Patent Publication Number: US-7596281-B2

Title: Apparatus and method for alignment of spatial or temporal non-overlapping images sequences

Description:
FIELD OF THE INVENTION 
   The present invention relates to methodologies and systems for combining and integrating visual information from various sources. 
   BACKGROUND OF THE INVENTION 
   The problem of image alignment or image registration has been extensively researched and successful methods for solving this problem have been developed. Some of these methods are based on matching extracted local image features. Other approaches are based on directly matching image intensities. However, all these methods share one basic assumption: that there is sufficient overlap between the images to allow extraction of common image properties, namely, that there is sufficient “similarity” between the images. The term similarity is used here in the broadest sense, for example: gray-level similarity, feature similarity, similarity of frequencies and statistical similarity, such as mutual information. Consequently, the prior art does not provide alignment between images where there is very little similarity between the images or where there is no spatial overlap between the images. 
   However, a sequence of images contains much more information than any individual image does. For example: a sequence of images of a scene contains temporal changes caused by dynamic changes in the scene, or temporal changes induced by the motion of the camera. This information is employed by the present invention to provide alignment between images of the two sequences of images even when the sequences have no spatial overlap or have very low similarity. 
   State of the art methods for sequence-to-sequence alignment, and other relevant technologies, are described in:
     [1] S. Avidan and A. Shashua. Threading fundamental matrices. In European Conference on Computer Vision, 1998.   [2] P. A. Beardsley, P. H. S. Torr and A. Zisserman. 3D model acquisition from extended image sequences. In Proc. 4 th  European Conference on Computer Vision, LNCS 1065, Cambridge, pages 683-695, 1996.   [3] P. R. Burt and R. J. Kolczynski. Enhanced image capture through fusion. In International Conference on Computer Vision, 1993.   [4] Y. Caspi and M. Irani. A step towards sequence-to-sequence alignment. In IEEE Conference on Computer Vision and Pattern Recognition, Hilton Head Island, S.C., June 2000.   [5] D. Demirdijian, A. Zisserman and R. Horaud. Stereo autocalibration from one plane. In European Conference on Computer Vision, 2000.   [6] Y. Dufournaud, C. Schmid and R. Horaud. Matching images with different resolutions. In IEEE Conference on Computer Vision and Pattern Recognition, Hilton Head Island, S.C., June 2000.   [7] C. E. Pearson (ed). Handbook of applied mathematics—Second Edition. Van Nostrand Reinhold Company, New York, 1983, pp 898.   [8] F. R. Gantmakher. The theory of matrices. Chapter VIII, Chelsea Pub., New York, 1959.   [9] Gene Golub and Charles Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore and London, pp. 123-127, 1989   [10] Richard I. Hartley. In Defence of the 8-point Algorithm. In Pattern Recognition and Machine Intelligence 19 (6) June pages 580-593 1997   [11] R. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Chapter 3, Cambridge university press, Cambridge, 2000.   [12] R. Horaud and G. Csurka. Reconstruction using motions of a stereo rig. In International Conference on Computer Vision, pages 96-103, 1998.   [13] R. Horaud and F. Dornaika. Hand-eye calibration. International Journal of Robotics Research, 14(3):195-210, June 1995.   [14] M. Irani and P. Anandan. About direct methods. In Vision Algorithms Workshop, pages 267-277, Corfu, 1999.   [15] M. Irani, P. Anandan and D. Weinshall. From reference frames to reference planes: Multi-view parallax geometry and applications. In European Conference on Computer Vision, Freiburg, June 1998.   [16] M. Irani, B. Rousso and S. Peleg. Computing occluding and transparent motions. International Journal of Computer Vision, 12(1):5-16, January 1994.   

   [17] R. Kumar, P. Anandan and K. Hanna. Direct recovery of shape from multiple views: parallax based approach. In International Conference on Pattern Recognition, 1994.
     [18] Harpreet Sawhney. 3D geometry from planar parallax. In IEEE Conference on Computer Vision and Pattern Recognition, June 1994.   [19] A. Shashua and N. Navab. Relative affine structure: Theory and application to 3D reconstruction from perspective views. In IEEE Conference on Computer Vision and Pattern Recognition, pages 483-489, Seattle, Wash., June 1994.   [20] G. P. Stein. Tracking from multiple view points: Self-calibration of space and time. In DARPA IU Workshop, pages 1037-1042, 1998.   [21] P. H. S. Torr and A. Zisserman. Feature based methods for structure and motion estimation. In Vision Algorithms Workshop, pages 279-329, Corfu, 1999.   [22] R. Y. Tsai and R. K. Lenz. A new technique for full autonomous and efficient. 3D robotics hand/eye calibration. IEEE Journal of Robotics and Automation, 5(3):345-358, June 1989.   [23] P. Viola and W. Wells III. Alignment by maximization of mutual information. In International Conference on Computer Vision, pages 16-23, 1995.   [24] A. Zisserman, P. A. Beardsley and I. D. Reid. Metric calibration of a stereo rig. In Workshop on Representations of Visual Scenes, pages 93-100, 1995.   [25] P. H. S. Torr and A. Zisserman. Feature based methods for structure and motion estimation. In Vision Algorithms Workshop, pages 279-290, Corfu, 1999.   [26] Z. Zhang, R. Deriche, O. Faugeras, and Q. Luong. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence, 78:87-119, 1995.   

   The disclosures of all publications mentioned in the specification and of the publications cited therein are hereby incorporated by reference. 
   SUMMARY OF THE INVENTION 
   This invention seeks to provide methodologies and systems for combining and integrating visual information from various sources. 
   The present invention replaces a requirement of “coherent appearance”, which is a fundamental assumption in prior art image alignment methods, with a requirement of “consistent temporal behavior”. The requirement of “consistent temporal behavior” is easier to satisfy, for example by producing two sequences of images using two cameras moved jointly in space. The present invention is therefore useful not only in cases of non-overlapping sequences, but also in cases which are inherently difficult for standard image alignment techniques. 
   In a preferred embodiment of the present invention the two cameras are attached closely to each other, so that their centers of projection are very close, and move jointly in space. In this case the induced frame-to-frame transformations within each sequence have correlated behavior across the two sequences. 
   A preferred embodiment of the present invention employs correlated temporal behavior to resolve both spatial and temporal transformations between images of the two sequences. 
   The present invention is therefore useful for a variety of real-world applications, including:
         (i) Multi-sensor alignment for image fusion. This requires accurate alignment of images obtained by sensors of different sensing modalities, such as Infra-Red and visible light. Such images differ significantly in their appearance due to different sensor properties.   (ii) Alignment of image sequences obtained at different zooms. The problem here is that different image features are prominent at different image resolutions. Alignment of a wide-field-of-view sequence with a narrow-field-of-view sequence is useful for detecting small zoomed-in objects inside or outside a zoomed-out view of the scene. This can be useful in surveillance applications.   (iii) Generation of wide-screen movies from multiple non-overlapping narrow FOV movies, such as in IMAX™ movies.       

