Journal of VLSI Signal Processing 42, 79–89, 2006 c 2006 Springer Science + Business Media, Inc Manufactured in The Netherlands DOI: 10.1007/s11265-005-4167-8 A Super-Resolution Imaging Method Based on Dense Subpixel-Accurate Motion Fields HA V LE Department of Electrical and Computer Engineering, Vietnam National University, Hanoi, 144 Xuan Thuy, Vietnam GUNA SEETHARAMAN Department of Electrical and Computer Engineering, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433-7765 Received September 12, 2003; Revised January 29, 2004; Accepted March 22, 2004 Abstract A super-resolution imaging method suitable for imaging objects moving in a dynamic scene is described The primary operations are performed over three threads: the computation of a dense inter-frame 2-D motion field induced by the moving objects at a sub-pixel resolution in the first thread Concurrently, each video image frame is enlarged by the cascode of an ideal low-pass filter and a higher rate sampler, essentially stretching each image onto a larger grid Then, the main task is to synthesize a higher resolution image from the stretched image of the first frame and that of the subsequent frames subject to a suitable motion compensation A simple averaging process and/or a simplified Kalman filter may be used to minimize the spatio-temporal noise, in the aggregation process The method is simple and can take advantage of common MPEG-4 encoding tools A few experimental cases are presented with a basic description of the key operations performed in the over all process Keywords: Super-resolution, motion compensation, optical flow Introduction The objective of super-resolution imaging is to synthesize a higher resolution image of objects from a sequence of images whose spatial resolution is limited by the operational nature of the imaging process The synthesis is made possible by several factors that effectively result in sub-pixel level displacements and disparities between the images Research on super-resolution imaging has been extensive in recent years Tsai and Huang were the first trying to solve the problem In [1], they proposed a frequency domain solution which uses the shifting property of the Fourier transform to recover the displacements between images This as well as other frequency domain methods like [2] have the advantages of being simple and having low computational cost However, the only type of motion between images which can be recovered from the Fourier shift is the global translation, therefore, the ability of these frequency domain methods is quite limited Motion-compensated interpolation techniques [3, 4] also compute displacements between images before integrating them to reconstruct a high resolution image The difference between these methods and the frequency domain methods mentioned above is that they work in the spatial domain Parametric models are usually used to model the motions The problem is, most parametric models are established to represent rigid motions such as camera movements, while in 80 Le and Seetharaman the real world motions captured in image sequences are often non-rigid, too complex to be described by a parametric model Model-based super-resolution imaging techniques such as back-projection [5] also face the same problem More powerful and robust methods such as the projection onto convex sets (POCS)-based methods [6], which are based on set theories, and stochastic methods like maximum a posteriori (MAP)-based [7] and Markov random field (MRF)-based [8] algorithms are highly complex in term of computations, hence unfit for applications which require real-time processing The objective of this research is a super-resolution imaging technique which is simple and fast enough to be used for camera surveillance systems requiring on-line processing We chose the motioncompensated interpolation approach because of its simplicity and low computational complexity Current motion-compensated interpolation methods suffer from the complexity of object motions captured in real-world image sequences, which makes it impossible to model the displacements with parametric models often used by these methods To overcome that problem, we proposed a technique for computing the flow fields between images The technique is fairly simple with the use of linear affine approximations, yet it is able to recover the displacements with a sub-pixel-level accuracy, thank to its multi-scale piecewise approach Our optical flow-based method assumes that the cameras not exhibit any looming effect, and there is no specular reflection over the zones covered by the objects of interest within each image We also assume there is no effect of motion blur in the images With the proliferation of cheap high-speed CMOS cameras and fast video capturing hardware, motion blur is no longer a serious problem in video image processing as it used to be We focus our experimental study on digital video images of objects moving steadily in the field of view of a camera fitted with a wide-angle lens These assumptions hold good for a class of video based security and surveillance