow fields between the video frames are utilised to guide blur kernel estimation. Levin et al. levin propose to capture a uniformly blurred image by controlling the camera motion along a parabolic arc. Such uniform blur can then be removed based on the speed or direction of the known arc motion. As a complement to Levin el al.’s levin hardwarebased deblurring algorithm, Joshi et al. Joshi apply inertial sensors to capture the acceleration and angular velocity of a camera over the course of a single exposure. This extra information is introduced as a constraint in their energy optimisation scheme for recovering the blur kernel. All the hardwareassisted solutions described provide extra information in addition to the blurry image, which significantly improves overall performance. However, the methods require complex electronic setups and the precise calibration.
Optical flow techniques are widely studied and adopted across computer vision because of dense image correspondences they provide. Such dense tracking is important for other fundamental research topics e.g. 3D reconstruction reflection and visual effects lv2013game ; lv2014multimodal , etc. In the last two decades, the optical flow model has evolved extensively – one landmark work being the variational model of Horn and Schunck HS where the concept of Brightness Constancy is proposed. Under this assumption, pixel intensity does not change spatiotemporally, which is, however, often weakened in realworld images because of natural noise. To address this issue, some complementary concepts have been developed to improve performance given large displacements Brox , taking advantage of featurerich surfaces Xu_deblur and adapting to nonrigid deformation in scenes APO ; LME ; APO_JIFS ; moBlur ; tang ; li2013nonrigid . However, flow approaches that can perform well given blurred scenes – where the Brightness Constancy is usually violated – are less common. Of the approaches that do exist, Schoueri et al. Schoueri perform a linear deblurring filter before optical flow estimation while Portz et al. Portz attempt to match ununiform camera motion between neighbouring input images. Whereas the former approach may be limited given nonlinear blur in realworld scenes; the latter requires two extra frames to parameterise the motioninduced blur. Regarding non opticalflow based methods, Yuan et al. Yuan align a blurred image to a sharp one by predefining an affine image transform with a blur kernel. Similarly HaCohen et al. HaCohen achieve alignment between a blurred image and a sharp one by embedding deblurring into the correspondence estimation. Li et al. moBlur present an approach to solve the image deblurring and optical flow simultaneously by using the RGBMotion imaging.
3 RGBMotion Imaging System
Camera shake blur within video footage is typically due to fast camera motion and/or long exposure time. In particular, such blur can be considered as a function of the camera trajectory supplied to image space during the exposure time . It therefore follows that knowledge of the actual camera motion between image pairs can provide significant information when performing image deblurring Joshi ; levin .
In this paper, we propose a simple and portable setup (Fig. 2(a)), combining an RGB sensor and a 3D pose&position tracker (SmartNav by NaturalPoint Inc.) in order to capture continuous scenes (video footage) along with realtime camera pose&position information. Note that the RGB sensor could be any camera or a Kinect sensor – A Canon EOS 60D is applied in our implementation to capture video at frame rate of 24 FPS. Furthermore, our tracker is proposed to provide the rotation (yaw, pitch and roll), translation and zoom information within a reasonable error range (2 mm). To synchronise this tracker data and the image recording, a real time collaboration (RTC) server JeeHang is built using the instant messaging protocol XMPP (also known as Jabber^{1}^{1}1http://www.jabber.org/) which is designed for messageoriented communication based on XML, and allows realtime responses between different messaging channels or any signal channels that can be transmitted and received in message form. In this case, a time stamp is assigned to the received message package by the central timer of the server. Those message packages are synchronised if they contain nearly the same time stamp. We consider the Jabber for synchronisation because of its opensource nature and the low respond delay (around 10 ms).
Assuming objects have similar depth within the same scene (a common assumption in image deblurring which will be discussed in our future work), the tracked 3D camera motion in image coordinates can be formulated as:
(1) 
where represents the average of the camera motion vectors from the image to image . X denotes the 3D position of the camera while is a pixel location and represents the number of pixels in an image. represents the 3D projection matrix while and denote the rotation and translation matrices respectively of tracked camera motion in the image domain. All these information , and is computed using Optitrack’s Camera SDK^{2}^{2}2http://www.naturalpoint.com/optitrack (version 1.2.1). Fig 2(b,c) shows sample data (video frames and camera motion) captured from our imaging system. It is observed that blur from the realworld video is near linear due to the relatively high sampling rate of the camera. The blur direction can therefore be approximately described using the tracked camera motion. Let the tracked camera motion be represented in polar coordinates where and denote the magnitude and directional component respectively. is a sharing index between tracked camera motion and frame number. In addition, we also consider the combined camera motion vector of neighbouring images as shown in Fig 2(d), e.g. where denotes the combined camera motion vector from image 1 to image 3. As one of our main contributions, these realtime motion vectors are proposed to provide additional constraints for blur kernel enhancement (Sec. 7) within our framework.