   It is appreciated that the term “correlated” as used in the specification and claims of the present application includes inter alia the following: mathematical correlation, statistical correlation and any other co-relationship. 
   There is thus provided in accordance with a preferred embodiment of the present invention a method for computing at least one of spatial and temporal relationships between at least first and second sequences of representations having respective first and second temporal progressions. The method includes employing the first and second temporal progressions to obtain at least one of the spatial and temporal relationships. 
   There is also provided in accordance with a preferred embodiment of the present invention a method for computing at least one of spatial and temporal relationships between at least first and second sequences of representations having respective first and second temporal progressions. The method includes employing only the first and second temporal progressions to obtain at least one of the spatial and temporal relationships. 
   There is further provided in accordance with a preferred embodiment of the present invention a system for computing at least one of spatial and temporal relationships between at least first and second sequences of representations having respective first and second temporal progressions. The system includes input functionality for receiving the first sequence of representations and the first temporal progression and the second sequence of representations and the second temporal progression and computation functionality employing at least one of the first sequence of representations and the first temporal progression and at least one of the second sequence of representations and the second temporal progression to obtain at least one of the spatial and temporal relationships. 
   There is further provided in accordance with another preferred embodiment of the present invention a system for computing at least one of spatial and temporal relationships between at least first and second sequences of representations having respective first and second temporal progressions. The system includes input functionality for receiving the first and second temporal progressions and computational functionality employing the first and second temporal progressions to obtain at least one of the spatial and temporal relationships. 
   Further in accordance with a preferred embodiment of the present invention the method also includes computing the first and second temporal progressions which are then employed to obtain at least one of the spatial and temporal relationships. 
   Preferably, the representations include visual representations, images and at least three-dimensional objects. 
   Still further in accordance with a preferred embodiment of the present invention the step of employing obtains the spatial relationship, and/or the temporal relationship. 
   Additionally in accordance with a preferred embodiment of the present invention the employing step obtains the spatial and temporal relationships in the absence of a spatial overlap between the representations 
   Additionally or alternatively, the employing step obtains the spatial and temporal relationships in the absence of common spatial information between the representations. 
   Further in accordance with a preferred embodiment of the present invention the step of employing obtains the spatial and temporal relationships in the absence of common information between individual ones of the representations belonging to different ones of the at least first and second sequences. 
   Still further in accordance with a preferred embodiment of the present invention the step of employing obtains the spatial and temporal relationships in the absence of common information between any individual ones of the representations belonging to different ones of the at least first and second sequences. 
   Additionally in accordance with a preferred embodiment of the present invention the spatial relationship includes at least one parameter of a geometric transformation between the at least first and second sequences of representations. 
   Preferably, the spatial relationship includes a 2-dimensional projective transformation. 
   Alternatively, the spatial relationship includes a 3-dimensional projective transformation. 
   Further in accordance with a preferred embodiment of the present invention the spatial relationship includes a fundamental matrix. 
   Still further in accordance with a preferred embodiment of the present invention the spatial relationship includes a 2-dimensional parametric transformation. 
   Additionally in accordance with a preferred embodiment of the present invention the spatial relationship includes an at least 3-dimensional parametric transformation. 
   Further in accordance with a preferred embodiment of the present invention the spatial relationship includes a 2-dimensional non-parametric transformation. 
   Additionally or alternatively, the spatial relationship includes an at least 3-dimensional non-parametric transformation. 
   Further in accordance with a preferred embodiment of the present invention the spatial relationship includes a spatial relationship between individual ones of the representations belonging to different ones of the at least first and second sequences. 
   Still further in accordance with a preferred embodiment of the present invention the spatial relationship includes a spatial relationship between individual ones of the representations belonging to different ones of the at least first and second sequences, at least one of the individual ones of the representations being an interpolated representation. 
   Additionally in accordance with a preferred embodiment of the present invention the spatial relationship includes a spatial relationship between individual ones of the representations belonging to different ones of the at least first and second sequences, one of the individual ones of the representations being an interpolated representation and the other of the individual ones of the representations being a non-interpolated representation. 
   Further in accordance with a preferred embodiment of the present invention the temporal relationship includes at least one parameter of a temporal transformation between the at least first and second sequences of representations. 
   Still further in accordance with a preferred embodiment of the present invention the temporal relationship includes a time shift between the at least first and second sequences of representations. 
   Preferably, the temporal relationship includes an affine transformation in time, a parametric transformation in time and/or a non-parametric transformation in time. 
   Further in accordance with a preferred embodiment of the present invention the temporal relationship includes a temporal relationship in time between individual ones of the representations belonging to different ones of the at least first and second sequences. 
   Still further in accordance with a preferred embodiment of the present invention the temporal relationship includes a temporal relationship in time between individual ones of the representations belonging to different ones of the at least first and second sequences, at least one of the individual ones of the representations being an interpolated representation. 
   Additionally in accordance with a preferred embodiment of the present invention the temporal relationship includes a temporal relationship in time between individual ones of the representations belonging to different ones of the at least first and second sequences. Typically, one of the individual ones of the representations is an interpolated representation and the other of the individual ones of the representations is a non-interpolated representation. 
   Preferably, the interpolated representation is interpolated in time and the interpolated representation is interpolated in space. 
   Further in accordance with a preferred embodiment of the present invention the first and second temporal progressions include ordered intra-sequence representation-to-representation transformations. 
   Still further in accordance with a preferred embodiment of the present invention the first and second temporal progressions include ordered intra-sequence representation-to-representation transformations resulting from relative motion between sensors and a scene. 
   Preferably, the intra-sequence representation-to-representation transformations include 2-dimensional projective transformations and/or 3-dimensional projective transformations. 
   Further in accordance with a preferred embodiment of the present invention the intra-sequence representation-to-representation transformations include fundamental matrices, 2-dimensional parametric transformations, at least 3-dimensional parametric transformations, 2-dimensional non-parametric transformations, at least 3-dimensional non-parametric transformations and camera matrices. 
   Still further in accordance with a preferred embodiment of the present invention the step of employing includes correlating the first and second temporal progressions. 
   Additionally in accordance with a preferred embodiment of the present invention the step of employing includes equating properties of the first temporal progression and the second temporal progression. 
   Further in accordance with a preferred embodiment of the present invention the step of employing includes correlating properties of the first temporal progression and the second temporal progression. 
   Still further in accordance with a preferred embodiment of the present invention the step of employing includes equating a sequential application and at least one of the intra-sequence representation-to-representation transformations of the first temporal progression and an unknown the spatial relationship between the at least first and second sequences with a sequential application of the unknown spatial relationship between the at least first and second sequences and at least one of the intra-sequence representation-to-representation transformations of the second temporal progression. 
   Further in accordance with a preferred embodiment of the present invention the step of employing includes equating a composition and at least one of the intra-sequence representation-to-representation transformations of the first temporal progression and an unknown the spatial relationship between the at least first and second sequences with a composition of the unknown spatial relationship between the at least first and second sequences and at least one of the intra-sequence representation-to-representation transformations of the second temporal progression. 
   Additionally in accordance with a preferred embodiment of the present invention the step of employing includes obtaining an unknown the spatial relationship between the at least first and second sequences by equating a sequential application and at least one of the intra-sequence representation-to-representation transformations of the first temporal progression and the unknown spatial relationship between the at least first and second sequences with a sequential application of the unknown spatial relationship between the at least first and second sequences and at least one of the intra-sequence representation-to-representation transformations of the second temporal progression. 
   Additionally in accordance with a preferred embodiment of the present invention the step of equating includes equating up to a scale factor. 
   Further in accordance with a preferred embodiment of the present invention the intra-sequence representation-to-representation transformations include multiple simple motions taking place at least partially at different times. 
   Preferably, the intra-sequence representation-to-representation transformations include multiple combinations of multiple simple motions taking place at least partially at different times. 
   Preferably, the intra-sequence representation-to-representation transformations include multiple complex motions taking place at least partially at different times. 
   Preferably, the intra-sequence representation-to-representation transformations include multiple combinations of multiple complex motions taking place at least partially at different times. 
   Further in accordance with a preferred embodiment of the present invention the first and second temporal progressions include ordered intra-sequence representation-to-representation transformations at least some of which result from relative motion between sensors and a scene. 
   Still further in accordance with a preferred embodiment of the present invention the step of employing uses multiple combinations of intra-sequence representation-to-representation transformations. 
   Further in accordance with a preferred embodiment of the present invention the spatial relationship between the at least first and second sequences of representations results from an acquisition relationship, between first and second sensors acquiring respective the at least first and second sequences, being fixed over time. 
   Still further in accordance with a preferred embodiment of the present invention the spatial relationship between the at least first and second sequences of representations results from an acquisition relationship, between first and second sensors acquiring respective the at least first and second sequences, changes in a known way over time. 
   Preferably, the acquisition relationship includes relative position, relative orientation and relative internal sensor parameters. 
   Further in accordance with a preferred embodiment of the present invention the acquisition relationship is known. 
   Alternatively, the acquisition relationship is not known. 
   Additionally in accordance with a preferred embodiment of the present invention the at least first and second sequences are acquired generally at the same time. 
   Preferably, the at least first and second sequences are acquired generally at different times. 
   Further in accordance with a preferred embodiment of the present invention the at least first and second sequences represent measurements from the same scene. 
   Additionally in accordance with a preferred embodiment of the present invention the at least first and second sequences represent measurements from different portions of the same scene. 
   Additionally or alternatively, the at least first and second sequences represent measurements from different overlapping portions of the same scene. 
   Further in accordance with a preferred embodiment of the present invention the at least first and second sequences represent measurements from different non-overlapping portions of the same scene. 
   Still further in accordance with a preferred embodiment of the present invention the at least first and second sequences represent measurements from different scenes. 
   Alternatively, the scene is two-dimensional. 
   Preferably, the scene is at least three-dimensional. 
   Further in accordance with a preferred embodiment of the present invention the scene is static. Alternatively, the scene is dynamic. 
   Further in accordance with a preferred embodiment of the present invention the measurements are generally the same for each sensor. 
   Still further in accordance with a preferred embodiment of the present invention the measurements are generally different for each sensor. 
   Preferably, the measurements include at least one of illumination, heat, radiance, electromagnetic radiation, color, distance, density, sound and speed. 
   Further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for sequence fusion. 
   Additionally in accordance with a preferred embodiment of the present invention the method also includes step of employing at least one of the spatial and temporal relationships for alignment of sequences obtained at different zooms. 
   Still further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for surveillance. 
   Further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for generation of wide-screen movies from multiple at least partially non-overlapping narrow field of view movies. 
   Additionally in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for image fusion. 
   Further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for integrating information contained in the at least first and second sequences. 
   Still further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for alignment of images obtained at different zooms. 
   Additionally in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for comparing information contained in the at least first and second sequences. 
   Further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for finding differences between information contained in the at least first and second sequences. 
   Still further in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for finding differences between information contained in the at least first and second sequences relating to the same scene at different times. 
   Additionally in accordance with a preferred embodiment of the present invention the method also includes the step of employing at least one of the spatial and temporal relationships for integrating information contained in the at least first and second sequences and thereby providing an information output which exceeds limitations of individual sensors. 
   Further in accordance with a preferred embodiment of the present invention the step of employing includes comparing properties of the first temporal progression and the second temporal progression. 
   Still further in accordance with a preferred embodiment of the present invention the computation functionality includes functionality for computing the first and second temporal progressions which are then employed to obtain at least one of the spatial and temporal relationships. 
   Additionally in accordance with a preferred embodiment of the present invention the system also includes temporal progression computation functionality for computing the first and second temporal progressions and supplying them to the input functionality. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS AND APPENDICES 
     The present invention will be understood and appreciated more fully from the following detailed description, taken in conjunction with the drawings in which: 
       FIG. 1A  is a simplified illustration of two cameras, fixed to each other, each taking a sequence of images of a portion of a scene, the portions of the scene photographed by the two cameras being non-overlapping; 
       FIG. 1B  illustrates inherent ambiguities in the relative spatial relationship of images in the sequences taken as shown in  FIG. 1A ; 
       FIG. 1C  illustrates portions of two sequences of images taken as shown in  FIG. 1A  and the unknown temporal relationship between the sequences; 
       FIG. 1D  illustrates pairs of images forming part of the sequences of images taken as shown in  FIG. 1A , whose spatial and temporal relationships are determined in accordance with the present invention; 
       FIG. 2A  is a simplified illustration of two cameras, fixed to each other, each taking a sequence of images of a portion of a scene, the portions of the scene being photographed at significantly different zooms by the two cameras; 
       FIG. 2B  illustrates inherent ambiguities in the relative spatial relationship of images in the sequences taken as shown in  FIG. 2A ; 
       FIG. 2C  illustrates portions of two sequences of images taken as shown in  FIG. 2A  and the unknown temporal relationship between the sequences; 
       FIG. 2D  illustrates pairs of images forming part of the sequences of images taken as shown in  FIG. 2A , whose spatial and temporal relationships are determined in accordance with the present invention; 
       FIG. 3A  is a simplified illustration of two sensors, fixed to each other, each taking a sequence of images of a portion of a scene, the portions of the scene being imaged by the two sensors employing different sensing modalities; 
       FIG. 3B  illustrates inherent ambiguities in the relative spatial relationship of images in the sequences taken as shown in  FIG. 3A ; 
       FIG. 3C  illustrates portions of two sequences of images taken as shown in  FIG. 3A  and the unknown temporal relationship between the sequences; 
       FIG. 3D  illustrates pairs of images forming part of the sequences of images taken as shown in  FIG. 3A , whose spatial and temporal relationships are determined in accordance with the present invention; 
       FIG. 4A  is a simplified illustration of a scene being photographed at two different times, producing two corresponding sequences of images, the path traveled by a camera used for photographing being generally identical for both sequences; 
       FIG. 4B  illustrates inherent ambiguities in the relative spatial relationship of images in the sequences taken as shown in  FIG. 4A ; 
       FIG. 4C  illustrates portions of two sequences of images taken as shown in  FIG. 4A  and the unknown temporal relationship between the sequences; 
       FIG. 4D  illustrates pairs of images forming part of the sequences of images taken as shown in  FIG. 4A , whose spatial and temporal relationships are determined in accordance with the present invention; 
       FIG. 5A  illustrates the relationships between image to image transformations within two sequences, induced by motion of two cameras along an axis, the two cameras being arranged at 180 degrees with respect to each other along the axis; 
       FIG. 5B  illustrates the relationships between image to image transformations within two sequences, induced by motion of two cameras along an axis, the two cameras being arranged at 90 degrees with respect to each other, one of the cameras being aligned along the axis of motion; 
       FIG. 5C  illustrates the relationships between image to image transformations within two sequences, induced by motion of two cameras along an axis, the two cameras being directed in the same direction perpendicular to the direction of motion but at different zooms; 