systems Typically, these systems routinely perform MPEG analysis to produce a compressed video for storage and offline processing In this context, the MPEG subsystem can be exploited to facilitate super-resolution imaging through a piecewise affine registration process which can easily be implemented with the MPEG-4 procedures The method is able to increase the effectiveness of camera security and surveillance systems Figure The schematic block diagram of the proposed superresolution imaging method Super-Resolution Imaging Based on Motion Compensation The flow of computation in the proposed method is depicted in Fig Each moving object will be separated from the background using standard image segmentation techniques Also, a set of feature points, called the points-of-interest, will be extracted These points include places were the local contrast patterns are well defined, and/or exhibit a high degree of curvature, and such geometric features We track their motions in the 2-D context of a video image sequence This requires image registration, or some variant of point correspondence matching The net displacement of the image of an object between any two consecutive video frames will be computed with sub-pixel accuracy Then, a rigid coordinate system is associated with the first image, and any subsequent image is modeled as though its coordinate system has undergone a piecewise affine transformation We recover the piecewise affine transform parameters between any video frame with respect to the first video frame to a sub-pixel accuracy Independently, all images will be enlarged to a higher resolution using a bilinear interpolation [9] by a scale factor The enlarged image of each subsequent frame is subject to an inverse affine transformation, to help register it with the previous enlarged image Given K A Super-Resolution Imaging Method Based on Dense Subpixel Figure Graph of mean square errors between reconstructed images and the original frame is performed to find the correspondences between feature points in two consecutive image frames Piecewise flow approximation: a mesh of triangular patches is created, whose vertices are the matched feature points For each triangular patch in the first frame there is a corresponding one in the second frame The affine motion parameters between these two patches can be determined by solving a set of linear equations formed over the known correspondences of their vertices Each set of these affine parameters define a smooth flow within a local patch 3.1 video frames, then, in principle, it will be feasible to synthesize K−1 new versions of the scaled and interpolated and inverse-motion-compensated image at the first frame instant Thus, we have K high resolution images to assimilate from We follow a framework proposed by Cho et al [10] for optical flow computation based on a piecewise affine model A surface moving in the 3-D space can be modeled as a set of small planar surface patches so that projected motion of each of those 3-D planar patches in a 2-D plane between two consecutive image frames can be described by an affine transform Basically, this is a mesh-based technique for motion estimation, using 2-D content-based meshes The advantage of contentbased meshes over regular meshes is their ability to reflect the content of the scene by closely matching boundaries of the patches with boundaries of the scene features [11], yet finding feature points and correspondences between features in different frames is a difficult task A multi-scale coarse-to-fine approach is utilized in order to increase the robustness of the method as well as the accuracy of the affine approximations An adaptive filter is used to smooth the flow field such that the flow appears continuous across the boundary between adjacent patches, while the discontinuities at the motion boundaries can still be preserved Many of these techniques are already available in MPEG-4 tools Optical Flow Computation Our optical flow computation method includes the following phases: Feature extraction and matching: in this phase the feature points are extracted and feature matching 81 The Multi-Scale Approach Affine motion is a feature of the parallel projection, yet it is common even in applications using the perspective imaging model to use a 2-D affine transform to approximate the 2-D velocity vector field produced by a small planar surface patch moving rigidly in the 3-D space, since the quadratic terms of the motion in such a case are very small A curved surface can be approximated with a set of small planar surface patches, then the motion of the curved surface can be described by a piecewise set of affine transforms, one for each planar patch, even if the surface is non-rigid, because a nonrigid surface can be approximated with a set of small rigid patches The more number of patches are used, the more accurate the approximation is Therefore, it is obvious that we would like to create the mesh in each image frame using as many feature points as possible The problem is, when the set of feature points in each frame is too dense, finding