4 Blind Deconvolution
The motion blur process can commonly be formulated:
(2) 
where is a blurred image and represents a blur kernel w.r.t. a specific Point Spread Function. is the latent image of ; denotes the convolution operation and represents spatial noise within the scene. In the blind deconvolution operation, both and are estimated from , which is an illposed (but extensively studied) problem. A common approach for blind deconvolution is to solve both and in an iterative framework using a coarsetofine strategy:
(3)  
(4) 
where represents a regularization that penalizes spatial smoothness with a sparsity prior FMD , and is widely used in recent stateoftheart work Shan ; Xu_deblur . Due to noise sensitivity, lowpass and bilateral filters Tai are typically employed before deconvolution. Eq. 5 denotes the common definition of an optimal kernel from a filtered image.
(5) 
where represents the ground truth blur kernel, is a filter, and denotes the optimal blur kernel from the filtered image . The lowpass filtering process improves deconvolution computation by removing spatiallyvarying high frequency noise but also results in the removal of useful information which yields additional errors over object boundaries. To preserve this useful information, we introduce a directional highpass filter that utilises our tracked 3D camera motion.
5 Directional Highpass Filter
Detail enhancement using directional filters has been proved effective in several areas of computer vision Zhong . Here we define a directional highpass filter as:
(6) 
where represents a pixel position and denotes a 1D Gaussian based highpass function. controls the filtering direction along . is a normalization factor defined as . The filter is proposed to preserve overall high frequency details along direction without affecting blur detail in orthogonal directions Chen . Given a directionally filtered image , the optimal blur kernel is defined (Eq 5) as . Fig. 3 demonstrates that noise or object motion within a scene usually results in low frequency noise in the estimated blur kernel (Cho&Lee FMD ). This low frequency noise can be removed by our directional highpass filter while preserving major blur details. In our method, this directional highpass filter is supplemented into the Cho&Lee FMD framework using a coarsetofine strategy in order to recover high quality blur kernels for use in our optical flow estimation (Sec. 7.2).
6 BlurRobust Optical Flow Energy
Within a blurry scene, a pair of adjacent natural images may contain different blur kernels, further violating Brightness Constancy. This results in unpredictable flow error across the different blur regions. To address this issue, Portz et al. proposed a modified Brightness Constancy term by matching the ununiform blur between the input images. As one of our main contributions, we extend this assumption to a novel Blur Gradient Constancy
term in order to provide extra robustness against illumination change and outliers. Our main energy function is given as follows:
(7) 
A pair of consecutively observed frames from an image sequence is considered in our algorithm. represents the current frame and its successor is denoted by where and represent rectangular images in the RGB channel. Here is latent image and denotes the relative blur kernel. The optical flow displacement between and is defined as . To match the ununiform blur between input images, the blur kernel from each input image is applied to the other. We have new blur images and as follows:
(8)  
(9) 
Our energy term encompassing Brightness and Gradient Constancy relates to and as follows:
(10) 
The term presents a spatial gradient and denotes a linear weight. The smoothness regulariser penalizes global variation as follows:
(11) 
where we apply the Lorentzian regularisation to both the data term and smoothness term. In our case, the image properties, e.g. small details and edges, are broken by the camera blur, which leads to additional errors in those regions. We suppose to apply strong boundary preservation even the nonconvex Lorentzian regularisation may bring the extra difficulty to the energy optimisation (More analysis can be found in Li et al. LME ). In the following section, our optical flow framework is introduced in detail.
7 Optical Flow Framework
Our overall framework is outlined in Algorithm 1 based on an iterative topdown, coarsetofine strategy. Prior to minimizing the BlurRobust Optical Flow Energy (Sec. 7.4), a fast blind deconvolution approach FMD is performed for preestimation of the blur kernel (Sec. 7.1), which is followed by kernel refinement using our Directional Highpass Filter (Sec. 7.2). All these steps are detailed in the following subsections.