       FIGS. 6A ,  6 B and  6 C together illustrate that employing one type of transformation may not be sufficient for resolving ambiguities in the spatial and temporal relationships between sequences but that employing multiple different types of transformations reduces ambiguities in the spatial and temporal relationships between sequences; 
       FIG. 7  is a simplified illustration of a complex motion of two cameras, fixed to each other, each taking a sequence of images of a portion of a scene, the portions of the scene photographed by the two cameras being non-overlapping; 
       FIG. 8  is a simplified illustration portions of two sequences of images taken as shown in  FIG. 7 , wherein the two sequences are spatially related by a fixed and unknown inter-camera homography and temporally related by a fixed and unknown time shift. 
       FIG. 9  is a simplified functional block diagram of a preferred process of creating two sequences of images and aligning them. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   Reference is now made to  FIGS. 1A-1D  which illustrate resolution of spatial and temporal relationships between sequences of images taken by two moving cameras fixed to each other in accordance with the present invention. 
   As seen in  FIG. 1A , two cameras, designated respectively by reference numerals  100  and  102  are fixed to each other in any suitable manner. Each camera takes a sequence of images of a portion of a scene as the cameras move while they are fixed together. The movement of the cameras may be any suitable movement, such as rotation and/or translation in one or more dimensions and relative to any suitable point. Thus, for example, the two cameras  100  and  102  may rotate about the optical axis of one of the cameras or about any other axis. Similarly, translation of the cameras may occur in any suitable direction. 
   In the present case ( FIGS. 1A-1D ) portions of the scene photographed by the two cameras are non-overlapping and are designated respectively by reference numerals  104  and  106 . 
   Turning now to  FIG. 1B , it is seen that the portions  104  and  106  of the scene of  FIG. 1A  may be represented by corresponding images  108  and  110 . As seen in  FIG. 1C , images  108  and  110  each belong to a sequence of images, each produced by one of the moving cameras  100  and  102 , the respective sequences being designated by reference numerals  112  and  114 . It is seen that sequence  112  also includes images  122 ,  124  and  126 . Likewise, sequence  114  also includes images  130 ,  132  and  134 . 
   A problem addressed by the present invention is that the visual information contained in individual pairs of images, one belonging to sequence  112  and the other belonging to sequence  114 , (e.g. ( 108 ,  110 ), ( 122 ,  130 ) or ( 122 ,  134 )), is sufficient to establish neither the spatial nor the temporal relationship between two images of a pair. More generally, the visual information contained in individual pairs of images, one of the pair belonging to sequence  112  and the other of the pair belonging to sequence  114 , is sufficient to establish neither the spatial nor the temporal relationship between the two sequences  112  and  114 . 
   In the context of  FIGS. 1A-1D , two images are in the same “temporal relationship” if they are taken at the same time. 
   The unknown spatial relationship of images  108  and  110  is seen graphically by considering three examples of possible relative spatial relationships shown in  FIG. 1B  and designated by reference numerals  116 ,  118  and  120 . In each example, the two images  108  and  110  are placed in a different spatial relationship, all of which are consistent with the visual content of the images  108  and  110 . 
   The unknown temporal relationship of sequences  112  and  114  is seen graphically by considering  FIG. 1C . It is appreciated that it is not apparent from a cursory examination of sequences  112  and  114 , which images in sequence  112  are taken at the same time as which images in sequence  114 . 
   The present invention provides a system and technique for determining the correct relationship between the images  108  and  110  and more generally the correct spatial and temporal relationships between sequences  112  and  114 , as shown in  FIG. 1D . Thus, it may be appreciated that the present invention determines which image in sequence  112  corresponds in time with which image in sequence  114  and further determines the spatial relationship between the images which correspond in time. 
   Thus it is seen in  FIG. 1D  that image A ( 108 ) in sequence  112  is found to correspond in time with image b ( 110 ) in sequence  114 . The correct spatial relationship between images A ( 108 ) and b ( 110 ) is shown in  FIG. 1D  at reference numeral  150 . Similarly, image B ( 122 ) in sequence  112  is found to correspond in time with image c ( 132 ) in sequence  114  and image C ( 124 ) in sequence  112  is found to correspond in time with image d ( 134 ) in sequence  114 . The correct spatial relationship between images B ( 122 ) and c ( 132 ) is shown in  FIG. 1D  at reference numeral  152  and the correct spatial relationship between images C ( 124 ) and d ( 134 ) is shown in  FIG. 1D  at reference numeral  154 . 
   The present invention employs an appreciation that despite there being no common static visual information in the two sequences, the image to image dynamics in each of the two sequences are nevertheless related both spatially and temporally. The present invention utilizes the relationship between the dynamics to correlate the two sequences in time and space. 
   As will be described hereinbelow in detail, the image to image dynamics preferably are expressed by spatial transformations over time between images in each sequence. The problem of aligning sequences of images in time and space is thus reduced to a problem of determining the relationship between sequences of such transformations in time and in space. 
   It is appreciated from a consideration of  FIG. 1D , that by employing the present invention aligned sequences may be realized, which sequences may then be utilized to localize two portions of a scene with respect to each other, despite the absence of spatial overlap therebetween. 
   Reference is now made to  FIGS. 2A-2D  which illustrate the resolution of spatial and temporal relationships between sequences of images taken by two moving cameras fixed to each other in accordance with the present invention, the portions of the scene being photographed at significantly different zooms by the two cameras. 
   As seen in  FIG. 2A , two cameras, designated respectively by reference numerals  200  and  202  are fixed to each other in any suitable manner. Each camera takes a sequence of images of a portion of a scene as the cameras move while they are fixed together. The movement of the cameras may be any suitable movement, such as rotation and/or translation in one or more dimensions and relative to any suitable point. Thus, for example, the two cameras may rotate about the optical axis of one of the cameras or about any other axis. Similarly, translation of the cameras may occur in any suitable direction. 
   In the present case ( FIGS. 2A-2D ) portions of the scene photographed by the two cameras are imaged at significantly different zooms, thus imaging various features in the scene at different spatial scales. Two portions of the scene, which need not necessarily be overlapping, are designated respectively by reference numerals  204  and  206 . 
   Turning now to  FIG. 2B , it is seen that the portions  204  and  206  of the scene of  FIG. 2A  may be represented by corresponding images  208  and  210 , respectively. As seen in  FIG. 2C , images  208  and  210  each belong to a sequence of images, each produced by one of the moving cameras  200  and  202 , the respective sequences being designated by reference numerals  212  and  214 . It is seen that sequence  212  also includes images  222 ,  224  and  226 . Likewise, sequence  214  also includes images  230 ,  232  and  234 . 
   It is appreciated that features imaged in the zoomed-in image  210  may be different from the features imaged in the zoomed-out image  208  due to the different spatial scales employed by the cameras. Thus, for example, individual leaves of a tree may be discerned in image  210 , while such leaves may not be discernible in image  208 . Similarly trees may be discerned in image  208 , while only leaves and branches are discernible in image  210 . 
   A problem addressed by the present invention is that the visual information contained in individual pairs of images, one belonging to sequence  212  and the other belonging to sequence  214 , (e.g. ( 208 ,  210 ), ( 222 ,  230 ) or ( 222 ,  234 )), is sufficient to establish neither the spatial nor the temporal relationship between two images of a pair. More generally, the visual information contained in individual pairs of images, one of the pair belonging to sequence  212  and the other of the pair belonging to sequence  214 , is sufficient to establish neither the spatial nor the temporal relationship between the two sequences  212  and  214 . 
   In the context of  FIGS. 2A-2D , two images are in the same “temporal relationship” if they are taken at the same time. 
   The unknown spatial relationship of images  208  and  210  is seen graphically by considering three examples of possible relative spatial relationships shown in  FIG. 2B  and designated by reference numerals  216 ,  218  and  220 . In each example, the two images  208  and  210  are placed in a different spatial relationship with each other, all of which are consistent with the visual content of the images  208  and  210 . 
   The unknown temporal relationship of sequences  212  and  214  is seen graphically by considering  FIG. 2C . It is appreciated that it is not apparent from a cursory examination of sequences  212  and  214 , which images in sequence  212  are taken at the same time as which images in sequence  214 . 
   The present invention provides a system and technique for determining the correct relationship between the images  208  and  210  and more generally the correct spatial and temporal relationships between sequences  212  and  214 , as shown in  FIG. 2D . Thus, it may be appreciated that the present invention determines which image in sequence  212  corresponds in time with which image in sequence  214  and further determines the spatial relationship between the images which correspond in time. 
   Thus it is seen in  FIG. 2D  that image A ( 208 ) in sequence  212  is found to correspond in time with image b ( 210 ) in sequence  214 . The correct spatial relationship between images A ( 208 ) and b ( 210 ) is shown in  FIG. 2D  at reference numeral  250 . Similarly, image B ( 222 ) in sequence  212  is found to correspond in time with image c ( 232 ) in sequence  214  and image C ( 224 ) in sequence  212  is found to correspond in time with image d ( 234 ) in sequence  214 . The correct spatial relationship between images B ( 222 ) and c ( 232 ) is shown in  FIG. 2D  at reference numeral  252  and the correct spatial relationship between images C ( 224 ) and d ( 234 ) is shown in  FIG. 2D  at reference numeral  254 . 
   The present invention employs an appreciation that despite there not necessarily being any common static visual information in the two sequences, the image to image dynamics in each of the two sequences are nevertheless related both spatially and temporally. The present invention utilizes the relationship between the dynamics to correlate the two sequences in time and space. 
   As will be described hereinbelow in detail, the image to image dynamics preferably are expressed by spatial transformations over time between images in each sequence. The problem of aligning sequences of images in time and space is thus reduced to a problem of determining the relationship between sequences of such transformations in time and in space. 
   It is appreciated from a consideration of  FIG. 2D , that by employing the present invention aligned sequences may be realized, which sequences may then be utilized to provide high resolution details of a portion of a scene, localized with respect to a zoomed-out view of the scene, providing a wide field of view context. 
   Reference is now made to  FIGS. 3A-3D  which illustrate resolution of spatial and temporal relationships between sequences of images taken by two moving sensors fixed to each other in accordance with the present invention, the portions of the scene being imaged by the two sensors employing different sensing modalities. 
   As seen in  FIG. 3A , two sensors, such as an infra red (IR) camera  300  and a visible light (VL) camera  302 , are fixed to each other in any suitable manner. Each sensor takes a sequence of images of a portion of a scene as the sensors move while they are fixed together. The movement of the sensors may be any suitable movement, such as rotation and/or translation in one or more dimensions and relative to any suitable point. Thus, for example, the two sensors may rotate about the optical axis of one of the sensors or about any other axis. Similarly, translation of the sensors may occur in any suitable direction. 
   In the present case ( FIGS. 3A-3D ) portions of the scene sensed by the two sensors are imaged at different wavelength ranges, thus imaging various features in the scene in different modalities. Two portions of the scene, which need not necessarily be overlapping or at the same zoom, are designated by reference numerals  304  and  306 . 
   Turning now to  FIG. 3B , it is seen that the portions  304  and  306  of the scene of  FIG. 3A  may be represented by corresponding images  308  and  310 . As seen in  FIG. 3C , images  308  and  310  each belong to a sequence of images, each produced by one of the moving sensors  300  and  302 , the respective sequences being designated by reference numerals  312  and  314 . It is seen that sequence  312  also includes images  322 ,  324  and  326 . Likewise, sequence  314  also includes images  330 ,  332  and  334 . 
   It is appreciated that features imaged in the infra-red (IR) image  308  may be different from the features imaged in the visible light (VL) image  310  due to the different sensing modalities employed by the sensors. Thus, for example, hot regions may be discerned in image  308 , while heat is not sensed in image  310 . Similarly the visual appearance of a bus may be discerned in image  310 , while features appearing only in visible light are not captured in image  308 . 
   A problem addressed by the present invention is that the visual information contained in individual pairs of images, one belonging to sequence  312  and the other belonging to sequence  314 , (e.g. ( 308 ,  310 ), ( 322 ,  330 ) or ( 322 ,  334 )), is sufficient to establish neither the spatial nor the temporal relationship between two images of a pair. More generally, the visual information contained in individual pairs of images, one of the pair belonging to sequence  312  and the other of the pair belonging to sequence  314 , is sufficient to establish neither the spatial nor the temporal relationship between the two sequences  312  and  314 . 
   In the context of  FIGS. 3A-3D , two images are in the same “temporal relationship” if they are taken at the same time. 
   The unknown spatial relationship of images  308  and  310  is seen graphically by considering three examples of possible relative spatial relationships shown in  FIG. 3B  and designated by reference numerals  316 ,  318  and  320 . In each example, the two images  308  and  310  are placed in a different spatial relationship, all of which are consistent with the visual content of the images  308  and  310 . 
   The unknown temporal relationship of sequences  312  and  314  is seen graphically by considering  FIG. 3C . It is appreciated that it is not apparent from a cursory examination of sequences  312  and  314 , which images in sequence  312  are taken at the same time as which images in sequence  314 . 
   The present invention provides a system and technique for determining the correct relationship between the images  308  and  310  and more generally the correct spatial and temporal relationships between sequences  312  and  314 , as shown in  FIG. 3D . Thus, it may be appreciated that the present invention determines which image in sequence  312  corresponds in time with which image in sequence  314  and further determines the spatial relationship between the images which correspond in time. 
   Thus it is seen in  FIG. 3D  that image A ( 310 ) in sequence  314  is found to correspond in time with image a ( 308 ) in sequence  312 . The correct spatial relationship between images A ( 310 ) and a ( 308 ) is shown in  FIG. 3D  at reference numeral  350 . Similarly, image B ( 330 ) in sequence  314  is found to correspond in time with image b ( 322 ) in sequence  312 , image C ( 332 ) in sequence  314  is found to correspond in time with image c ( 324 ) in sequence  312  and image D ( 334 ) in sequence  314  is found to correspond in time with image d ( 326 ) in sequence  312 . The correct spatial relationship between images B ( 330 ) and b ( 322 ) is shown in  FIG. 3D  at reference numeral  352 , the correct spatial relationship between images C ( 332 ) and c ( 324 ) is shown in  FIG. 3D  at reference numeral  354  and the correct spatial relationship between images D ( 334 ) and d ( 326 ) is shown in  FIG. 3D  at reference numeral  356 . Thus it may be realized that the hot regions captured in the IR image sequence  312  are in fact the radiator and the exhaust gases of the bus. 
   The present invention employs an appreciation that despite there not necessarily being any common static visual information in the two sequences, the image to image dynamics in each of the two sequences are nevertheless related both spatially and temporally. The present invention utilizes the relationship between the dynamics to correlate the two sequences in time and space, without requiring understanding or interpretation of the features captured by the sequences. 
   As will be described hereinbelow in detail, the image to image dynamics preferably are expressed by spatial transformations over time between images in each sequence. The problem of aligning sequences of images in time and space is thus reduced to a problem of determining the relationship between sequences of such transformations in time and in space. 
   It is appreciated from a consideration of  FIG. 3D , that by employing the present invention aligned sequences may be realized, which sequences may then be integrated to provide a composite sequence displaying the totality of information captured by both sequences. In this example, the composite sequence thus captures information over a wavelength range which is beyond that of the individual sensors  300  and  302 . 
   Reference is now made to  FIGS. 4A-4D  which illustrate resolution of spatial and temporal relationships between sequences of images of a scene taken at two different times, producing two corresponding sequences of images. A camera used for producing the sequences of images is fixed to a element which travels along a path that is generally identical for both sequences. 
   As seen in  FIG. 4A , a camera mounted onto an element, traveling along a generally identical path, such as a railroad track, images a scene at two different times, here typically summer and winter. The camera used in the summer is designated by reference numeral  400 , while the camera used in the winter is designated by reference numeral  402 , it being appreciated that the same camera or different cameras may be employed. The cameras may have the same or different settings. 
   Each camera takes a sequence of images of the scene. In this case the scene being imaged at the two different times, has two different appearances, as designated by reference numerals  404  and  406 . Reference numeral  404  designates a view of a train bearing camera  400 , traveling along railroad track  407  in the summer. Reference numeral  406  designates a view of a train bearing camera  402  traveling along railroad track  407  in the winter. 
   Turning now to  FIG. 4B , it is seen that cameras  400  and  402  image portions of the views designated by reference numerals  404  and  406  of  FIG. 4A , generating respective images  408  and  410 . As seen in  FIG. 4C , images  408  and  410  each belong to a sequence of images, each produced by one of the moving cameras  400  and  402 , the respective sequences being designated by reference numerals  412  and  414 . It is seen that sequence  412  also includes images  422 ,  424  and  426 . Likewise, sequence  414  also includes images  430 ,  432  and  434 . It is appreciated that the images in sequence  412  are not necessarily spaced in time (taken at the same time differences) as the images in sequence  414 . 
   It is appreciated that features imaged in the summer may be different from the features imaged in the winter due, for example to snow cover. Thus, for example, a house seen in sequence  412  is not visible in sequence  414 , because it is covered by snow. Similarly snow seen in sequence  414  is not visible in sequence  412 . 
   A problem addressed by the present invention is that the visual information contained in individual pairs of images, one belonging to sequence  412  and the other belonging to sequence  414 , (e.g. ( 408 ,  410 ), ( 422 ,  430 ) or ( 422 ,  434 )), is sufficient to establish neither the spatial nor the temporal relationship between two images of a pair. More generally, the visual information contained in individual pairs of images, one of the pair belonging to sequence  412  and the other of the pair belonging to sequence  414 , is sufficient to establish neither the spatial nor the temporal relationship between the two sequences  412  and  414 . 
   In the context of  FIGS. 4A-4D , two images are the to be in the same “temporal relationship” if an element to which the camera is fixed is at the same location along the path. It is appreciated that the time differences between images  422 ,  424 ,  408  and  426  in sequence  412  may or may not be the same as the time differences between the images  430 ,  432 ,  410  and  434  in sequence  414 . 