correspondences between points in two consecutive frames is very difficult, especially when the displacements are relatively large Our solution for this problem is a multi-scale scheme It starts at a coarse level with only a few feature points, so matching them is fairly simple A piecewise set of affine motion parameters, which gives an approximation of the motion field, is computed from these matching points At the next finer scale, more feature points are extracted Each of the feature points in the first frame has a target in the second frame, which is given by an affine transform estimated in the previous iteration To find a potential match for a feature point in the first frame, the algorithm has to consider only those feature points in the second frame, which are close to its target point This iterative process guarantees convergence, i.e the errors of the piecewise affine approximations get smaller after each iteration 82 3.2 Le and Seetharaman Feature Point Extraction 3.3 As we mentioned earlier, edge and corner points are the most commonly-used features for motion estimation methods which require feature matching It is due to the availability of numerous advanced techniques for edge and corner detection Besides, it has been known that most of the optical flow methods are bestconditioned at edges and edge corners We follow the suit by looking for points located at curved parts (corners) of edges Edge points are identified first by using Canny edge detection method Canny edge detector [12] applies a low-pass filter on the input image, then performs non-maxima suppression along the gradient direction at each potential edge point to produce thin edges Note that the scale of this operation is specified by the width σ e of the 2-D Gaussian function used to create the low-pass filter Using a Gaussian with a smaller value of σ e means a finer scale, giving more edge points and less smooth edges To find the points located at highly-curved parts of the edges, a curvature function introduced by Mokhtarian and Mackworth [13] is considered Their method allows the curvature measurement along a 2-D curve (s) = (x(s), y(s)), s is the arc length parameter, at different scales by first convolving the curve with an 1-D Gaussian function g(s, σk ) = the width of the Gaussian X (s, σk ) = Y(s, σk ) = +∞ −∞ +∞ −∞ σk √ −s 2 e 2σk , where σ k is 2π x(s1 )g(s − s1 , σk ) ds1 (1) y(s1 )g(s − s1 , σk ) ds1 (2) The curvature function κ(s, σ k ) is given by κ(s, σk ) = Xs (s, σk )Yss (s, σk ) − Xss (s, σk )Ys (s, σk ) [Xs (s, σk )2 + Ys (s, σk )2 ]3/2 (3) The first and second derivatives of X (s, σk ) and Y(s, σk ) can be obtained by convolving x(s) and y(s) with the first and second derivatives of the Gaussian function g(s, σ k ), respectively The feature points to be chosen are the local maxima of |κ(s, σk )| whose values must also exceed a threshold value tk At a finer scale, a smaller value of σ k is used, resulting in more corner points to be extracted Feature Point Matching Finding the correspondences between feature points in consecutive frames is the key step of our method We devised a matching technique in which the crosscorrelation, curvature, and displacement are used as matching criteria The first step is to find an initial estimate for the motion at every feature point in the first frame Some matching techniques such as that in [14] have to considered all possible pairs, hence M × N pairs needed to be examined, where M and N are the number of feature points in the first and second frames, respectively Some others assume the displacements are small to limit the search for a match to a small neighborhood of each point By giving an initial estimate for the motion at each point, we are also able to reduce the number of pairs to be examined without having to constrain the motion to small displacements Remember that we are employing a multi-scale scheme, in which the initial estimation of the flow field at one scale is given by the piecewise affine transforms computed at the previous level, as mentioned in 3.1 At the starting scale, a rough estimation can be made by treating the points as if they are under a rigid 2-D motion It means the motion is a combination of a rotation and a translation Compute the centers of gravity, C1 and C2 , the angles of the principal axes, α and α , of the two sets of feature points in two frames The motion at every feature point in the first frame can be roughly estimated by a rotation around C1 with the angle φ = α2 − α1 , followed by a translation represented by the vector t = xC2 − xC1 , where xc1 and xc2 are the vectors representing the coordinations of C1 and C2 in their image frame Let it and jt+1 be two feature points in two frames t and t+1, respectively Let i t+1 be the estimated match of it in frame t + 1, d(i , j) be the Euclidean distance between i t+1 and j t+1 , c(i, j) be the cross-correlation between it and jt+1 , ≤ c(i, j) ≤ 1, and K(i, j) be the difference between the curvature measures at it and jt+1 A matching score between it and jt+1 is defined as follows d(i , j) > dmax : s(i, j) = d(i , j) ≤ dmax : s(i, j) = wc c(i, j) + sk (i, j) + sd (i, j), (4) where sk (i, j) = wk (1 + κ(i, j))−1 sd (i, j) = wd (1 + d(i , j))−1 (5) A Super-Resolution Imaging Method Based on Dense Subpixel The quantity dmax specifies the maximal search distance from the estimated match point w c , w k , and w d are the weight values, determining the importance of each of the matching criteria The degree of importance of each of these criteria changes at different scales At a finer scale, the edges produced by Canny edge detector become less smooth, meaning the curvature measures are less reliable Thus, w k should be reduced On the other hand, wd should be increased, reflecting the assumption that the estimated match becomes closer to the true match For each point it , its optimal match is a point jt+1 such that s(i, j) is maximal and exceeds a threshold value ts Finally, inter-pixel interpolation and correlation matching are used in order to achieve sub-pixel accuracy in estimating the displacement of the corresponding points 3.4 Affine Flow Computation Consider a planar surface patch moving under rigid motion in the 3-D space In 2-D affine models, the change of its projections in an image plane from frame t to frame t + is approximated by an affine transform x t+1 y t+1 t t = a c t+1 b d xt yt + e f , (6) t+1 where (x , y ) and (x , y ) represent the coordinations of a moving point in frames t and t + 1, a, b, c, d, e, and f are the affine transform parameters Let x be vector [x, y]T The point represented by x is said to be under an affine motion from t to t + Then the velocity vector v = [d x/dt, d x/dt]T of that point at time t is given by vt = xt+1 − xt a−1 b e = xt + c d −1 f = Axt + c (7) A and c are called the affine flow parameters Using the constrained Delaunay triangulation [15] for each set of feature points, a mesh of triangular patches is generated to cover the moving part in each image frame A set of line segments, each of which connects two adjacent feature points on a same edge, is used to constrain the triangulation, so that the generated mesh closely matches the true content 83 of the image From (7), two linear equations of six unknowns are formed for each pair of corresponding feature points Therefore, for each pair of matching triangular patches, a total of six linear equations is established from their corresponding vertices Solving these equations we obtain the affine motion parameters, which define the affine flow within the small triangular region 3.5 Evaluation of Optical Flow Computation Technique We conducted experiments with our optical flow estimation technique using some common image sequences created exclusively for testing optical flow techniques and compared the results with those in [16, 17] The image sequences used for the purpose of error evaluation include the Translating Tree sequence (Fig 3), the Diverging Tree sequence (Fig 4), and the Yosemite sequence (Fig 5) These are simulated sequences for which the ground truth is provided As in [16, 17], an angular measure is used for error measurement Let v = [u v]T be the correct 2-D motion vector and ve be the estimated motion vector at a point in the image plane Let v˜ be a 3-D unit vector created from a 2-D vector v: v˜ = [v 1]T |[v 1]| (8) The angular error ψ e of the estimated motion vector ve with respect to the correct motion vector v is defined as follows: ψe = arccos(˜v.˜ve ) (9) Using this angular error measure, bias caused by the amplification inherent in a relative measure of vector differences can be avoided For the Translating Tree and Diverging Tree sequences, the performance of the piecewise affine approximation technique is comparable to most other methods shown in [16] (Tables and 2) The lack of features led to large errors at some parts of the images in these two sequences, especially near the center in the Diverging Tree sequence where the velocities are very small, increasing the average errors significantly, even though the estimated flow fields are accurate for most parts of the images The Yosemite sequence is a complex test There are diverging motions due to the movement of the camera 84 Le and Seetharaman Figure Top: two frames of the Translating Tree sequence Middle: generated triangular meshes Bottom: the correct flow (left) and the estimated flow (right) and translating motions of the clouds While all the techniques analyzed in [16] show significant increases of errors in comparison with the results from the previous two sequences, the performance of our technique remains consistent (Table 3) Only those methods of Lucas and Kanade [18], Fleet and Jepson [19], and Black and Anandan [17] are able to produce smaller errors than ours on this sequence And among them, Lucas and Kanade’s and Fleet and Jepson’s methods could manage to recover only about one third of the flow field in average, while the piecewise affine approximation technique recovers nearly 90 percent of the flow field To verify if the accuracies are indeed sub-pixel, we use the distance error de = |v−ve | For the Translating Tree sequence, the mean