7.1 Iterative Blind Deconvolution
Cho and Lee FMD describe a fast and accurate approach (Cho&Lee) to recover the unique blur kernel. As shown in Algorithm 1, we perform a similar approach for the preestimation of the blur kernel within our iterative process, which involves two steps of prediction and kernel estimation. Given the latent image estimated from the consecutively coarser level, the gradient maps of are calculated along the horizontal and vertical directions respectively in order to enhance salient edges and reduce noise in featureless regions of . Next, the predicted gradient maps as well as the gradient map of the blurry image are utilised to compute the preestimated blur kernel by minimizing the energy function as follows:
(12) 
where denotes the weight of Tikhonov regularization and represents a linear weight for the derivatives in different directions. Both and are propagated from the nearest coarse level within the pyramid. To minimise this energy Eq. (12), we follow the inneriterative numerical scheme of FMD which yields a preestimated blur kernel .
7.2 Directional Highpass Filtering
Once the preestimated kernel is obtained, our Directional Highpass Filters are applied to enhance the blur information by reducing noise in the orthogonal direction of the tracked camera motion. Although our RGBMotion Imaging System provides an intuitively accurate camera motion estimation, outliers may still exist in the synchronisation. We take into account the directional components of two consecutive camera motions and as well as their combination (Fig. 2(d)) for extra robustness. The preestimated blur kernel is filtered along its orthogonal direction as follows:
(13) 
where linearly weights the contribution of filtering in different directions. Note that two consecutive images and are involved in our framework where the former accepts the weight set while the other weight set is performed for the latter. This filtering process yields an updated blur kernel which is used to update the latent image within a nonblind deconvolution Zhong . Note that the convolution operation is computationally expensive in the spatial domain, we consider an equivalent filtering scheme in the frequency domain in the following subsection.
7.3 Convolution for Directional Filtering
Our proposed directional filtering is performed as convolution operation in the spatial domain, which is often highly expensive in computation given large image resolutions. In our implementations, we consider a directional filtering scheme in the frequency domain where we have the equivalent form of filtering model Eq. (6) as follows:
(14) 
where is the optimal blur kernel in the frequency domain while and present the Fourier Transform of the blur kernel and our directional filter respectively. Thus, the optimal blur kernel in the spatial domain can be calculated as using Inverse Fourier Transform. In this case, the equivalent form of our directional highpass filter in the frequency domain is defined as follows:
(15) 
where the line function controls the filtering process along the direction while
is the standard deviation for controlling the strength of the filter. Please note that other more sophisticated highpass filters could also be employed using this directional substitution
. Even though this consumes a reasonable proportion of computer memory, convolution in the frequency domain is faster than equivalent computation in the spatial domain .Having performed blind deconvolution and directional filtering (Sec. 7.1, 7.2 and 7.3), two updated blur kernels and on the th level of the pyramid are obtained from input images and respectively, which is followed by the uniform blur image and computation using Eq. (9). In the following subsection, BlurRobust Optical Flow Energy optimisation on and is introduced in detail.
7.4 Optical Flow Energy optimisation
As mentioned in Sec. 6, our blurrobust energy is continuous but highly nonlinear. minimisation of such energy function is extensively studied in the optical flow community. In this section, a numerical scheme combining EulerLagrange Equations and Nested Fixed Point Iterations is applied Brox to solve our main energy function Eq. 7. For clarity of presentation, we define the following mathematical abbreviations:
At the first phase of energy minimization, a system is built based on Eq. 7 where EulerLagrange is employed as follows:
(16)  
(17) 
An level image pyramid is then constructed from the top coarsest level to the bottom finest level. The flow field is initialized as on the top level and the outer fixed point iterations are applied on w. We assume that the solution converges on the level. We have:
(18)  
(19) 
Because of the nonlinearity in terms of , , the system (Eqs. 18, 19) is difficult to solve by linear numerical methods. We apply the first order Taylor expansions to remove these nonlinearity in , which results in:
Based on the coarsetofine flow assumption of Brox et al. Brox w.r.t. and where the unknown flow field on the next level can be obtained using the flow field and its incremental from the current level . The new system can be presented as follows:
(20)  
(21) 
where the terms and contained provide robustness to flow discontinuity on the object boundary. In addition, is also regularizer for a gradient constraint in motion space. Although we fixed in Eqs. 20 and 21, the nonlinearity in leads to the difficulty of solving the system. The inner fixed point iterations are applied to remove this nonlinearity: and are assumed to converge within iterations by initializing and . Finally, we have the linear system in and as follows:
(22)  
(23) 
where denotes a robustness factor against flow discontinuity and occlusion on the object boundaries. represents the diffusivity of the smoothness regularization.