   The unknown spatial relationship of images  408  and  410  is seen graphically by considering three examples of possible relative spatial relationships shown in  FIG. 4B  and designated by reference numerals  416 ,  418  and  420 . In each example, the two images  408  and  410  are placed in a different spatial relationship, all of which are consistent with the visual content of the images  408  and  410 . 
   The unknown temporal relationship of sequences  412  and  414  is seen graphically by considering  FIG. 4C . It is appreciated that it is not apparent from a cursory examination of sequences  412  and  414 , which images in sequence  412  are taken from the same position along the railroad track  407  as images in sequences  414 . 
   The present invention provides a system and technique for determining the correct relationship between the images  408  and  410  and more generally the correct spatial and temporal relationships between sequences  412  and  414 , as shown in  FIG. 4D . Thus, it may be appreciated that the present invention determines which image in sequence  412  corresponds temporally with which image in sequence  414  and further determines the spatial relationship between the images which correspond temporally. 
   Thus it is seen in  FIG. 4D  that image A ( 422 ) in sequence  412  is found to correspond temporally with image a ( 430 ) in sequence  414 . The correct spatial relationship between images A ( 422 ) and a ( 430 ) is shown in  FIG. 4D  at reference numeral  450 . Similarly, image B ( 424 ) in sequence  412  is found to correspond temporally with image b ( 432 ) in sequence  414 , image C ( 408 ) in sequence  412  is found to correspond temporally with image c ( 410 ) in sequence  414  and image D ( 426 ) in sequence  412  is found to correspond temporally with image d ( 434 ) in sequence  414 . The correct spatial relationship between images B ( 424 ) and b ( 432 ) is shown in  FIG. 4D  at reference numeral  452 , the correct spatial relationship between images C ( 408 ) and c ( 410 ) is shown in  FIG. 4D  at reference numeral  454  and the correct spatial relationship between images D ( 426 ) and d ( 434 ) is shown in  FIG. 4D  at reference numeral  456 . Thus the location of the house buried under the snow may be determined in sequence  414  although the house is not visible in that sequence. 
   The present invention employs an appreciation that despite there not necessarily being any common static visual information in the two sequences, the image to image dynamics in each of the two sequences are nevertheless related both spatially and temporally. The present invention utilizes the relationship between the dynamics to correlate the two sequences spatially and temporally, without requiring understanding or interpretation of the features captured by the sequences. 
   As will be described hereinbelow in detail, the image to image dynamics preferably are expressed by spatial transformations over time between images in each sequence. The problem of aligning sequences of images spatially and temporally is thus reduced to a problem of determining the relationship between sequences of such transformations. 
   It is appreciated from a consideration of  FIG. 4D , that by employing the present invention aligned sequences may be realized, which sequences may then be employed to determine the location of objects visible in only some sequences across other sequences. The aligned sequences may also be employed to detect changes in a scene, such as man-made changes, which take place during a time interval between acquisitions of different sequences. 
   It is also appreciated that the spatial relationship may also be produced by moving at least one camera at different times and/or in different locations, along generally different trajectories, wherein the at least two trajectories are correlated as if they were produced by two cameras that are mounted rigidly on the same device or rig, and moved jointly in space. 
   It is appreciated that  FIGS. 4A-4D  show one example of a possible visualization of the output of the method of the present invention. However, the method is not limited to this particular output or this particular visualization. It is appreciated that, for example, the output can be numerical, in the form of spatial and temporal transformations, or the output can be visual in the form of aligned and/or integrated video sequences. 
   When numerical outputs are provided, the output transformations can be provided in various possible coordinate systems and when visual outputs are provided, there are many possible visualizations. 
   Reference is now made to  FIGS. 5A-5C , which illustrate an important inventive principle underlying the present invention. As noted hereinabove with reference to  FIGS. 1A-4D , a difficulty exists in establishing spatial and temporal correspondence between images that have little or no visual information in common. 
   Noting that each of the images forms part of a sequence of images which have substantial visual information in common, the present invention overcomes this difficulty by considering intra-sequence spatial transformations between images in each sequence and by then correlating series of such intra-sequence spatial transformations across the sequences. 
   It is appreciated that while visual information common to images within a sequence is used to compute the intra-sequence spatial transformations, these transformations, once computed, do not include any visual information and can therefore be readily correlated across sequences. Thus, once series of intra-sequence transformation are found, there is no longer any need to refer back to the visual information in the images themselves. 
   It is a particular feature of the present invention that the series of intra-sequence spatial transformations of images taken by cameras whose motion is spatially and temporally interrelated, such as described in any of  FIGS. 1A-4D , are spatially and temporally correlated.  FIGS. 5A-5C  and the following description present examples of these correlations which can be employed in accordance with the present invention to derive the spatial and temporal relationship, notwithstanding a lack of common visual information across the image sequences. 
   Specific reference is now made to  FIG. 5A , which illustrates the relationships between image to image transformations within two sequences, induced by motion of first and second cameras  500  and  502  along an axis  504 , the two cameras being arranged at 180 degrees with respect to each other along the axis. Camera  500  captures a sequence of images  506 , while camera  502  captures a sequence of images  508 . 
   Sequence  506  includes inter alia images  510 ,  512  and  514 , while sequence  508  includes inter alia images  520 ,  522  and  524 . It is seen that the motion of camera  500  along axis  504  in a direction indicated by arrow  526  causes an imaged scene including a house to appear smaller in image  512  than in image  510 . This relationship is represented by an intra-sequence image-to-image spatial transformation and the graphical illustration of the transformation from image  510  to image  512  is designated by reference numeral  530 . The arrows shown in the transformation represent the displacement of the corresponding points of the image  512  as the image  512  is transformed to image  510 . Similarly, it is seen that the motion of camera  500  along axis  504  in a direction indicated by arrow  526  causes the imaged scene including the house to appear smaller in image  514  than in image  512 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  532 . 
   It is also seen that the motion of camera  502  along axis  504  in a direction indicated by arrow  526  causes an imaged scene including a tree to appear larger in image  522  than in image  520 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  540 . Similarly, it is seen that the motion of camera  502  along axis  504  in a direction indicated by arrow  526  causes the imaged scene including the tree to appear larger in image  524  than in image  522 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  542 . 
   It is appreciated from a comparison of the series of transformations  530  and  532  with the series of transformations  540  and  542  that cameras  500  and  502  appear to be directed in opposite directions as they move jointly. 
   Specific reference is now made to  FIG. 5B , which illustrates the relationships between image to image transformations within two sequences, induced by motion of first and second cameras  550  and  552  along an axis  554 , the two cameras being arranged at 90 degrees with respect to each other, one of the cameras being aligned substantially parallel to the axis of motion  554 . Camera  550  captures a sequence of images  556 , while camera  552  captures a sequence of images  559 . 
   Sequence  556  includes inter alia images  560 ,  562  and  564 , while sequence  558  includes inter alia images  570 ,  572  and  574 , It is seen that the motion of camera  550  along axis  554  in a direction indicated by arrow  576  causes an imaged scene including a house to appear shifted sideways to the left in image  562  as compared with image  560 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  580 . Similarly, it is seen that the motion of camera  550  along axis  554  in a direction indicated by arrow  576  causes the imaged scene including the house to appear shifted sideways to the left in image  564  as compared with image  562 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  582 . 
   It is also seen that the motion of camera  552  along axis  554  in a direction indicated by arrow  576  causes an imaged scene including a tree to appear larger in image  572  than in image  570 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  590 . Similarly, it is seen that the motion of camera  552  along axis  554  in a direction indicated by arrow  576  causes the imaged scene including the tree to appear larger in image  574  than in image  572 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  592 . 
   It is appreciated from a comparison of the series of transformations  580  and  582  with the series of transformations  590  and  592  that cameras  550  and  552  appear to be directed perpendicularly to each other as they move jointly. 
   Specific reference is now made to  FIG. 5C , which illustrates the relationships between image to image transformations within two sequences, induced by motion of first and second cameras  600  and  602  along an axis  604 , the two cameras being directed in the same direction perpendicular to the direction of motion along axis  604  but at different zooms. Camera  600  captures a sequence of images  606 , while camera  602  captures a sequence of images  668 . 
   Sequence  606  includes inter alia images  610 ,  612  and  614 , while sequence  608  includes inter alia images  620 ,  622  and  624 . It is seen that the motion of camera  600  along axis  604  in a direction indicated by arrow  626  causes an imaged scene including a house to appear shifted sideways to the left in image  612  as compared with image  610 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  630 . Similarly, it is seen that the motion of camera  600  along axis  604  in a direction indicated by arrow  626  causes the imaged scene including the house to appear shifted sideways to the left in image  614  as compared with image  612 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  632 . 
   It is also seen that the motion of camera  602  along axis  604  in a direction indicated by arrow  626  causes an imaged scene including the house to appear shifted sideways to the left in image  622  as compared with image  620 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  640 . Similarly, it is seen that the motion of camera  602  along axis  604  in a direction indicated by arrow  626  causes the imaged scene including the house to appear shifted sideways to the left in image  624  as compared with image  622 . This relationship is represented by an intra-sequence image-to-image spatial transformation designated by reference numeral  642 . It is noted that the sideways shifts represented by transformations  640  and  642  are larger than those represented by transformations  630  and  632  due to the difference in zooms. 
   It is appreciated from a comparison of the series of transformations  630  and  632  with the series of transformations  640  and  642  that cameras  600  and  602  appear directed in the same direction perpendicular to the direction of motion along axis  604  but at different zooms as they move jointly. 
   Reference is now made to  FIGS. 6A ,  6 B and  6 C, which together illustrate that computational functionality employing one type of transformation may not be sufficient for resolving ambiguities in the spatial and temporal relationships between sequences but that employing multiple different types of transformations reduces ambiguities in the spatial and temporal relationships between sequences. 
   Turning to  FIG. 6A , it is seen that intra-sequence spatial transformations  650  and  652  representing two oppositely directed sideways movements, each belonging to a sequence taken by a different camera, can represent at least two different arrangements of the cameras. One such arrangement, designated by reference numeral  654  is of first and second cameras,  656  and  658  respectively, moving in a direction  660  along an axis  662  and being arranged in a mutual upside down arrangement and both directed in parallel perpendicular to axis  662 . 
   Another possible arrangement, designated by reference numeral  664 , is of cameras  656  and  658  being arranged at 180 degrees with respect to each other, directed perpendicular to the axis  662  and moving along axis  662  in a direction  660  therealong. 
   Turning to  FIG. 6B , it is seen that intra-sequence spatial transformations  670  and  672  representing forward and backward motion respective, each belonging to a sequence taken by a different camera, can represent at least two different arrangements of the cameras. One such arrangement, designated by reference numeral  664  is of first and second cameras,  656  and  658  respectively, moving in a direction  680  along an axis  682  and being arranged at 180 degrees with respect to each other along axis  682 . 
   Another possible arrangement, designated by reference numeral  684 , is of cameras  656  and  658  being arranged at 180 degrees with respect to each other along axis  682  for motion in a direction  680  therealong, wherein one of the cameras  656  is rotated about axis  682  with respect to the other one of the cameras  658 . 
   Turning to  FIG. 6C , it is seen that two series of intra-sequence spatial transformations  690  and  692 , each taken by a different moving camera, and containing transformations of different types, are employed by computational functionality to resolve ambiguities of the type described hereinabove with reference to  FIGS. 6A and 6B . Series  690  comprises transformation  650  ( FIG. 6A ) followed by transformation  670  ( FIG. 6B ), while series  692  comprises transformation  652  ( FIG. 6A ) followed by transformation  672  ( FIG. 6B ). 
   It is seen that only one of the three camera configurations shown in  FIGS. 6A ,  6 B and  6 C is possible, namely that configuration designated by reference numeral  664  in  FIGS. 6A and 6B . 
     FIGS. 1A to 4D  described hereinabove present four typical applications of the present invention. In these four applications the present invention provides alignment between images of two sequences of images notwithstanding that there is insufficient similarity between any image of one sequence to any image of the other sequence to enable the prior art to provide image to image alignment. “Similarity” of images is used here in a broad sense and includes, inter alia, gray-level similarity, feature similarity, similarity of frequencies and statistical similarity, such as mutual information. Consequently, in the prior art, at least partial overlap between the images is required to obtain similarity. 
   In contrast to the prior art, which requires overlap between images in order to mutually align them, the present invention employs correlated temporal behavior between the two sequences, as shown and described with reference to  FIGS. 5A to 6C , to resolve both the spatial and the temporal transformations between images of the two sequences. A feature of the present invention is the ability to replace the requirement for “coherent appearance”, which is a fundamental assumption in the prior art, with the requirement for “consistent temporal behavior” which is often easier to satisfy, for example by moving the two cameras jointly in space. 
   In one application of the present invention, shown and described hereinabove with reference to  FIGS. 1A to 1D , the two sequences are produced concurrently by two cameras moving jointly in space. The term ‘concurrently’ here designates that the two sequences are produced at least partially during the same period of time. The concurrency does not require that images, or frames, of the two sequences be produced simultaneously or even at the same rate. ‘Moving jointly’ here means that the two cameras have, for practical purposes, the same center of projection throughout the sequences. This may be realized, for example by fixedly attaching both cameras to the same moving rig. In reality, the centers of projection of the two cameras are not at precisely the same location. However in the mathematical treatment which follows, the centers of projection of both cameras are considered as if they were at the same location. It is further appreciated that the mathematical principles employed in the mathematical treatment are also applicable to situations where the centers of projection of the cameras are clearly not at the same location but are at locations whose spatial relationship is fixed. It is additionally appreciated that the mathematical principles employed in the mathematical treatment are also applicable to situations where the centers of projection of the cameras are clearly not at the same location but are at locations having an at least partially known spatial/temporal relationship. 
   In another application of the present invention, shown and described hereinabove with reference to  FIGS. 2A to 2D  and  FIGS. 3A to 3D , the two sequences contain images that are partially overlapping. However the similarity between the images is very low or even nonexistent, namely, no detail of any image of one sequence can be identified in any image of the second sequence. 
   In the applications shown and described hereinabove with reference to  FIGS. 2A to 2D , the two sequences are produced by cameras employing significantly different zoom. Similar applications may relate to sequences produced by cameras having significantly different magnification or resolution. 
   In the applications shown and described hereinabove with reference to  FIGS. 3A to 3D , the two sequences are produced by cameras employing different sensing modalities. Typical sensing modalities include: using a photographic camera, such as a still film camera, a digital camera, a video camera or a motion film camera; employing infra-red imaging equipment, radar, x-ray, CAT-scan, MRI or other electromagnetic imaging equipment, acoustic imaging equipment, such as ultrasound, sonar and sub-acoustic geophysical surveillance equipment, satellite remote sensing equipment. 
   In yet another application of the present invention shown and described hereinabove with reference to  FIGS. 4A to 4D , the two sequences are produced at different times but the two cameras that produce the respective two sequences are moving generally along the same path. The sequences may not have any spatial overlap or may have a partial overlap but still lack a degree of similarity required by the prior art. Such lack of similarity may result from differences in internal camera calibration, such as magnification, zoom and resolution, or from employing different sensing modalities. This lack of similarity may also result from changes in the scene, such as changes caused by the seasons, as illustrated in  FIGS. 4A to 4D , or due to differences in visibility between the two sequences. 
   The mathematical treatment which follows employs an appreciation that despite there not necessarily being any common static visual information in the two sequences, the image to image dynamics in each of the two sequences are nevertheless related both spatially and temporally. This mathematical treatment utilizes the relationship between the dynamics to correlate the two sequences in time and space, without requiring understanding or interpretation of the features captured by the sequences. 
   As will be described hereinbelow in detail, the image to image dynamics preferably are expressed by spatial transformations over time between images in each sequence. The problem of aligning sequences of images in time and space is thus reduced to a problem of determining the relationship between sequences of such transformations in time and in space. 
   The motions of the two cameras induce motions to the two sequences. The induced motions may be different for the two sequences but are correlated. Although the motions of the cameras are not known, the correlation between the induced motions serves to correlate between sequences both in time and in space. 
   To correlate two images using the image to image dynamics, it is required that:
         (1) Each of the images must be a part of a temporal sequence of images;   (2) The two sequences must be produced during relative motion between cameras and a scene;   (3) The optical parameters of the two cameras must not change other than in a known manner throughout the production of the two sequences.       