distance error is 11.40% of a pixel and the standard deviation of errors is 15.69% of a pixel The corresponding figures for the Diverging Tree sequence are 17.08% and 23.96%, and for the Yosemite sequence are 31.31% and 46.24% It is obvious that the flow errors at most points of the images are sub-pixel 3.6 Utilizing MPEG-4 Tools for Motion Estimation MPEG-4 is an ISO/IEC standard (ISO/IEC 14496) developed by the Moving Picture Experts Group A Super-Resolution Imaging Method Based on Dense Subpixel 85 Figure Top: two frames of the Diverging Tree sequence Middle: generated triangular meshes Bottom: the correct flow (left) and the estimated flow (right) (MPEG) Among many other things, it provides solutions in the form of tools and algorithms for contentbased coding and compression of natural images and video Mesh-based compression and motion estimation are important parts of image and video compression standards in MPEG-4 [20] Some functions of our optical flow computation technique are already available in MPEG-4, including: • Mesh generation: MPEG-4 2-D meshing functions can generate regular or content-based Delaunay triangular meshes from a set of points Methods for selecting the feature points are not subject to standardization 2-D meshes are used for meshbased image compression with texture mapping on meshes, as well as for motion estimation • Computation of piecewise affine motion fields: MPEG-4 tools allow construction of continuous motion fields from 2-D triangular meshes tracked over video frames MPEG-4 also has functions for standard × or 16 × 16 block-based motion estimation, and for global motion estimation techniques Overall, utilizing 2-D content-based meshing and motion estimation functions of MPEG-4 helps ease the implementation tasks 86 Le and Seetharaman Figure Top: two frames of the Yosemite sequence Middle: generated triangular meshes Bottom: the correct flow (left) and the estimated flow (right) for our optical flow technique On the other hand, our technique makes improvements over MPEG-4’s meshbased piecewise affine motion estimation method, thank to its multi-scale scheme Super-Resolution Image Reconstruction Given a low-resolution image frame bk (m, n), we can reconstruct an image frame fk (x, y) with a higher resolution as follows [9]: fk (x, y) = bk (m, n) m,n × sin π (xλ−1 − m) π (xλ−1 − m) sin π (yλ−1 − n) π (yλ−1 − n) (10) where sinθ θ is the ideal interpolation filter, and λ is the desired resolution step-up factor For example, if bk (m, n) is a 50 × 50 image and λ = 4, then, fk (x, y) will be of the size 200 × 200 A Super-Resolution Imaging Method Based on Dense Subpixel Table Performance of various optical flow techniques on the Translating Tree sequence Average errors Standard deviations Horn and Schunck (original) 38.72◦ 27.67◦ Horn and Schunck (modified) 2.02◦ 2.27◦ 100.0% Lucas and Kanade (modified) 0.66◦ 0.67◦ 39.8% Uras et al 0.62◦ 0.52◦ 100.0% Nagel 2.44◦ 3.06◦ 100.0% Anandan 4.54◦ 3.10◦ Singh 1.64◦ 2.44◦ Heeger 8.10◦ 12.30◦ 77.9% Waxman et al 6.66◦ 10.72◦ 1.9% Fleet and Jepson 0.32◦ 0.38◦ 74.5% Piecewise affine approximation 2.83◦ 4.97◦ 86.3% Techniques 87 Table Performance of various optical flow techniques on the Yosemite sequence Average errors Standard deviations Horn and Schunck (original) 32.43◦ 30.28◦ 100.0% Horn and Schunck (modified) 11.26◦ 16.41◦ 100.0% 4.10◦ 9.58◦ 35.1% Uras et al 10.44◦ 15.00◦ 100.0% Nagel 11.71◦ 10.59◦ 100.0% 100.0% Anandan 15.84◦ 13.46◦ 100.0% 100.0% Singh 13.16◦ 12.07◦ 100.0% Heeger 11.74◦ 19.04◦ 44.8% Waxman et al 20.32◦ 20.60◦ 7.4% Fleet and Jepson 4.29◦ 11.24◦ 34.1% Black and Anandan 4.46◦ 4.21◦ 100.0% Piecewise affine approximation 7.97◦ 11.90◦ 89.6% Densities 100.0% Techniques Lucas and Kanade Densities Table Performance of various optical flow techniques on the Diverging Tree sequence Average errors Standard deviations Horn and Schunck (original) 12.02◦ 11.72◦ 100.0% Horn and Schunck (modified) 2.55◦ 3.67◦ 100.0% Lucas and Kanade 1.94◦ 2.06◦ 48.2% Uras et al 4.64◦ 3.48◦ 100.0% Nagel 2.94◦ 3.23◦ 100.0% Anandan 7.64◦ 4.96◦ 100.0% Singh 8.60◦ 4.78◦ 100.0% Heeger 4.95◦ 3.09◦ 73.8% 11.23◦ 8.42◦ 4.9% Fleet and Jepson 0.99◦ 0.78◦ 61.0% Piecewise affine approximation 9.86◦ 10.96◦ 77.2% Techniques Waxman et al Densities Each point in the high-resolution grid corresponding to the first frame can be tracked along the video sequence from the motion fields computed between consecutive frames, and the super-resolution image is updated sequentially: x (1) = x, y (1) = y, f1(1) (x, y) = f1 (x, y) (11) x (k) = x (k−1) + u k x (k−1) , y (k−1) , y (k) = y (k−1) + vk x (k−1) , y (k−1) k − (k−1) fk(k) (x, y) = fk−1 (x, y) + fk x (k) , y (k) k k (12) (13) for k = 2, 3, The values uk and v k represent the dense velocity field between bk−1 and bk This sequential reconstruction technique is suitable for online processing, in which the super-resolution images can be updated every time a new frame comes Experimental Results In the first experiment we used a sequence of 16 frames capturing a slow-moving book (Fig 6) Each frame was down-sampled by a scale of four High resolution