7.5 Alternative Implementation with Automatic Camera Motion Estimation
Alternative to using our assisted tracker, we also provide an additional implementation by using the camera motion estimated generically from the flow field. As shown in Algorithm 2, the system does not take the camera motion () as input but computes it (CameraMotionEstimation) generically at every level of the image pyramid.
(24) 
On each level, we calculate the Affine Matrix from to using the correspondences and RANSAC. The translation information from is then normalized and converted to the angle format . In this case, our is also downgraded to consider one direction for each level. In the next section, we quantitatively compare our method to other popular baselines.
8 Evaluation
In this section, we evaluate our method on both synthetic and realworld sequences and compare its performance against three existing stateoftheart optical flow approaches of Xu et al.’s MDP Xu_deblur , Portz et al.’s Portz and Brox et al.’s Brox (an implementation of liu ). MDP is one of the best performing optical flow methods given blurfree scenes, and is one of the top 3 approaches in the Middlebury benchmark Middlebury . Portz et al.’s method represents the current stateoftheart in optical flow estimation given object blur scenes while Brox et al.’s contains a similar optimisation framework and numerical scheme to Portz et al.’s, and ranks in the midfield of the Middlebury benchmarks based on overall average. Note that all three baseline methods are evaluated using their default parameters setting; all experiments are performed using a 2.9Ghz Xeon 8cores, NVIDIA Quadro FX 580, 16Gb memory computer.
In the following subsections, we compare our algorithm (moBlur) and four different implementations (auto, nonGC, nonDF and nonGCDF) against the baseline methods. auto denotes the implementation using the automatic camera motion estimation scheme (Algorithm 2); nonGC represents the implementation without the Gradient Constancy term while nonDF denotes an implementation without the directional filtering process. nonGCDF is the implementation with neither of these features. The results show that our BlurRobust Optical Flow Energy and Directional Highpass Filter significantly improve algorithm performance for blur scenes in both synthetic and realworld cases.
8.1 Middlebury Dataset with camera shake blur
One advance for evaluating optical flow given scenes with object blur is proposed by Portz et al. Portz where synthetic Ground Truth (GT) scenes are rendered with blurry moving objects against a blurfree static/fixed background. However, their use of synthetic images and controlled object trajectories lead to a lack of global camera shake blur, natural photographic properties and real camera motion behaviour. To overcome these limitations, we render four sequences with camera shake blur and corresponding GT flowfields by combining sequences from the Middlebury dataset Middlebury with blur kernels estimated using our system.
In our experiments we select the sequences Grove2, Hydrangea, RubberWhale and Urban2 from the Middlebury dataset. For each of them, four adjacent frames are selected as latent images along with the GT flow field (supplied by Middlebury) for the middle pair. blur kernels are then estimated FMD from realworld video streams captured using our RGBMotion Imaging System. As shown in Fig. 4, those kernels are applied to generate blurry images denoted by , , and while the camera motion direction is set for each frame based on the 3D motion data. Although the between latent images can be utilised for the evaluation on relative blur images Sintel ; SintelWS , strong blur can significantly violate the original image intensity, which leads to a multiple correspondences problem: a point in the current image corresponds to multiple points in the consecutive image. To remove such multiple correspondences, we sample reasonable correspondence set to use as the GT for the blur images where denotes a predefined threshold. Once we obtain , both Average Endpoint Error (AEE) and Average Angle Error (AAE) tests Middlebury are considered in our evaluation. The computation is formulated as follows:
(25)  
(26) 
where and denotes the baseline flow field and the ground truth flow field (by removing multiple correspondences) respectively while presents the number of ground truth vectors in . The factor in AAE is an arbitrary scaling constant to convert the units from pixels to degrees Middlebury . Fig. 5(a) Left shows AEE (in pixel) and AAE (in degree) tests on our four synthetic sequences. moBlur and nonGC lead both AEE and AAE tests in all the trials. Both Brox et al. and MDP yield significant error in Hydrangea, RubberWhale and Urban2 because those sequences contain large textureless regions with blur, which in turn weakens the inner motion estimation process as shown in Fig. 5(b). Fig. 5(a) also illustrates the average time cost (second per frame) of the baseline methods. Our method gives reasonable performance (45 sec. per frame) comparing to the stateoftheart Portz et al. and MDP even an inner image deblurring process is involved. Furthermore, Fig 5(a) Right shows the AEE metric for RubberWhale by varying the distribution of Salt&Pepper noise. It is observed that a higher noise level leads to additional errors for all the baseline methods. Both moBlur and nonGC yield the best performance while Portz et al. and Brox et al. show a similar rising AEE trend when the noise increases.