   The mathematical treatment which follows also imposes the following requirement in the examples presented therein:
         (4) The cameras producing the two sequences must have approximately the same center of projection throughout the production of the two sequences.       

   The present invention does not require that any single image in one sequence of images have any spatial or visual similarity with any single image in the other sequence. Consequently and in addition, the present invention provides alignment between two images belonging to two sequences of images when the level of similarity between these images is insufficient to provide alignment by employing the prior art. 
   The present invention normally does not require any prior internal or external camera calibration at any time. 
   Each of the two preferred embodiments of the present invention described hereinbelow is applicable to one of the following two situations:
         (a) A substantial portion of the scene is either planar or distant from the cameras producing the two sequences of images;   (b) The scene is effectively three-dimensional.       

   It is appreciated that the two situations each pertain to all the applications described hereinabove in accordance with  FIGS. 1A to 4D . 
   As shown and described hereinabove with reference to  FIGS. 6A to 6C , simple camera motion may not be sufficient to resolve the spatial and the temporal transformations between images of the two sequences. Therefore several different types of camera motions may be required. Three examples of such camera motion are shown in  FIGS. 5A to 5C , it being understood that other, perhaps better, examples of camera motion may exist. An embodiment of the present invention described hereinbelow provides assessment of the required complexity of the motion. 
   Reference is now made to  FIG. 7 , which is a simplified illustration of a complex motion  700  of two cameras  702  and  704 , fixed to each other, each camera producing a sequence of images of a portion of a scene  703 , the sequences being  706 ,  708 ,  710  and  712 ,  714  and  716  correspondingly, the portions of the scene photographed by the two cameras being non-overlapping. 
   Reference is now made to  FIG. 8 , which is a simplified illustration portions of two sequences  720  and  722  of images taken as shown in  FIG. 7 , wherein the two sequences are spatially related by a fixed and unknown inter-camera homography and temporally related by a fixed and unknown time shift Δt. The symbols I i , T i , T′ i  and H are defined hereinbelow. 
   Reference is now made to  FIG. 9 , which is a functional block diagram of a preferred sequence of steps of implementation of a preferred embodiment of the present invention. 
   In step  900  two cameras  910  and  920  are employed to produce two sequences of images  930  and  940 . 
   Let S be a first sequence of images I i  produced by a first camera and let S′ be a second sequence of images I′ i  produced by a second camera, for example as shown and described in accordance with any of the  FIGS. 1A to 8  hereinabove, and wherein S=I i , . . . , I n+1  and S′=I′ i , . . . , I′ n+1 . 
   The two input sequences have an unknown temporal relation and an unknown spatial relation between them. 
   In a preferred embodiment of the present invention the spatial relation may represent the fact that the two cameras are firmly attached to each other. 
   In another preferred embodiment of the present invention the two cameras are mounted rigidly on the same device or rig, and moved jointly in space. 
   In yet another preferred embodiment of the present invention the spatial relation may represent the fact that the cameras are moving in a generally identical trajectory but at different times. 
   In an additional preferred embodiment of the present invention the spatial relation is produced by moving at least one cameras in a generally identical trajectory but in at least one of a different time and a different location, wherein there is a fixed relative position and a fixed relative orientation between the two cameras and wherein the optical properties of each of the two cameras do not change between the images I i  and I i+1  and between I′ i  and I′ i+1 . 
   In a further preferred embodiment of the present invention the spatial relationship is produced by moving at least one cameras in a generally identical trajectory but in at least one of a different time and a different location, wherein the relative position and a relative orientation between the two cameras may change in a known way and wherein the optical properties of each of the two cameras may change between the images I i  and I i+1  and between I′ i  and I′ i+1  in a known way. 
   In a further preferred embodiment of the present invention the spatial relationship is produced by moving at least one camera at different times and/or in different locations, along generally different trajectories, wherein the at least two trajectories are correlated as if they were produced by two cameras that are mounted rigidly on the same device or rig, and moved jointly in space. 
   The temporal relation between the two sequences represents the time when the two cameras were in the spatial relation as described hereinabove. 
   In a preferred embodiment of the present invention the two sequences are produced by employing synchronization equipment to synchronize between the two cameras. In this case the temporal relation between the two sequences represents the fact that the image I i  and the corresponding image I′ i  are produced together. 
   In another preferred embodiment of the present invention the temporal relation between the two sequences represents the fact the image I i  and the corresponding image I′ i  are produced with a fixed time delay between them. 
   In yet another preferred embodiment of the present invention the temporal relation applies to image I i  and a point in sequence S′ that is between a corresponding I′ i  and image I′ i+1 . 
   In yet another preferred embodiment of the present invention the temporal relation applies to two sequences produced employing two cameras employing two different frame rates. 
   In a preferred embodiment of the present invention the temporal relation between the two sequences may represent the fact that the two images are the to be in the same “temporal relationship” if the cameras producing the sequences were in the same location in the same time. 
   It is appreciated that temporal relation between sequences may exist even if the temporal relation between images may hold only for virtual images, that is, for images that have not been produced but could have been produced if the camera was producing an image at that time. 
   It is also appreciated, that S and S′ are not necessarily the original sequences, but may be subsequences of the original sequences, where subsequence may represent, a portion of the field-of-view, temporal portion of the sequence, sub-sampling of pixels, and sub-sampling of frames. 
   The sequences S and S′ may be produced in several methods as described hereinbelow: 
   The first and the second camera may have a different 3D orientation and move jointly in space, such as the camera pairs shown in  FIG. 7 . In a preferred embodiment of the present invention none of the images I i  contains any feature contained in any of the images I′ i , however, there is at least a partial temporal overlap between S and S′. 
   In another preferred embodiment of the present invention the first camera and the second camera have at least partially overlapping fields of view, wherein images I i  comprises details that are not comprised in the overlapping parts of the corresponding images I′ j  and wherein the two cameras move jointly in space, such as the camera pair shown and described in accordance with  FIGS. 3A to 3D  hereinabove. 
   In yet another preferred embodiment of the present invention the sequence S and the sequence S′ may be produced by the same camera moving in at least approximately the same pattern in space and at different times. 
   Also in another preferred embodiment of the present invention the sequence S and the sequence S′ may be produced by two cameras moving in at least approximately the same pattern in space and at different times. 
   It is appreciated that the sequences S and S′ may be produced by any image producing instrument that produces at least two dimensional images of space, such as photographic camera, such as still film camera, digital camera, video camera or motion film camera, infra-red imaging equipment, radar, x-ray, CAT-scan, MRI and other electromagnetic imaging equipment, acoustic imaging equipment, such as ultrasound, sonar and sub-acoustic geophysical surveillance equipment and satellite remote sensing equipment. It is also appreciated that the two sequences S and S′ may be produced by image producing instruments of the same type or of different types. 
   It is appreciated that neither of the two sensors has to be calibrated. 
   The temporal correspondence can be achieved electronically, for example by employing Genlock, or recovered using the method describe hereinbelow with reference to element  960  of  FIG. 9 . 
   Furthermore, a method for computing temporal relation described hereinbelow can compute temporal relation at sub-frame time units. In this case a temporal relation is computed between a transformation T i  and a transformation T′ j+δ  wherein T j+δ  is interpolated from T j  and T j+1 . The sequence of transformations T j+δ  applies to virtual images I j+δ  wherein in the virtual images were not actually produced by the camera producing image sequence S′, but could have been produced by the camera if the images produced at the appropriate time points and are therefore interpolated representations of the scene. 
   Reference is now made to step  990  of  FIG. 9 . It can now be assumed that the transformations and images are temporally synchronized. 
   Let H be a spatial transformation function that transforms between a coordinate system of sequence S and a coordinate system of sequence S′. Applying the transformation H on the sequence S′ results in the content of images I′ i  in the coordinates of the sequence S, as if the transformed sequence S′ has been produced by the camera that has produced the sequence S. Therefore:
 