images were reconstructed from the down-sampled ones, using 2, 3, 16 frames, respectively The graph in Fig shows errors between reconstructed images and their corresponding original frame keep decreasing when the number of low-resolution frames used for reconstruction is increased, until the accumulated optical flow errors become significant Even though this is a simple case because the object surface is planar and the motion is rigid, it nevertheless presented the characteristics of this technique The second experiment was performed on images taken from a real surveillance camera In this experiment we tried to reconstruct high-resolution images of faces of people captured by the camera (Fig 7) Results show obvious improvements of reconstructed super-resolution images over original images For the time being, we are unable to conduct a performance analysis of our super-resolution method 88 Le and Seetharaman Figure Top: parts of an original frame (left) and a down-sampled frame (right) Middle: parts of an image interpolated from a single frame (left) and an image reconstructed from frames (right) Bottom: parts of images reconstructed from frames (left) and 16 frames (right) Figure Left: part of an original frame containing a human face Center: part of an image interpolated from a single frame Right: part of an image reconstructed from frames in comparison with others’, because: (1) There has been no study on quantitative evaluation of the performance of super-resolution techniques so far; and (2) There are currently no common metrics to measure the performance of super-resolution techniques (in fact, most of the published works on this subject did not perform any quantitative performance analysis at all) The number of super-resolution techniques are so large that a study on comparison of their performances could provide enough contents for another paper Conclusion We have presented a method for reconstructing superresolution images from sequences of low-resolution video frames, using motion compensation as the basis for multi-frame data fusion Motions between video frames are computed with a multi-scale piecewise affine model which allows accurate estimation of the motion field even if the motion is non-rigid The reconstruction is sequential—only the current frame, the frame immediately before it and the last 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Department of Electrical and Computer Engineering, Vietnam National University, Hanoi He received the B.S degree in Computer Science from the Hanoi University of Technology in 1993 He was employed at the Institute of Information Technology, Vietnam, from 1993 to 1997, as a researcher, working to develop software tools and applications in the areas of Computer Graphics and Geographical Information Systems He received the M.S degree from the California State Polytechnic University, Pomona, in 2000, and the Ph.D degree from the University of Louisiana at Lafayette in 2003, both in Computer Science His research interests include Computer Vision, Robotics, Image Processing, Computer Graphics, and Neural Networks hvle@hn.vnn.vn Guna Seetharaman is currently with The Air Force Institute of Technology, where he is an associate professor of computer engineering and computer science He has been with the Center for Advanced Computer Studies, University of Louisiana at Lafayette since 1988 He was also a CNRS Visiting Professor at The Institute for Electronics Fundamentals, University of Paris XI, His current focus is on Three Dimensional Displays, Digital Light Processing, Nano and Micro sensors for imaging applications He has earned his Ph.D in electrical and computer engineering in 1988 from University of Miami FL; M.Tech in Electrical Engineeing (1982) from Indian Institute of Technology, Chennai; and, B.E Electronics and Telecommunications from University of Madras, Guindy Campus He served as the Technical Program Chair, and The local organizations chair for The Sixth IEEE Workshop on Computer Architecture for Machine Perception, New Orleans, May 2004; and Technical Committee member and editor for The Second International DOEONR-NSF Workshop on Foundations of Decision and Information Fusion, Washington DC, 1996 He served on the program committees of various International Conferences in the areas of Image Processing and Computer Vision His works have been widely cited in industry and research He is a member of Tau Beta Pi, Eta Kappa Nu, ACM, and IEEE guna.seetharaman@afit.edu ... 3.1 At the starting scale, a rough estimation can be made by treating the points as if they are under a rigid 2-D motion It means the motion is a combination of a rotation and a translation Compute... Super-Resolution Imaging Method Based on Dense Subpixel M Elad and Y Hel-Or, A Fast Super-Resolution Reconstruction Algorithm for Pure Translational Motion and Common Space-Invariant Blur,” IEEE Trans on. .. Second International DOEONR-NSF Workshop on Foundations of Decision and Information Fusion, Washington DC, 1996 He served on the program committees of various International Conferences in the areas