Fig. 6 shows our quantitative measure by comparing our two implementations which use the RGBMotion Imaging (moBlur) and automatic camera motion estimation scheme (auto, see Sec. 7.5) respectively. For better observation, we also give the Portz et al. in this measure. We observe that both our implementations outperform Portz et al. in the AEE and AAE tests. Especially the moBlur gives the best accuracy in all trials. The implementation auto yields the less accurate results than the moBlur. It may be because the auto camera motion estimation is affected by ambiguous blur that often caused by multiple moving objects. To investigate this issue, we plot the angular error by comparing the autoestimated camera motion to the ground truth on all the sequences (Fig. 6, right end). We observe that our automatic camera motion estimation scheme leads to higher errors on the upper/coarser level of the image pyramid. Even the accuracy is improved on the finer levels but the error may be accumulated and affect the final result.
In practice, the system may be used in some challenge scenes, e.g. fast camera shaking, super high frame rate capture, or even infrared interference, etc. In those cases, the wrong tracked camera motion may be given to some specific frames. To investigate how the tracked camera motion affects the accuracy of our algorithm, we compare moBlur to nonDF (our method without directional filtering) and Portz et al. by varying the direction of input camera motion. As shown in Fig. 7(a), we rotate the input camera motion vector with respect to the GT blur direction by an angle of degrees. Here represents the ideal situation where the input camera motion has the same direction as the blur direction. The increasing simulates more errors in the camera motion estimation. Fig. 7(b,c) shows the AEE metric by increasing the . We observe that the AEE increases during this test. moBlur outperforms the nonDF (moBlur without the directional filter) in both Grove2 and RubberWhale while nonDF provides higher performance in Hydrangea when is larger than . In addition, moBlur outperforms Portz et al. in all trials except Hydrangea where Portz et al. shows a minor advantage (AEE 0.05) when . The rationale behind this experiment is that the wrong camera motion may yield significant information loss in the directional highpass filtering. Such information loss harms the deblurring process and consequently leads to errors in the optical flow estimation. Thus, obtaining precise camera motion is the essential part of this system, as well as a potential future research.
8.2 Realworld Dataset
To evaluate our method in the realworld scenes, we capture four sequences warrior, chessboard, LabDesk and shoes with tracked camera motion using our RGBMotion Imaging System. As shown in Fig. 8, both warrior and chessboard contain occlusions, large displacements and depth change while the sequences of LabDesk and shoes embodies the object motion blur and large textureless regions within the same scene. Fig. 9 shows visual comparison of our method moBlur against Portz et al. on these realworld sequences. It is observed that our method preserves appearance details on the object surface and reduce boundary distortion after warping using the flow field. In addition, our method shows robustness given cases where multiple types of blur exist in the same scene (Fig.9(b), sequence shoes).
9 Conclusion
In this paper, we introduce a novel dense tracking framework which interleaves both a popular iterative blind deconvolution; as well as a warping based optical flow scheme. We also investigate the blur papameterization for the video footages. In our evaluation, we highlight the advantages of using both the extra motion channel and the directional filtering in the optical flow estimation for the blurry video footages. Our experiments also demonstrated the improved accuracy of our method against large camera shake blur in both noisy synthetic and realworld cases. One limitation in our method is that the spatial invariance assumption for the blur is not valid in some realworld scenes, which may reduce accuracy in the case where the object depth significantly changes. Finding a depthdependent deconvolution and deep datadriven model would be a challenge for future work as well.
10 Acknowledgements
We thank Ravi Garg and Lourdes Agapito for providing their GT datasets. We also thank Gabriel Brostow and the UCL Vision Group for their helpful comments. The authors are supported by the EPSRC CDE EP/L016540/1 and CAMERA EP/M023281/1; and EPSRC projects EP/K023578/1 and EP/K02339X/1.
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