 p′=H ( p ) where p′ is the feature p of I i  in the coordinates of I′ i .
 
   In one preferred embodiment of the present invention, typically where the two cameras have approximately the same center of projection, H is a 3×3 inevitable matrix. 
   Such H is denoted here as a “homography”, the terms collination and 2D projective transformation may be equally used. In this case p and p′ are given in homogeneous coordinates and the equation p′=H(p) becomes p′≅Hp where ≅ denotes equation up to a scale factor. 
   In another preferred embodiment of the present invention, typically where the two cameras have a different center of projection and are firmly attached to the same rig H is a 4×4 inevitable Euclidean or inevitable projective matrix. 
   However, the two sequences S and S′ do not share sufficient common feature or any other properties of the images. Therefore we cannot compute H from common properties of the images I i . Instead H is computed by correlating between properties of the temporal progressions within each sequence 
   In a preferred embodiment of the present invention H is computed from the induced frame to frame transformation within each of the sequences S and S′. 
   Let T be a first sequence of frame-to-frame transformations T i  and let T′ be a second sequence of frame-to-frame transformations T′ i  wherein T=T 1 , . . . , T n+1  and T′=T′ i , . . . , T′ n+1  and wherein I i +1 ≅T i I i  and I′ i+1 ≅T′ i I′ i . 
   In a preferred embodiment of the present invention, a large portion of the scene that is photographed by the two cameras, producing the sequences S and S′, is planar or distant from the cameras. In this case T i  and T′ i  are 3×3 non-singular matrices. 
   Let P be a three dimensional point in the planar or distant scene and let p i  and p′ i  be the homogeneous coordinates of the projection of the point P in the images I i  and I′ i , respectively. It is noted that there is no need for P to appear in both or in any of the images I i  and I′ i , that is, within the field of view of the cameras when images I i  , or I′ i  are produced. Let p i+1 , and p′ i+1  be the coordinates of the projection of the point P in images I i+1  and I′ i+1  respectively. 
   Then:
 
 P   i+1   ≅T   i   p   i  and  p′   i+1   ≅T′   i   p′   i  
 
and
 
 p′   i   ≅Hp   i  and  p′   i+1   ≅Hp   i+1  
 
and therefore Eq. (1) is:
 
 HT   i   p   i   ≅Hp   i+1   ≅p′   i+1   ≅T′   i   p′   i   ≅T′   i   Hp   i  
 
and therefore Eq. (2) is:
 
 HT   i ≅T′ i H
 
This operation is also referred to as “matrix composition”.
 
   Since H is inevitable then Eq. (3) is:
 
 T′   i   =s   i   HT   i   H   −1   where s is a frame dependent scale factor.
 
   Eq.(3) is true for all the images, that is, for any pair of temporally corresponding transformations T i  and T′ i , where i=1, . . . , n. This equation shows that there is a similarity relation between the two matrices T i  and T′ i  up to a scale factor. A matrix A is the to be “similar” to a matrix B if there exists an inevitable matrix M such that A=MBM −1 . The term “conjugate matrices” can be used instead of “similar matrices”. 
   Let eig(A)=[λ 1 , λ 2 , μ 3 ] t  be a 3×1 vector containing, in decreasing order, eigenvalues of a 3×3 matrix A. □ t  denotes the transpose vector. It is known in the art, as described for example in C. E. Pearson (ed.). Handbook of applied mathematics Second Edition. Van Nostrand Reinhold Company, New York, 1983, pp 898, that: If A and B are similar matrices then they have the same eigenvalues: eig(A)=eig(B), and, the eigenvalues of a scaled matrix are scaled: eig(sA)=s(eig(A)). 
   Using these two facts and Eq. (3) we obtain Eq. (4) hereinbelow:
 
eig( T′   i )= s   i eig( T   i ) where si is the scale factor defined by Eq. (3).
 
   Equation (4) implies that eig(T i ) and eig(T′ i ) are “parallel”. This gives rise to a measure of similarity between two matrices T i  and T′ i , denoted by sim(T i , T′ i ) and presented in Eq. (5) hereinbelow: 
               sim   ⁡     (       T   l     ,     T   l   ′       )       =           eig   ⁡     (     T   l     )       i     ⁢           ⁢     eig   ⁡     (     T   l   ′     )                  eig   ⁡     (     T   l     )            ⁢          eig   ⁡     (     T   l   ′     )                  ,         
where ∥·∥ is the vector norm.
 
   For each pair of the temporally corresponding transformations T i  and T′ i , in the sequences S and S′ the eigenvalues eig(T i ) and eig(T′ i ) are first computed. The scale factor s i  which relates the eigenvalues eig(T i ) and eig(T′ i ) is then estimated from Eq. (4). Eq. (4) is a set of three equations with one unknown and can be solved using least squares minimization. Alternatively, the input homographies T i  and T′ i  can be normalized to have determinant equal 1 and to avoid the need to compute s i  . 
   Once s i  is estimated, Eq. (3) or Eq. (2) can be rewritten in the form of Eq. (6) hereinbelow:
 
 s   i   HT   i   −T′   i   H= 0
 
   Equation (6) is linear in the unknown components of H. Rearranging the components of H in a 9×1 column vector {right arrow over (h)}=[H 11 H 12 H 13 H 21 H 22 H 23 H 31 H 32 H 33 ] t , Eq. (6) can be rewritten as a set of linear equations in h in the form of Eq. (7) hereinbelow:
 
M i {right arrow over (h)}={right arrow over (0)}
 
wherein M i  is a 9×9 matrix defined by T i , T′ i  and s i :
 
             M   i     =       [               s   i     ⁢     T   i   i       -       T     i   11     ′     ⁢   I               -     T     i   12     ′       ⁢   I             -     T     i   13     ′       ⁢   I                 -     T     i   21     ′       ⁢   I               s   i     ⁢     T   i       -       T     i   22     ′     ⁢   I               -     T     i   23     ′       ⁢   I                 -     T     i   31     ′       ⁢   I             -     T     i   32     ′       ⁢   I               s   i     ⁢     T   i       -       T     i   33     ′     ⁢   I             ]       9   ×   9             
and wherein I is the 3×3 identity matrix.
 
   Eq. (7) implies that each pair of corresponding transformations T i  and T′ i  contributes 9 linear constrains in the unknown homography H (i.e., {right arrow over (h)}). 
   The constraints from all the transformations T 1 , . . . , T n  and T′ 1 , . . . , T′ n  can be combined into a single set of linear equations in {right arrow over (h)} provided by Eq (8) hereinbelow:
 
A{right arrow over (h)}={right arrow over (0)} where A is a 9n×9 matrix:
 
   
     
       
         
           A 
           = 
           
             [ 
             
               
                 
                   
                     M 
                     1 
                   
                 
               
               
                 
                   ⋮ 
                 
               
               
                 
                   
                     M 
                     n 
                   
                 
               
             
             ] 
           
         
       
     
   
   Equation (8) is a homogeneous set of line equations in {right arrow over (h)} that can be solved in a variety of ways. In a preferred embodiment of the present invention {right arrow over (h)} is solved by computing the eigenvector which corresponds to the smallest eigenvalue of the matrix A t A. 
   In another preferred embodiment of the present invention the scene for which the sequences S and S′ have been produced is neither planar nor distant. In this case the temporal progression between any two consecutive images I i  and I i+1  is described by a fundamental matrix F i . 
   The fundamental matrix Fdefines the relation between corresponding image points p i  so that:
         if p i  in I i  and p i+1  in I i+1  are corresponding image points, then   p t   i+1 F i p i =0 wherein F i  are each a 3×3 matrix of rank 2.       

   Let F be a sequence of fundamental matrices F i  wherein F=F 1 , . . . , F n  Respectively, the temporal progression between any two consecutive images I′ i  and I′ i +1  is described by F′ i , wherein F′=F′ 1 , . . . , F′ n . 
   Many methods for computing fundamental matrices are known in the prior art. One such method is taught by P. H. S. Torr and A. Zisserman in “Feature based methods for structure and motion estimation” in the proceedings of the Vision Algorithms Workshop pages 279-290, Corfu, 1999. Another method is taught by Z. Zhang, R. Deriche, O. Faugeras, and Q. Luong in “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry” in the journal of Artificial Intelligence, 78:87-119, 1995. 
   The two sequences S and S′ may not share sufficient common features or any other properties of the images. Therefore the spatial and the temporal relations between the sequences S and S′ are computed by correlating between properties of the temporal progressions F=F 1 , . . . , F n  and F′ 1 , . . . , F′ n  within each sequence. 
   It is appreciated that neither of the two cameras has to be calibrated. 
   It is appreciated that because the two cameras share the same center of projection the inter-camera transformation remains a homography H although the relations within each sequence are characterized by fundamental matrices, 
   In a preferred embodiment of the present invention, where the scene for which the sequences S and S′ have been produced is neither planar nor distant, H is computed from properties of the temporal progression output as expressed by the sequences F and F′ of the matrices F i  and F′ i . 
   Each fundamental matrix F, can be decomposed into a homography+epipole as described by Eq. (9) hereinbelow:
 
F i   =[e   i ] x   T   i  wherein e i  is the epipole relating frames I i  and I i+1 ;
         wherein the matrix T i  is the induced homography from I i  to I i+1  via any plane, real or virtual; and   wherein [·] x  is the cross product matrix [v] x  {right arrow over (w)}={right arrow over (v)}×{right arrow over (w)}.       

   The homographies, T 1 , . . . , T n  and T′ 1 , . . . , T′ n , and the epipoles e 1 , . . . , e n  and e′ 1 , . . . , e′ n , impose separate constraints on the inter-camera homography H. These constraints can be used separately or jointly to recover H as described. hereinbelow. 
   (i) Homography-Based Constraints: 
   The homographies T 1 , . . . , T n  and T′ 1 , . . . , T′ n  extracted from the fundamental matrices F 1 , . . . , F n  and F′ 1 , . . . , F′ n , respectively may correspond to different three dimensional planes. In order to apply the algorithm described above for the case of a planar or a distant scene, using these homographies, we have to impose plane-consistency across the two sequences, to guarantee that temporally corresponding homographies correspond to the same plane in the three dimensional world. 
   In a preferred embodiment of the present invention the “Plane+Parallax” method which is well known in the prior art, is used to impose plane-consistency across and within the two sequences T 1 , . . . , T n  and T′ 1 , . . . , T′ n . The Plane+Parallax method describes the image-to-image transformation using two components: homography and residual parallax. The homography is associated with a selected plane and relates points on this plane between the two images. The parallax component describes the residual displacement of off-plane points. The “Plane+Parallax” method requires that a real physical planar surface be visible in all images I i  and I′ i . 
   One method for computing a Plane+Parallax transformation is taught by M. Irani, P., Anandan and D. Weinshall in “From reference frames to reference planes: Multi-view parallax geometry and applications” in the proceedings of the European Conference on Computer Vision, Freiburg, June 1998. 
   Another method for computing a Plane+Parallax transformation is taught by R. Kumar, P. Anandan and K. Hanna in “Direct recovery of shape from multiple views: parallax based approach” in the proceedings of the International Conference on Pattern Recognition, 1994. 
   Yet another method for computing a Plane+Parallax transformation is taught by Harpreet Sawhney in “3D geometry from planar parallax” in the proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, June 1994. 
   Another method for computing a Plane+Parallax transformation is taught by A. Shashua and N. Navab in “Relative affine structure: Theory and application to 3D reconstruction from perspective views” in the proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 483-489, Seattle, Wash., June 1994. 
   The homography part of the Plane+Parallax transformation is then used in the same manner as in the planar or distant case described hereinabove. 
   In another preferred embodiment of the present invention the “threading” method, as described for example in S. Avidan and A. Shashua. Threading fundamental matrices. In European Conference on Computer Vision, 1998, which is also well known in the prior art, is used to impose plane-consistency across and within the two sequences T 1 , . . . , T n  and T′ 1 , . . . , T′ n . The “threading” method can impose plane-consistency within each sequence, even if no real physical plane is visible in any of the frames. 
   It is appreciated that there are other methods in the prior art for computing a consistent set camera matrices. It is appreciated that the term camera matrix refers to a 4×3 matrix, which is comprised of a 3×3 left minor. matrix and a 1×3 vector, wherein the 3×3 left minor matrix represents a 3×3 homography. The consistency of the camera matrices imposes plane-consistency on the 3×3 homographies. 
   In a preferred embodiment of the present invention plane consistency between the two sequences is provided by applying the threading method on each of the two sequences S and S′, by initiating the threading method at frames which are known to simultaneously view the same real plane in both sequences. 
   It is appreciated that the two cameras can see different portions of the plane, allowing for non-overlapping fields of view, and need not see the plane at any of the other images. 
   (ii) Epipole-Based Constraints: 
   In a preferred embodiment of the present invention the homography H is computed from epipoles e i  and e′ i , which are computed from the fundamental matrices F i  and F′ i . Given two images I i  and I i+1 , an epipole denotes the spatial location of the center of projection of the camera for image I i+1  in the coordinates system of image I i . The fundamental matrices F 1 , . . . , F n  and F′ 1 , . . . , F′ n  provide a list of epipoles e 1 , . . . , e n  and e′ 1 , . . . , e′ n , wherein an epipole e 1  is the null space of a fundamental matrice F i  and an epipole e′ i  is the null space of a fundamental matrice F′ i . 
   It is appreciated that the epipoles e i  and e′ i  can be computed by other methods than from the null space of the fundamental matrices F i.  and F′ i.  respectively. There are many other methods known in the prior art to find epipoles between two images. 
   The epipoles e i  and e′ i  are uniquely defined and therefore there is no issue of plane consistency. Since the two cameras have the same center of projection, then for any pair of images e i  and e′ i  H satisfies the equation:
 
e′ i ≅He i ,
 
which yields Eqs. (10a and 10b) hereinbelow:
 
   
     
       
         
           
             
               
                 
                   
                     
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                             4 
                           
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   Multiplying by the denominator and rearranging terms yields two new linear constrains on H for every pair of corresponding epipoles e i  and e′ i  as described in Eq. (11) hereinbelow: 
                 [           e   l   i             0   _     i               (     e   l   ′     )     x     ⁢     e   l   i                   0   _     i           e   l   i             (     e   l   ′     )     ⁢     e   l   i             ]       2   ×   9       ⁢     h   →       =   0         
where {right arrow over (0)} t =[0,0,0].
 
   Therefore, every pair of temporally corresponding epipoles e i  and e′ i  imposes two linear constraints on H. The two sequences e 1 , . . . , e n  and e′ 1 , . . . , e′ n  provide a homogeneous linear system of 2n equations, wherein Eq(11) is one pair of these 2n equations. 
   The system of 2n equations can be solved for H in a variety of ways. 
   In a preferred embodiment of the present invention {right arrow over (h)} is solved by computing the eigenvector which corresponds to the smallest eigenvalue. Theoretically, four pairs of corresponding epipoles e i  and e′ i  are sufficient, provided that no three epipoles are on the same line. 
   Alternatively, H can be solved by adding the system of 2n equations to the set of linear equations in Eq. (8) which are imposed by the homographies. 
   Reference is now made to step  960  of  FIG. 9 . 
   In yet another preferred embodiment of the present invention the sequences S and S′ are not temporally synchronized. Namely, the image I i  in sequence S corresponds to image I′ i+Δt  in sequence S′ and Δt is unknown. Therefore the transformation T i  corresponds to transformation T i+Δt  and not to T′ i . If time stamping, or similar additional information, is available for each image I i  and I′ i  then synchronization can be recovered. When there is no additional information to recover the synchronization, the synchronization can be recovered in using the method hereinbelow. 
   Given two unsynchronized sequences of transformations T 1 , . . . , T n  and T′ 1 , . . . , T′ m , wherein T i  and T′ i+Δt  are temporally corresponding transformations. According to Eq. (4) eig(T i )∥eig(T′ i+Δt ) meaning that the 3×1 vectors of eigenvalues are parallel, namely there is an angle of 0° between the two vectors. Therefore the similarity measure sim(T t     i   ,T′ t′     i+Δt   ) of Eq. (5) is equal to 1, corresponding to cos(0). All pairs of temporally corresponding transformations T i  and T i+Δt , must simultaneously satisfy this constraint. When the unknown temporal shift Δt can be modeled by a set of parameters Δt i =f(i,a i  . . . a n ), then the parameters can be recovered by maximizing the following objective function presented hereinbelow as Eq. (12): 
             SIM   ⁡     (       a   l     ,   …   ⁢           ,     a   n       )       =       ∑   i     ⁢       sim   ⁡     (       T   i     ,     T     i   +     Δ   ⁢           ⁢     t   i             )       2             
For example, when the unknown Δt is a simple global temporal shift, it can be recovered by maximizing the following objective function presented hereinbelow as Eq. (12.a):
 
   
     
       
         
           
             SIM 
             ⁡ 
             
               ( 
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
               
               ) 
             
           
           = 
           
             
               ∑ 
               i 
             
             ⁢ 
             
               
                 sim 
                 ⁡ 
                 
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                           ⁢ 
                           
                               
                           
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                   ) 
                 
               
               2 
             
           
         
       
     
   
   In a preferred embodiment of the present invention the maximization is performed by an exhaustive search over a finite range of valid time shifts Δt. 
   Alternatively, to address larger temporal shifts, a hierarchical search is applied. Coarser temporal levels are constructed by composing transformations to obtain fewer transformation between more distant frames. 
   In still another preferred embodiment of the present invention the images in sequence S are produced at a different rate than the images of sequence S′. For example, when sequence S is produced by an NTSC camera at 30 frame per second and sequence S′ is produced by a PAL camera at 25 frames per second. The ratio between the rate of recording of the two sequences S and S′ is fixed and in the case of PAL and NTSC cameras it is 25:30=5:6. Therefore, the objective function that should be maximized for an PAL-NTSC pair of sequences as presented by Eq. (13) hereinbelow: 
             SIM   ⁡     (     Δ   ⁢           ⁢   t     )       =       ∑   i     ⁢       sim   (       T     5   ⁢   l       5   ⁢     (     i   +   1     )         ,     T       6   ⁢   l     +     Δ   ⁢           ⁢   t           ′6   ⁡     (     i   +   1     )       +     Δ   ⁢           ⁢   t           )     2             
where T i   j  is the transformation from frame I i  to frame I j .
 
   PAL and NTSC video cameras produce each image as pair of fields, wherein the first field comprises the odd lines of the image and the second field comprises the even lines of the picture. The fields are produced in a rate that is double the rate of the images, or frames, that is at 60 and 50 fields per second respectively. The method described above with respect to Eq. (12.a) can recover synchronization to the field level. Sub-field accuracy can be recovered by interpolating the values of SIM(Δt) obtained at discrete time shifts. 
   In a preferred implementation of the present invention the transformations are interpolated according to the computed temporal correspondence and SIM(Δt) is re-estimated, in an iterative manner, to increase the temporal accuracy. 
   In order to solve H it is required to have a sufficient number N min  of corresponding pairs of frame-to-frame transformations T i  and T′ i . 
   In a preferred embodiment of the present invention the number N min  is assessed by examining via the similarity equation Eq. (3) the number of constraints imposed on H by a single pair of transformations T i  and T′ i . 
   The scale factor s i  can be extracted directly from T i  and T′ i  as described hereinabove with reference to element  950  of  FIG. 9  and therefore can be omitted from Eq. (3). Therefore, the number of constraints imposed on H by a single pair of transformations via the similarity equation Eq. (3) can be estimated by examining the number of constraints imposed on H by an equation of the form G=HBH −1  where B=T i  and G=T′ i . A general analysis of matrix equations of the form GH=HB may be found in The theory of matrices. Chapter VIII, by F. R. Gantmakher from Chelsea Pub., New York, 1959. 
   The following notations are used hereinbelow:
         denote by λ 1 , λ 2 , λ 3  the eigenvalues of the matrix B in decreasing order (|λ 1 |≧|λ 2 |≧|λ 3 |);   denote by {right arrow over (u)} b1 , {right arrow over (u)} b2 , {right arrow over (u)} b3  the corresponding eigenvectors with unit norm (∥{right arrow over (u)} b1 ∥=∥{right arrow over (u)} b2 ∥=∥{right arrow over (u)} b3 ∥);   denote by r j  the algebraic multiplicity of the eigenvalue λ j ; and   denote by V j ={{right arrow over (v)}∈R n :B{right arrow over (v)}=λ j {right arrow over (v)}} the corresponding eigen subspace.       

   It is noted that:
         If λ 1 ?λ 2 ?λ 3  then the algebraic multiplicity of all eigenvalues is 1 (r j =1);   If λ 1 =λ 2 ?λ 3  then the algebraic multiplicity of λ 1  and λ 2  is 2, and the algebraic multiplicity of λ 3  is 1 (r 1 =r 2 =2 and r 3 =1); and that   If λ 1 =λ 2 =λ 3  then the algebraic multiplicity of λ 1 , λ 2 , and λ 3  is 3 (r 1 =r 2 =r 3 =3).       

   Similar matrices, also known as conjugate matrices, such as B and G, have the same eigenvalues but different eigenvectors. The eigenvectors of the similar matrices are related by H. If ub is an eigenvector of B with corresponding eigenvalue λ, then Hu b  is an eigenvector of G with the same eigenvalue λ so that G(Hu b )=λ(Hu b ). The same applies for eigen subspaces, wherein if V is an eigen subspace of B corresponding to an eigenvalue λ, then H(V) is an eigen subspace G with the same eigenvalue λ. The number of constraints imposed on H by B and G is therefore investigated according to the dimensionality of the eigen subspaces of B and G. Let V be the eigen subspace corresponding to an eigenvector u b  of B. Three possible cases are valid, one case for each possible dimensionality of V, i.e., dim(V)=1, 2, 3.
 
dim( V )=1.   Case I
 
   This case typically occurs when all three eigenvalues are distinct. The case can also occur when some eigenvalues have algebraic multiplicity of two or three. In all these cases, V is spanned by the single eigenvector u b . Similarly H(V) is spanned by the eigenvector u g  of G. Therefore Eq. (13):
 
Hu b =au g  
 
wherein α is an unknown scale factor
 
   Eq. (13) provides three linear equations in Hand one new unknown a and therefore Eq. (13) provides two new linearly independent constrains on H.
 
dim( V )=2.   Case II
 
   This case typically occurs in one of the following two cases:
         (a) when there exists an eigenvalue with algebraic multiplicity two; or   (b) when there is only one eigenvalue with algebraic multiplicity three.       

   However, the eigen subspace spanned by all eigenvectors has dimensionality of two. For example, a homography defined by pure shift (Δx, Δy) has the form: 
   
     
       
         
           H 
           = 
           
             
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
             . 
           
         
       
     
   
   This matrix has a single eigenvalue λ 1 =λ 2 =λ 3 =1 with algebraic multiplicity three. The corresponding eigen subspaces has dimensionality 2. It is spanned by two linearly independent eigenvetors [1,0,0] t  and [0,1,0] t . 
   When dim(V)=2 the two eigenvectors span V, such that, for example, w.l.o.g., u b1  and u b2 . Therefore every linear combination of u b1  and u b2  is also an eigenvector of B with the same eigenvalues. Similarly, every linear combination of u g1  and u g2  is an eigenvector of G with the same eigenvalue. Therefore Eq. (14):
 
 Hu   hj   =α   j   u   g1 +β j   u   g2  where α b  and β j  are unknown scalars (j=1,2)
 
   Therefore, each of the two eigenvectors ub 1  and ub 2  provides 3 linear equations and 2 new unknowns. Therefore, together the two eigenvectors u b1  and u b2  provide 2 new linear constraints on H.
 
dim( V )=3.   Case III
 
   In this case any vector is an eigenvector and all the eigenvectors have the same eigenvalues λ. This is the case when B≅G≅λI are the identity transformation up to a scale factor, which means that there is no motion. In this case B and G provide no additional constraints on H. 
   The number of constraints imposed on H were counted hereinabove for a single eigen subspace. In order to count the total number of linear constraints that B and G impose on H every possible combination of eigen subspaces is analyzed according to the algebraic multiplicity of the eigenvalues. There are three such combinations:
         1. λ i ≠λ j ≠λ k , which implies that:
           V i ≠V j ≠V k:  and   dim(V i ) dim(V j )=dim(V k )=1.   2. λ i =λ j ≠λ k , which implies that V i =V j ≠V k . There are two such cases:   (a) dim(V i =V j )=2 and dim(V k )=1.   (b) dim(V i =V j )=1 and dim(V k )=1.   
           3. λ i =λ j =λ k          

   In this case there is only a single eigen subspace V=V i =V j =V k , which dimensionality may be 1, 2, or 3. 
   The following table summarizes the number of linearly independent constraints of each of the above cases: 
   
     
       
         
             
             
             
             
             
           
             
                 
                 
             
             
                 
                 
               Eigenvalues 
                 
               # of Linearly 
             
             
                 
                 
               Algebraic 
               Eigen Subspaces 
               independent 
             
             
                 
               Case 
               Multiplicity 
               dimensionality 
               constraints 
             
             
                 
                 
             
           
          
             
                 
               (1) 
               λ i  ≠ λ j  ≠ λ k   
               |V i | = |V j | = |V k | = 1 
               6 
             
             
                 
               (2.a) 
               λ i  = λ j  ≠ λ k   
               |V i  = V j | = 2, |V k | = 1 
               4 
             
             
                 
               (2.b) 
               λ i  = λ j  ≠ λ k   
               |V i  = V j | = 1, |V k | = 1 
               4 
             
             
                 
               (3.a) 
               λ i  = λ j  = λ k   
               |V i  = V j  = V k | = 1 
               2 
             
             
                 
               (3.b) 
               λ i  = λ j  = λ k   
               |V i  = V j  = V k | = 2 
               2 
             
             
                 
               (3.c) 
               λ i  = λ j  = λ k   
               |V i  = V j  = V k | = 3 
               0 
             
             
                 
                 
             
          
         
       
     
   
   Therefore, when B and G have either two or three distinct eigenvalues, which is typical of general frame-to-frame transformations, then two independent pairs of transformations suffice to uniquely determine H, since each pair of transformations imposes 4 to 6 linearly independent constraints, and, according to the prior art, 8 independent linear constraints suffice to uniquely resolve H, up to an arbitrary scale factor. In a preferred embodiment of the present invention all available constraints from all pairs of transformation, for increased numerical stability. 
   In a preferred embodiment of the present invention two cameras are employed to produce two input sequences, preferably several seconds long to provide significant motion. 
   Referring to step  950 , frame-to-frame input transformations, T 1 , . . . , T n  and T′ 1 , . . . , T′ n.  are preferably extracted, for the case of the planar or distant scene, using the method described in, for example, M. Irani, B. Rousso and S. Peleg. Computing occluding and transparent motions. International Journal of Computer Vision, 12(1):5-16, January 1994. 
   Reference is now made to step  970  of  FIG. 9 . Step  970  comprises methods of robust statistics to enhance the accuracy of the transformations T i  and T′ i . In a preferred embodiment of the present invention step  970  comprises two optional methods:
         (a) Outlier rejection, designated by reference numeral  972 ; and   (b) Coordinate renormalization, designated by reference numeral  974 .       

   In step  972  inaccurate frame-to-frame transformations T i  are preferably pruned out by employing any of the two outlier detection mechanisms known in the prior art as described hereinbelow:
             (i) In a preferred embodiment of the present invention, the transformation between successive frames within each sequence are computed in both directions. The distance of the composed matrix T i T i   Reverse  from the identity matrix in the image space is computed, in terms of the maximal residual misalignment of pixels.           

   
     
       
         
           
             
               
                 
                   Reliability 
                   ⁡ 
                   
                     ( 
                     
                       T 
                       i 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     max 
                     
                       p 
                       ∈ 
                       
                         l 
                         i 
                       
                     
                   
                   ⁢ 
                   
                     
                        
                       
                         
                           
                             T 
                             i 
                           
                           ⁢ 
                           
                             T 
                             i 
                             Reverse 
                           
                           ⁢ 
                           p 
                         
                         - 
                         p 
                       
                        
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
       
       
         
           
             
               (ii) Alternatively, the similarity criterion of Eq. (5) is employed to verify the degree of “similarity” between T i  and T′ i . An unreliable pair of transformations can thus be pruned out. 
             
           
         
       
     
  
   In step  974 , optionally, any one of two methods of coordinate renormalization known in the prior art is used as described hereinbelow:
         (a) The input matrices are re-normalized to increase the numerical stability, as described for example in Richard L. Hartley. In Defence of the 8-point Algorithm. In Pattern Recognition and Machine Intelligence 19 (6) June pages 580-593 1997.   (b) Input matrices are normalized so that the rows of M have approximately the same norm, using the heuristic provided in for example, Gene Golub and Charles Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore and London, pp. 123-127, 1989.       

   Reference is now made to step  980  of  FIG. 9 . 
   In a preferred embodiment of the present invention, when the frame-to-frame transformations are too. small, temporal sub-sampling of the sequences can provide more significant transformations between successive frames. Preferably, temporal sub-sampling is done after recovering the temporal synchronization, in order to guarantee that temporally corresponding frames are sampled from the two video sequences. 
   In another preferred embodiment of the present invention the three dimensional joint motions of the two cameras are planned in advance to increase the accuracy of the computed result. 
   For example, in a preferred embodiment of the present invention input data are sequences of transformation T i  and T′ i . The transformation T i  and T′ i  are extracted from two video sequences using one of a variety of algorithms provided in the prior art. The reliability of the input data also depends on the specific algorithm used to extract the input transformations. Employing a rough approximation to the accuracy. of the different transformations T i  and T′ i  it is possible to analyze the combinations of the transformations T i  and T′ i  that participate in the matrix M (Eq.7). Hence by analyzing the pure 6 motions, comprising thee translation and three rotations, the stability of the derived equations can be predicted. It is appreciated that a higher stability of the derived equations results in a higher accuracy of the alignment between the sequences. For example, for the method described, for example, in M. Irani, B. Rousso and S. Peleg. Computing occluding and transparent motions. International Journal of Computer Vision, 12(1):5-16, January 1994, the accuracy of the projective parameters T 3,1  and T 3,2  is poor, the accuracy of the translation parameters T 1,3  and T 2,3  is good, and the. other parameters have fair accuracy. Therefore it is predicted that rotation within the image plane produces the most reliable set of equations. 
   It is appreciated that various features of the invention which are, for clarity, described in the contexts of separate embodiments may also be provided in combination in a single embodiment. Conversely, various features of the invention which are, for brevity, described in the context of a single embodiment may also be provided separately or in any suitable subcombination. 
   It will be appreciated by persons skilled in the art that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention is defined only by the claims that follow: