digitization of cultural heritage objects has gained great attention around the world due to the importance and awareness of what they represent for each culture .researchers have been trying to achieve the same goal : capturing a 3d digital representation together with its color information to be able to pass them down safely to future generations .the recovery and generation of the 3d digital representation requires high geometric accuracy , availability of all details and photo realism .any single 3d imaging technique is unable to fulfil all of these requirements and the only way to solve this problem is through the fusion of multiple techniques .there have been a number of recent studies which have tried to map automatically , semi - automatically or manually a photorealistic appearance onto a 3d model .some of these have used only photogrammetry , , which provides poor geometric precision .however for cultural heritage applications , especially for conservation , a high density of the 3d point cloud is needed . in order to satisfy the demanding needs of cultural heritage , the combination of both photogrammetry and range scans have been considered .these approaches generally start by computing an image - to - geometry registration , followed by an integration strategy .the first one generally seeks to find the calibration parameters of the set of images , while the second tries to select the best color for each of the images .there has been research focusing on improving the alignment in all the images ( global registration ) .however , the visual results show significant blurring and ghosting artefacts .others have proved that a perfect global registration is not possible because the two geometries come from different devices and consequently the only solution available is to consider a local registration refinement .this paper proposes a solution for a full end - to - end pipeline in order to process data from different acquisition techniques to generate both a realistic and accurate visual representation of the object .our solution recovers the 3d dimension from 2d images to align the 3d recovered object with a second more geometrically accurate scan . the input 2d imagesare enhanced to improve the feature detection by the structure from motion algorithm ( sfm ) which provides the position and orientation of each image together with a sparse 3d point cloud .the idea behind the 3d reconstruction is to perform the alignment in 3 dimensions through the scale iterative closes point ( sicp ) algorithm obtaining the transformation parameters to be applied in the extrinsic ones of the cameras .even though , the alignment is performed minimizing the distance between both 3d models , it is approximate for different reasons ( sparseness , noise ) and a local registration refinement is needed . in the last stage of our pipeline ,color projection , an algorithm to correct the local color error displacement is performed .our local correction algorithm works in an image space finding the correct matches for each point in the 3d model is deviated from image to image .the main related issues taken into account in our pipeline can be divided into 3 major fields : ( 1 ) 2d/3d registration , ( 2 ) color projection , and ( 3 ) registration refinement process .the important related work in these fields is outlined below .image to geometry registration consists of registering the images with the 3d model defining all the parameters of the virtual camera ( intrinsic and extrinsic ) whose position and orientation gives an optimal inverse projection of the image onto the 3d model .numerous techniques exist and a number of different ways exist to try to solve this problem .the methods can be classified into ( i ) manual , ( ii ) automatic or semi - automatic depending mainly on matches or features . in the ( i )manual methods the registration is performed manually selecting correspondences between each image and the 3d geometry . this technique is often used for medical applications .others instead , have used features in order to automate the process , but finding consistent correspondences is a very complex problem . due to the different appearance of photographs and geometric models , ( ii )automatic methods are limited to some specific models and information .for example , line features are mostly used for urban environments , ; and silhouette information is used when the contour of the objects is visible in the images and the 3d model projected onto an image plane .nevertheless there are 3d scanners which provide also reflectance images and the registration is performed in a 2d space , . on the other hand ,some authors perform their registration in a 3d space reconstructing the 3d object from the 2d images and aligning both 3d objects ,, .this procedure is carried out in two steps : ( 1 ) 3d reconstruction and ( 2 ) point cloud alignment . through the widely used structure from motion technique ( sfm ) ,a 3d reconstruction and intrinsic and extrinsic camera parameters are recovered without making any assumptions about the images in the scene .the registration is usually performed by selecting correspondences that minimize the distances between a set of points .our work is based on sfm approach and the use of the sicp algorithm to register both point clouds with the only constraint being to locate them relatively close to each other .once the images are registered onto the 3d model , the next step is to exploit the photo - realistic information ( color , texture ) obtained by an optical sensor , together with the geometric details ( dense point cloud ) obtained by some type of 3d scanner ( laser scanner , structured light ) .the aim is to construct a virtual realistic representation of the object .  as a point in the 3d model projects onto different images and imagesmay possess some artifacts ( highlights , shadows , aberrations ) or small calibration errors , the selection of the correct color for each point is a critical problem . in order to deal with this task , research has been based on different solutions , each one with its own pros and cons .   in , ,the authors assign the best image to each portion of the geometry .this assignment relies on the angle between the viewing ray and the surface normal . as the color of a group of 3d points comes from one image ,seams are produced when adjacent groups are mapped with different images and also artifacts such as differences in brightness and specularities are visible . even though some research has dealt with the seams by smoothing the transitions , important and critical detail can be lost .+   in these kind of approaches ,, , an specific weight is assigned to each image or to each pixel in the images according to different quality metrics .the metrics vary between authors considering visible points , borders or silhouettes , depth , or the distance to the edge of the scan .all these methods , in comparison with orthogonal view , are able to eliminate the artifacts previously mentioned but instead introduce blurring / ghosting when the calibration of the images is not sufficiently accurate .alternatively , approaches such as attempt to make an estimation of the lighting environment .this approach is able to remove possible illumination artifacts presented in the images ( shadows / highlights ) .unfortunately , in real scenarios it is difficult to accurately recover the position and contribution of all the light sources in the scene .  due to the evaluation criteria used and advantages provided by all of these approaches , a weighting procedure was selected as the best option for our work .we used the approach by callieri et al . because of its robustness , availability and the good results obtained with it from our data set .since the data comes from 2 different devices and the geometry and camera calibration is imprecise after the 2d/3d registration ; blurring or ghosting artifacts appear once the color projection is performed . in order to remove them ,a global or local refinement is necessary .some approaches try to correct the small inaccuracies in a global manner , by computing a new registration of the camera parameters according to the dense 3d model obtained with the scanner .the goal is to distribute the alignment error among all the images to minimize the inaccuracies and improve the quality of the final color of the model .unfortunately as the registration is mostly based on features , an exact alignment will not be possible due to image distortions or low geometric features .nevertheless even if the global alignment refinement finds the best approximate solution , the matches will not be exactly the same . asa consequence blurry details ( ghosting effects ) will appear after the color projection , especially when the color details are in areas with low geometric features .the only straightforward solution to correct these small inaccuracies , is to perform a local refinement .a number of studies have been carried out based on local refinements which locally analyze the characteristics of each point and try to find the best correspondence in the image series .finding these correspondences locally has been addressed in the literature by using optical flow .some studies have computed dense optical flow , but the results depend on the image resolution and the amount of mismatch ( displacement ) together with the computational power available .on the other hand , others instead of working in the image space , have tried to optimize the 3d geometry in order to deform textures more effectively . as our 3d geometry can not be modified these kind of approachesare not feasible for our purpose .computing dense optical flow in our datasets was impossible due to relatively high resolution of the images , e.g. 4008 pixels compared to the 1024 pixels used in the literature in .for this reason we decided to use sparse optical flow to compute the color for each point in the 3d geometry limiting the number of images to the best three , evaluated according to the quality metrics of callieri et al .our goal is to fuse the information provided by the two different devices ( 3d scanner and 2d camera ) in order to recreate a high resolution realistic digital visualization with both very accurate geometric and visual detail .the procedure to achieve this result needs to take into account various problems which will be solved in essentially four main stages ( see figure [ fig : pipeline ] ) : ( 1 ) image preprocessing , ( 2 ) camera calibration through structure from motion , ( 3 ) cloud registration to align the images to the geometry , and ( 4 ) color projection which involves the most correct images to project the color onto the 3d geometry .the whole process is designed to consider as input a set of uncalibrated images and a dense 3d point cloud or a 3d triangulated mesh . by uncalibrated images we refer to images in which the intrinsic and extrinsic camera parameters are unknown .    2   even though a number of studies have used a set of uncalibrated images to perform camera calibration and 3d reconstruction through some structure from motion algorithm ,,, ,very few have considered a preprocessing step .this stage is performed in order to improve the camera calibration procedure ( stage 1 ) and consequently obtain more accurate camera parameters together with a better 3d representation .three preprocessing steps were considered .the first two had already been applied by the c2rmf to their data sets and the introduction of a third preprocessing step also enabled an improvement for the next stage .1 .  color calibration . performed to accurately record the colorimetric appearance of the object in the set of color images and to eliminate mis - matches caused by varying lighting conditions . in order to calibrate ,a color chart is used during the image acquisition to determine a color transformation between the captured values and the reference color target .2 .  background subtraction . as a sfm procedureis based on feature matching , features will be detected in the foreground as well as in the background . in order to avoid the reconstruction of unwanted points ( outliers ) and have a clean 3d object , the background was removed manually .there are many segmentation techniques available in the literature but in order to be precise the manual method was considered by the c2rmf .3 .  image enhancement . through histogram equalization , we enhance the image contrast in order to find a larger number of features and generate more 3d points in the next stage. the original principle applies to gray - scale images , but we used it in color , changing from rgb to the hsv color space and equalizing the value ( v ) channel in order to avoid hue and saturation changes .this step is very useful especially when the object lacks texture details .the same idea was exploited in with a wallis filter .the second stage of our pipeline consists of a self - calibration procedure .it is assumed that the same camera , which is unknown , is used throughout the sequence and that the intrinsic camera parameters are constant .the task consists of ( i ) detecting feature points in each image , ( ii ) matching feature points between image pairs , and ( iii ) running an iterative robust sfm algorithm to recover the camera parameters and a 3d structure of the object .  for each image ,sift keypoints are detected to find the corresponding matches using approximate nearest neighbors ( ann ) kd - tree package from arya et al . and the ransac algorithm to remove outliers .then a structure from motion ( sfm ) algorithm , is used to reconstruct a sparse 3d geometry of the object and obtain the intrinsic ( i.e. focal length , principal point and distortion coefficients ) and extrinsic ( i.e. rotation and translation ) camera parameters .  in order to achieve a more geometrically complete surface of the 3d object , clustering views from multi - view stereo ( cmvs ) and patch - based multi - view stereo ( pmvs ) tools are used .this aims to increase the density of the 3d geometry and be able to obtain a more precise parameter estimation during the cloud registration ( stage 2 ) .   after the 3d geometry obtained with the sfm algorithm and from the 3d scanner ,a 3d-3d registration process is performed .as both points clouds possess different scales and reference frames , we will need to find the affine transformation that determines the scale ( s ) , rotation ( r ) and translation ( t ) parameters which aligns better both 3d geometries .usually a 3d-3d registration refers to the alignment between multiple point clouds scanned with the same device .algorithms like iterative closest point ( icp ) and 4 point congruent set evaluate the similarity and minimize the distance between the 3d point clouds considering only the rotation and translation parameters .but when a scale factor is involved it can be solve separately or together from the registration procedure .calculating a bounding box for both 3d geometries and applying the ratio found between them seems to solve the scale problem , but if some outliers are present in one of the geometries the result will not be correct .therefore zhu et al . , extended the iterative closest point algorithm to consider also the scale transformation ( sicp ) , introducing a bidirectional distance measurement into the least squared problem .this algorithm works as follows : ( i ) define a target ( fixed ) and source ( transforming ) point clouds , which will be the scanned and reconstructed point clouds respectively in order to bring the camera parameters from the image space to the real object space ; and ( ii ) perform iteratively the distance error minimization using the root mean square error ( rmse ) , until the best solution is found .the output is a set of 3d points aligned to the object coordinate system ( real scale ) by means of a 3 rotational matrix , 3 scale matrix and vector indicating the translation in x , y and z axis .color projection is the last and the core of our proposed pipeline .the aim is to project accurate color information onto the dense 3d model to create a continuous visual representation from the photographic image set .selecting the best color is not an easy task ; first because a single point in the 3d geometry is visible in multiple images and those may present differences in illumination .secondly small errors in the camera parameters cause small misalignments between the images , and in consequence , a point which projects onto a specific are in one image plane will project onto a slightly different area in another one .this can result in different colors for each 3d point projected from several 2d images .  in order to address this problem some research based on the color selection from the most orthogonal image for a certain part of the 3d geometry , generating artifacts like highlights , shadows and visible seams .others project all images onto the 3d mesh and assign some weight to each image , as , for example , in ,, , which can remove artifacts that the orthogonal view is not capable of removing , but this can produce some ghosting artifacts when the alignment is not perfect .  in order to deal with less artifacts ,we consider the approach based on callieri et al . , which weights all the pixels in the images according to geometric , topological and colorimetric criteria .the general procedure to perform the color projection in our pipeline takes into account two steps : ( 1 ) a color selection considering weights assigned according to the quality of each pixel , and ( 2 ) a local error correction in the image space in order to produce sharp results .   through the approach by callieri et al . it is possible to compute the weights for each 3d point in the number of images they are visible .the three metrics are based on : angle , depth and borders . for each of these metrics , a mask is created which has the same resolution as the original image .the aim is , therefore , to create a unique mask combining the three masks through multiplication .the result is a weighting mask for each image that represents a per - pixel quality ( see an example in figure [ fig : qualitymasks ] ) .    2  once the weights are defined for each image , and knowing the camera parameters obtained in stage 1 , it is possible to perform a perspective projection from the 3d world onto an image plane .this projection allows us to know the color information for each 3d point in the geometry .the final color for each point is a weighted mean obtained by multiplying the rgb values from the pixels with their respective weights .the results obtained projecting the color information with the quality metrics of callieri et al . into the 3d geometry , generated blurring / ghosting effects in some parts of the mesh .these problems appear due to small inaccuracies introduced in the image - to - geometry registration .research such as that by , have considered these kind of artifacts ; but their origin , is explained by dellepiane et al . in figure [ fig : displacement ] .    2  the simplest way to correct these inaccuracies which generate the blurring artifacts ,consists of finding for each 3d point , the local displacement in the best 3 image planes where it is visible .this local error estimation algorithm , based on , is performed through a template matching algorithm shown in figure [ fig : localerroralgo ] .the reason for considering only the best three images for each point , instead of all where it is visible , is to speed up the process in the final color calculation . instead of computing _ ( n-1)p _ evaluations , we reduce them to _ ( 3 - 1)p _ where n is the number of images and p the points .dellepiane et al . affirmed that three images are enough to correct illumination artifacts .[ h ]   2  the size of the block and template was defined according to some experimental evaluation performed on the highest resolution dataset ( fz36147 ) .normally if the cloud registration step in stage 2 is accurate enough , the different 3d points projected in the image plane will not be so far from each other .for this reason the same parameters can be applied to lower resolution images , but they can not be considered for even higher ones . the most straightforward solution is to tune the parameters depending on the image resolution and the 3d cloud registration output .the matching procedure is done on a pixel - by - pixel basis in a luminance - chrominance color space .the conversion of the rgb values into ycbcr color space was performed directly with the built - in matlab function rbg2ycbcr and the similarity measurement mean square error ( mse ) was defined considering also changes in brightness for each block by subtracting the average value in each channel . through this subtractionwe account for big changes in illumination between images .the notation is the following :    where n is the total number of pixels in each block , s is the block in the source / reference image , t is the block inside the template of the target image , and are the mean values of their respective channels . at the endthe error with the minimum value is considered as the best match .  in the case where there is more than one block matching the same criterion , a comparison of the colors from the center points will decide which block in the template is the closest to the block from the reference image .when the three rgb color values are found for each point , we proceed with the multiplication of them with their respective weights to average the results and assign final color values to each point in the 3d geometry .in this section we present experiments performed on real data from the c2rmf with scans from objects from the department of roman and greek antiquities at the louvre museum in order to assess the performance of the proposed pipeline .the data had been captured at different times using different equipment .each object had data from a structured light scanner and a set of color images used for photogrammetry .the 2 data sets ( fz36147 and fz36152 ) contain information with different qualities and sizes and a small description of the datasets used , is listed below together with a brief explanation of the criteria used for a visual quality evaluation .  * *  dataset fz36147 .this greek vase , an oenocho from around 610bc , contains 1,926,625 points ( pts ) and 35 high resolution images ( 4008 pixels).the images were acquired under an uncalibrated setup , but our method was capable to remove the lighting artifacts and preserve details in its decorations . for the final evaluation of our proposed local error estimation algorithm implemented as part of the color projection procedure ( stage 3 ) , three small patches selected manually from the 3d geometry were extracted .each patch was carefully selected according to visual details where mis - registration of the camera parameters led to blurring artifacts .  * *  dataset fz36152 .this greek vase , an oenocho from between 600 - 625bc , represented by a 3d model which contains 1,637,375 points and 17 images of resolution 2152 pixels . with this dataset ,the registration in the second stage of our pipeline , is not accurate enough to avoid blurring effects which appear in the whole 3d geometry .the local error correction in our method , brings sharp results in the three patches extracted in the same way as in the previous dataset .due to the fact that the images of the datasets are uncalibrated ( no ground truth is available ) only qualitative , meaning visually , evaluations were performed , as found also in the state of the art , .the reason for performing the evaluation only in small patches , refers to the high density of points each 3d geometry contains and the programming language used for the implementation ( cpu programming ) .our algorithm , implemented in stage 3 , corrects the small errors in the projected image planes , converging to good results regardless the initial rough alignment obtained during stage 2 .figure [ fig : localresults ] shows the results of the color projection once the small errors are corrected .the quality of the appearance in the new projections ( down red squares ) is improved , removing the unpleasant blurring artifacts .table [ tab : finalresults ] shows a summary of the characteristics of the datasets used , together with the patches evaluated and their corresponding computational time .    [cols="^,^,^,^,^,^ " , ]   2  the time required to perform the local error estimations , depends on the amount of 3d points the geometry has , and on the displacement found for every projected point in the 2d image plane . if the density increases , the computational time will be higher .[ h ]   2  a visible comparison with the state of the art is presented in figure [ fig : comparisonstateofart ] .dellepiane et al .also evaluated their method with one dataset from the louvre museum with different resolution characteristics .the implementation in is based on dense optical flow and gpu programming for which really high resolution images are a disadvantage . the maximum resolution tested by dellepiane et al . was 3000 ( 5,988,000 pixels ) which took around 5 hours in 6 images . in our dataset fz36147, its resolution is 4008 and contains 35 images , the pixels needed to be evaluated with will be 21,418,152 which is at least 5.57 times more than in their dataset with maximum resolution , and 6 times the number of images . only with extremely powerful processing capable of handling such computations can their approach be applied , otherwise their method is not a feasible solution with our data set .  in general the state of the art methods , are based on dense optical flow which is the main reason there is no possible comparison with our datasets .  even though our implementation has proven to be robust and reliable , some limitations still remain .the main one relates to the programming language for the acceleration of the computational time ( from cpu to gpu programming ) .also , in the evaluations performed , the maximum local displacement found was not large ( 10 pixels ) ; but for other cases ( e.g. images with higher resolution ) , the displacement can be bigger and in consequence the parameters for the template matching algorithm in stage 3 , have to be adjusted .there are also some limitations related to lighting artifacts .factors like highlights / shadows may complicate the estimation of the local error displacement , and inclusive mislead the motion to totally wrong values .nevertheless , these drawbacks are shared with every method based on optical flow calculations .we have proposed a semi - automatic 2d-3d registration pipeline capable to provide extremely accurate realistic results from a set of 2d uncalibrated images and a 3d object acquired through laser scanning .the main advantage of our pipeline is the generality , since no assumption is made about the geometric characteristics or shape of the object .our pipeline is capable of handling registration with any kind of object , since the algorithm used is a brute force ( sicp ) which evaluates every single point and finds the best position .the only requirements needed are a set of 2d images containing sufficient overlapping information to be able to use the structure from motion ( sfm ) technique in stage 1 ; and a user intervention during stage 2 to locate the dense point cloud , coming from the scanner , closer to the one obtained by sfm , in order to provide the input that the scale iterative closest point ( sicp ) algorithm needs to converge .this user intervention during the second stage in our pipeline is what makes our approach semi - automatic .  in conclusion ,our main contribution is the local error correction algorithm in stage 3 which proved to be :  1 .robust : it works with low and high resolution images , as it considers the interest points ( projected 3d points into the image plane ) for the matching .not even the state of the art , is capable of dealing with as high resolution images as our algorithm .2 .  accurate: it finds the best possible matching for the small error displacements considering luminance and chrominance channels . through the best match, it removes the unpleasant blurring artifacts and produces sharp results .3 .  photo - realistic : with the point cloud generated by sfm , and the registration algorithm sicp , the color information from the 2d images is projected onto the 3d object transferring the appearance .  an interesting direction forfuture research would be to define a criterion with a respective threshold to identify the possible borders where the sharp results start to blur ( in the cases where only some parts of the 3d object are visible with ghosting effects ) .this identification has to be based on the depth difference between the 2 registered point clouds , and probably a segmentation according to depth may help to optimized our proposed local error estimation algorithm .special thanks to centre de recherche et de restauration des musees de france ( c2rmf ) for the data provided to evaluate the pipeline . thanks to j.h .zhu , n.n .zheng , z.j .yuan , s.y .du and l. ma for sharing their sicp code implementation in matlab .el - hakim , s. , beraldin , j. a. : detailed 3d reconstruction of monuments using multiple techniques . in proceedings of the international workshop on scanning for cultural heritage recording - complementing or replacing photogrammetry , 58 - 64 ( 2002 ) .liu , l. , stamos , i. , yu , g. , wolberg , g. , zokai , s. : multiview geometry for texture mapping 2d images onto 3d range data . in computer vision and pattern recognition , 2006 ieee computer society conference , vol .2 , pp . 22932300 . ieee ( 2006 ) .dellepiane , m. , callieri , m. , ponchio , f. , scopigno , r. : mapping highly detailed colour information on extremely dense 3d models : the case of david s restoration . in computer graphics forum vol .27 , no . 8 , pp . 21782187 .blackwell publishing ltd ( 2008 ) .liu , l. , stamos , i. : automatic 3d to 2d registration for the photorealistic rendering of urban scenes . in computer vision and pattern recognition , 2005 .cvpr 2005 .ieee computer society conference on vol .137143 . ieee ( 2005 ) .dellepiane , m. , scopigno , r. : global refinement of image - to - geometry registration for color projection on 3d models . in digital heritageinternational congress , pp .3946 . the eurographics association ( 2013 ) .dellepiane , m. , marroquim , r. , callieri , m. , cignoni , p. , scopigno , r. : flow - based local optimization for image - to - geometry projection . visualization and computer graphics ,ieee transactions on , 18(3 ) , 463 - 474 ( 2012 ) .eisemann , m. , de decker , b. , magnor , m. , bekaert , p. , de aguiar , e. , ahmed , n. , sellent , a. : floating textures . in computer graphics forum ,2 , pp . 409418 . blackwell publishing ltd ( 2008 ) .lensch , h. , heidrich , w. , seidel , h. p. : automated texture registration and stitching for real world models . in computer graphics and applications , 2000 .proceedings . the eighth pacific conference , pp . 317452 . ieee ( 2000 ) .ikeuchi , k. , oishi , t. , takamatsu , j. , sagawa , r. , nakazawa , a. , kurazume , r. , okamoto , y. : the great buddha project : digitally archiving , restoring , and analyzing cultural heritage objects .international journal of computer vision , 75(1 ) , 189 - 208 ( 2007 ) .corsini , m. , dellepiane , m. , ganovelli , f. , gherardi , r. , fusiello , a. , scopigno , r. : fully automatic registration of image sets on approximate geometry .international journal of computer vision , 102(1 - 3 ) , 91 - 111 ( 2013 ) .dellepiane , m. , callieri , m. , corsini , m. , cignoni , p. , scopigno , r. : improved color acquisition and mapping on 3d models via flash - based photography .journal on computing and cultural heritage ( jocch ) , 2(4 ) , 9 ( 2010 ) .

Abstract: in this paper we present a semi - automatic 2d-3d local registration pipeline capable of coloring 3d models obtained from 3d scanners by using uncalibrated images . the proposed pipeline exploits the structure from motion ( sfm ) technique in order to reconstruct a sparse representation of the 3d object and obtain the camera parameters from image feature matches . we then coarsely register the reconstructed 3d model to the scanned one through the scale iterative closest point ( sicp ) algorithm . sicp provides the global scale , rotation and translation parameters , using minimal manual user intervention . in the final processing stage , a local registration refinement algorithm optimizes the color projection of the aligned photos on the 3d object removing the blurring / ghosting artefacts introduced due to small inaccuracies during the registration . the proposed pipeline is capable of handling real world cases with a range of characteristics from objects with low level geometric features to complex ones .   heritage , 3d reconstruction , 2d-3d registration , local error  2


the spectroscopic pipeline for the fors instruments can handle long - slit and multi - object spectroscopic data using slitlets . in order to be as flexible as possible most information is obtained directly from the observed calibration data , minimizing the need for assumptions ( for details of the underlying principles see the contribution by izzo et al . ) .a first guess of the positions and lengths of the slitlets ( or long slit ) is obtained from an arc lamp frame ( see fig . [moehler : fig : arc_raw ] ) .the positions of arc lamp spectra on the ccd are determined by a pattern matching technique applied to the detected emission lines .for that an estimate of the linear dispersion as well as a line catalog ( which may both be provided by the user ) are required .adjacent slitlets with no offsets in dispersion direction are not distinguished and will form one longer slitlet .the spatial curvature is determined by tracing the edges of the corresponding flat field spectra ( cf .[ moehler : fig : spat_curv ] ) when possible .the curvature is fit by default with a second order polynomial .the final wavelength calibration is obtained taking into account the spatial curvature of the spectra .the pipeline provides the following output calibration files :  * slit positions on the ccd at the central wavelength of the grism * coefficients for the dispersion relation and the spatial curvature * master bias * master flat field ( normalized and not - normalized ) * wavelength map and spatial map * reduced arc lamp frame * spectral resolution and line widths  the wavelength map specifies at the position of each original ccd pixel its corresponding wavelength .thus the user can avoid the re - sampling of the spectra usually done after the wavelength calibration .the spatial map provides in the same manner the position of the ccd pixel within its associated slitlet .the reduced arc lamp frame allows to judge the quality of the calibration ( straight arc lines , no wriggles , no offsets ) . if successful ( see sect .[ moehler : sec : weak ] ) the pipeline achieves an accuracy of 0.1 pixel both in spatial and dispersion direction .the default reduction of spectroscopic science data is as follows : the data are first corrected for bias .then the extraction mask derived above ( i.e. the positions of the slitlets and the curvatures of the resulting spectra ) is applied to the science data ; they are flat - fielded and re - mapped eliminating the optical distortions .afterward they are rebinned to constant wavelength steps .the wavelength calibration can be adjusted using sky emission lines .this allows to correct for shifts between night - time science and day - time calibration data .such shifts ( of the order of 1 pixel ) are known to happen due to flexure for mxu observations .finally object spectra are detected and extracted , together with the corresponding error spectra and sky spectra .for long - slit spectroscopy the sky background is determined using a median .this is a valid approach if not more than half of the pixels contain flux from objects _ and _ there is no spatial gradient in the sky background . for multi - object spectroscopy instead the sky is subtracted before remapping , i.e. when the spectra are still in the original ccd coordinate system .the sky is determined with a robust linear fitting , which allows for a linear spatial gradient in the background .also in this case , however , not more than half of the pixels may contain flux from the object(s ) .the output science files are  * positions and widths of object spectra * extracted object spectra , error spectra , and sky spectra * unmapped ( corrected for spatial distortions , not rebinned ) science frames * mapped ( i.e. corrected for spatial distortions and rebinned ) science frames * adjusted dispersion coefficients and wavelength mapthe major strength of the new pipeline is its flexibility and robustness .it requires only very limited input information ( mainly dispersion estimate and line catalog ) and can therefore be applied to a large variety of instrument configurations .thus it is now possible for the first time to _ automatically _ reduce long - slit ( lss ) and multi - object spectroscopic data ( mos / mxu ) from the two fors instruments for all grisms and both collimators ( standard and high - resolution ) . to verify its flexibilitythe pipeline was applied to data from the low - resolution spectrograph of the hobby eberly telescope , which it handled without any problems .the pipeline does have problems if there are too few arc lines , e.g. for slitlets with large offsets .in such cases it sometimes fails to correctly identify the slitlet .this can be most easily recognized by comparing the not - normalized and the normalized screen flat ( cf .[ moehler : fig : bad_mos ] one slitlet is partly missing , another one was not processed at all ) .large gaps between arc lines can be problematic as non - linear dispersion terms can become important , while the pipeline currently uses only linear dispersion estimates for the pattern matching .the pipeline will fail in case of regularly spaced arc lines , like those created by a laser comb , as its pattern matching ( see contribution by izzo et al . )does not work for such data .in order to check the quality of observational data a pipeline is of great importance . otherwise it is difficult to distinguish between well - known and correctable instrument effects and real problems .quality control shall ensure that the data observed with the forses can be calibrated .in addition the trending of certain parameters allows to have an eye on the instrument health ( see http://www.eso.org/dfo/quality for more details ) .therefore the resolution , central wavelength , and number of identified arc lines are regularly monitored .a change in the number of identified lines can indicate reduced flux of the arc lamp and possibly impending failure of the lamp .a change in resolution could indicate focus problems .079 0 with respect to chip 1 ( the known value being 0 ) . the reduced arc lamp image corresponding to the coefficients in the dashed boxis shown in fig .[ moehler : fig : pccx_bad ] ]  the situation for the quality control of fors data has vastly improved with the new pipeline , as now almost all data can be reduced ( with polarimetric data being the only exception ) .moreover , the new pipeline also provides additional information on the instrument .for example , the combination of the curvature coefficients can be used to monitor grism alignment : the coefficients c1 and c2 describe the slope and the curvature of a slitlet , respectively . if the curvature is 0 , then the slope should be 0 as well otherwise this indicates that the grism is not well aligned .[ moehler : fig : coeff ] shows the coefficients obtained for the 600z fors2 grism .obviously the correlation for the majority of solutions passes through zero .some solutions , however , show significant offsets from the general trend .[ moehler : fig : pccx_bad ] shows a reduced arc lamp image for such a deviating case ( marked by the dashed box in fig . [ moehler : fig : coeff ] ) . obviously the wavelength calibration obtained for several slitlets is rather bad ( wriggles instead of straight lines ) .thus diagrams like fig .[ moehler : fig : coeff ] allow to look for bad solutions efficiently in case of large data volume , which may prohibit checking all solutions individually .  .]  other new parameters become available if there are more than 12 slitlets distributed across the ccd . in this casethe pipeline calculates a global distortion model , which contains good indicators for the monitoring of the instrument s health , like the instrument scale .the new spectroscopic pipeline for the fors instruments has been a great success in that it allows to reduce _ automatically _ long - slit and multi - object spectroscopic data for all fors grisms . the fact that it does not need a first guess from an instrument model makes it a promising candidate for the reduction of spectroscopic data from a variety of instruments .in addition this approach is extremely helpful in the presence of instrument instabilities as it allows to reduce data whose configurations deviate from the expected one ( e.g. due to instrument aging or earthquakes ) .i would like to thank c. izzo and u. hopp for their valuable comments on this manuscript .

Abstract: since october 1 , 2006 , spectroscopic data from the two fors instruments have been reduced with a new pipeline , which is based on a bottom - up calibration approach . i give a short description of the pipeline and discuss first experiences with automatic data reduction using this software , which has significantly increased the percentage of processed data for both instruments . i will also describe possible new options for quality control .


swarm optimization ( pso ) is a typical swarm intelligence optimization algorithm inspired by animal social behaviors , such as bird flocking and fish schooling .a group of particles in pso fly in the search space , aiming to find the optimum cooperatively .each particle exchanges information with others and learns useful information to improve its performance .due to its ease to implement and outstanding performance , pso has been widely used to solve real - world schedule or engineering problems such as antennas , system control , electronics and electromagnetics .  in the original pso , each particle learns from the best historical experience of the whole population .the concepts of structure and neighbors in pso were first introduced in , where each particle learns from the best historical experience of its neighbors .however , in both versions the particles learn from the best individual , hence some useful information of other individuals is neglected . to take the advantage of full information , the fully informed particle swarm optimization ( fipso )was proposed where all neighbors are information sources .though fipso can rapidly converge , it may miss some promising regions in the search space of the optimization problem .most previous works treated all individuals as the same , neglecting the individual heterogeneity .actually , the individual heterogeneity plays an important role in swarm intelligence and has been verified to be able to significantly improve the performance of pso . here , we propose a heterogeneous strategy particle swarm optimization ( hspso ) , in which a proportion of particles are single informed , while others are fully informed .our experimental results show that hspso obtains satisfactory solutions and outperforms fipso and canonical singly informed pso ( sipso ) , because in hspso fully - informed particles can adequately utilize the global information and guide the swarm while singly - informed particles can maintain the diversity .the rest of the paper is organized as follows .section ii introduces hspso in detail and shows its relation to sipso and fipso .section iii compares the results of three psos .section iv employs hspso to solve the problem of 2-dimensional recursive filter design .section v makes a conclusion .in hspso , particles fly in a -dimensional space to search the optimum .the particle updates its velocity and position of dimension by \\ & v_{i}^d:=\chi[v_{i}^d+\frac{\varphi}{k_i}\sum_{m=1}^{k_i}r_{m}(p_{i_m}^d - x_{i}^d)]\end{aligned}\ ] ]  where ( 2 ) is for singly - informed ( si ) particles while ( 3 ) is for fully - informed ( fi ) particles , =4.1 and =0.729 according to common practices ,  ] denotes the historical best position in all neighbors of particle , is the number of the particle s neighbors , is the neighbor of the particle ,  ] .note that there are two strategies of updating the velocity , i.e. fi and si .each particle employs the alternative velocity formula according to its property .here we use a parameter  ] + & ^d ] + & ^d ] + & ^d ] . the task of filter designing is to adjust the coefficients of to approximate the desired amplitude response of the -d filter . in this brief , the desired amplitude response follows as hence , the design of -d filter can be formalized as an optimization problem of minimizing the cost function ^p } } \end{aligned}\ ] ] s.t . where , and .the cost function describes the difference of and in points .     as the function of , , and , we construct a vector =[ , , , , , , , , , , , , , , ] for hspso .the parameter of hspso is set as follow : population size is set as , all variables in vector are in the range of $ ] , the evolution lasts for iterations .table [ t2 ] lists the parameters optimized by hspso and other competitors , including genetic algorithm ( ga) , neural network ( nn) , sipso ( hspso with , ) and fipso ( hspso with , ) .[ f7 ] shows the frequency response of the required filter and the designed filters with the parameters in table [ t2 ] .one can see that hspso performs better than ga and nn methods .note that the high frequency region of filters designed by sipso and fipso are flat , yet the low frequency region are not satisfactory , due to its elliptical transverse section rather than a circle .therefore , hspso outperforms both sipso and fipso due to the cooperation of singly- informed particles and fully - informed particles ..the results of optimized filter coefficients [ cols="^,^,^,^,^,^",options="header " , ]   [ t2 ]in this brief we propose hspso , a swarm optimization algorithm composed of two types of particles with different learning strategies .we test the performance of hspso on six widely - used benchmark functions .our results show that hspso is superior to canonical pso and fipso .our investigation on the impact of network topology and the underlying mechanism of hspso reveals that the heterogeneity of the swarm results in the division and cooperation between different particles , leading to a more effective optimization process .the successful application of hspso to -d filter design problem demonstrates its applicability in solving real - world optimization problems .this paper is supported by the national natural science foundation of china ( grant nos .61425014 , 61521091 ) , national key research and development program of china ( grant no .2016yfb1200100 ) , and national key technology r&d program of china ( grant no .2015bag15b01 ) .   j. kennedy and r. c. eberhart , particle swarm optimization , " _ in proc .neural netw .4 , pp . 1942 - 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free interactions , " _plos one _ , vol . 9 , e97822 , 2014 .y. gao , w. b. du , g. yan , selectively - informed particle swarm optimization , " _ sci ._ , vol . 5 , 9295 , 2015 .w. b. du , y. gao , c. liu , z. zheng , z. wang , adequate is better : particle swarm optimization with limited - information , " _ appl .comput . _ , vol .832 - 838 , 2015 .x. yao , y. liu , g. m. lin . evolutionary programming made faster , " _ ieee comput .82 - 102 , 1999 .x. li , x. f. wang , g. r. chen . pinning a complex dynamical network to its equilibrium , " _ ieee trans .circuits syst .i , regu . paper _ , vol .2074 - 2087 , 2004. q. j. zhang , j. n. lu , c. k. tse . adaptive feedback synchronization of a general complex dynamical network with delayed nodes , " _ ieee trans .circuits syst .ii , exp . brief _ ,183 - 187 , 2008 .z. h. rong , z. x. wu , g. r. chen . coevolution of strategy - selection time scale and cooperation in spatial prisoner s dilemma game , " _ epl _ , vol .102 , pp . 68005 , 2013 . h. y. liu , y. x. xia . optimal resource allocation in complex communication networks , " _ ieee trans . circuits syst .ii , exp . brief _ ,706 - 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Abstract: pso is a widely recognized optimization algorithm inspired by social swarm . in this brief we present a heterogeneous strategy particle swarm optimization ( hspso ) , in which a proportion of particles adopt a fully informed strategy to enhance the converging speed while the rest are singly informed to maintain the diversity . our extensive numerical experiments show that hspso algorithm is able to obtain satisfactory solutions , outperforming both pso and the fully informed pso . the evolution process is examined from both structural and microscopic points of view . we find that the cooperation between two types of particles can facilitate a good balance between exploration and exploitation , yielding better performance . we demonstrate the applicability of hspso on the filter design problem .   ying : heterogeneous strategy particle swarm optimization  optimization , complex networks , filter design , pso .


the phase space of a typical hamiltonian system is a mixture of periodic , quasiperiodic and chaotic trajectories , and the stable and unstable manifolds of these trajectories .the analysis of the trajectories of those systems is central to many branches in physics , ranging from celestial mechanics to chemistry .several methods have been designed to discriminate between regular ( periodic or quasiperiodic ) and chaotic trajectories . for two degrees of freedom ,the most commonly used method is a poincar section which is a plane section of the three dimensional energy surfaces .these sections give pictures of phase space and allow to analyze qualitatively the stability properties of the system by revealing , e.g. , the invariant tori , the stable islands surrounding elliptic periodic orbits and the spatial extent of chaotic trajectories .however , these sections do not provide very accurate information and the visual inspection of these sections for higher dimensional systems is less straightforward since , e.g. , for hamiltonian systems with three degrees of freedom , these sections are four dimensional .+ much of our understanding of nonlinear systems is based on their frequencies and especially resonances between them ( perturbation theorems , intramolecular dynamics  , laskar s work in celestial mechanics  and particle accelerators  ) .one way to obtain accurate information on the stability of the system is to associate with a given trajectory an indicator such as a frequency ( in general , several indicators are associated with a given trajectory ) and construct , e.g. , a frequency representation of phase space . in this frequency space , regular regions are characterized by a very small variation of the frequency as the time interval on which this frequency is computed increases ( diffusion constant in laskar s work  ) .the main advantage of this method is that it is generalizable to systems with any number of degrees of freedom  .+ however , a frequency representation of the phase space does not provide the history and the local properties of a trajectory .the computation of asymptotic quantities ( such as lyapunov exponents , entropy or fractal dimensions ) does not provide dynamical information .these quantities only reflect the asymptotic behavior of the system . for many purposes infinite - time quantities are not sufficient : rather than knowing whether or not a trajectory is chaotic , it is naturally desirable to say when , where , and to what degree a trajectory is chaotic .+ in this article , we describe a method that gives a _ time - frequency _ representation of trajectories , i.e. , that provides dynamical information on the system .the idea is to follow a given trajectory and see if the phase space structures where the trajectory passes by are revealed by analyzing a single trajectory . for quasiperiodic and chaotic systems, time - frequency representation is revealing because it shows when and how motion can be trapped in some resonance zones associated with quasiperiodicity .it also show the degree of chaoticity of some part of phase space visited by the trajectory .+ in a nutshell , we associate a set of time - varying frequencies with the trajectory by decomposing it on a set of elementary functions , the wavelets , which are localized in time and frequency . wavelet decomposition has been used to analyze trajectories of hamiltonian systems in celestial mechanics  and in molecular dynamics  . in refs . , the main frequency was extracted by computing the frequency curve where the modulus of the wavelet transform is maximum . however , stopping at this main frequency can give misleading or wrong information about resonance transitions .instead , we have found that the relevant information of the trajectory is contained in the _ ridges _ of the landscape formed by the magnitude of the coefficients of the time - frequency decomposition . a ridge curve ( which has a certain time length ) corresponds to time - frequency energy localization and is also called _ instantaneous frequency _ curve .we show that the instantaneous frequencies point to resonance trappings and transitions , and give a clear characterization of the degree of chaoticity .+ in sec .[ sec2 ] , we describe the time - frequency method and the extraction of ridges by two methods : the windowed fourier transform and the continuous wavelet transform . in sec .[ sec3 ] , we present briefly the two models which are used to illustrate the method in sec .[ sec4 ] : the first one is the standard map which is used as a benchmark model for the method since the phase space structures of this model are well - known , and the second one is the hydrogen atom driven by a elliptically polarized microwave field which is a basic and realistic problem in atomic physics with three degrees of freedom . + we show how time - frequency analysis reveals the phase space structures : for two degrees of freedom , we compare this analysis with the poincar surface of section , and we show that the time - frequency analysis can be carried out for higher dimensional systems .+ in appendix a , we give an approximation of the ridge curves for a quasiperiodic trajectory and in appendix b , we compute the effect of finite time on the computation of the ridges .given a real signal which can be , e.g. , one coordinate of the system ( one position or one momentum for instance ) , two methods are used to compute a time - frequency ( denoted - ) representation of : windowed fourier transform and continuous wavelet transform .we assume that we have computed the trajectory for all times .the effect of finite time interval is analyzed in appendix b. for more details on time - frequency analysis , we refer to refs .    the windowed fourier transform ( also called gabor transform ) of is given by where the window is chosen to be a gaussian window .we notice that in laskar s work  , the hanning filter over a time interval  ] .the time - frequency representation is obtained by the relation between the scale and the frequency : we will consider the normalized scalogram which can be interpreted as the energy density in the time - frequency plane . again , we notice that .for the computation of the wavelet decomposition , we only consider the positive frequency part of the time - frequency plane .   in order to use a fast wavelet algorithm, we consider that and are periodic ( and hence is periodic in the variable ) with period , the time span of the trajectory . by taking the fourier transform of eq .( [ eqn : cwt ] ) , one has : since is known exactly , one has to compute by fft and then retrieve by an inverse fft . for this computation, the scales are chosen according to a dyadic sampling for and where is the number of octaves , is the number of voices per octave and is the number of points in the time series on the interval  ] .the approximation of the main frequency , called is obtained by maximizing the modulus of where is a hanning filter . the amplitude of the component with frequency is obtained by projection on .laskar proved that this method gives the main frequency with an accuracy of order  ( i.e. , ) instead of an ordinary fourier transform which gives an accuracy of order .the approximation of the next frequency is obtained by iterating the above procedure using .since the basis is not orthogonal for the scalar product , a gramm - schmidt orthogonalization is necessary to obtain the next frequencies for . + for a chaotic trajectory , a ridge is a curve or a segment of curve in the time - frequency plane which is at each time a local maximum of the normalized scalogram , i.e. , each ridge has a weight which is the value of the normalized scalogram on this ridge ( it varies continuously in time ) .we will refer this value as the amplitude of the ridge .we call main ridge or main frequency , the ridge ( or a set of ridges ) where the normalized scalogram is maximum : in what follows , we will detect ridges by analyzing one coordinate of the system .we will show that the set of ridges of one coordinate contains all the frequency content of the trajectory , that is to say , the ridge plot will not be dependent on the choice of the coordinate used for the computation of the time - frequency transform .the respective amplitudes of the ridges depend on the specific coordinate chosen for the computation of the normalized scalogram .for instance , the main ridge depends in general on the chosen coordinate whereas the ridge plot does not .+ _ detection of the ridges ._ for each time , we are searching for the _ local _ maxima of the spectrogram or the normalized scalogram .in general , there is an infinite number of these ridges .we only detect the ones with sufficient amplitude according to a threshold , i.e. , we determine the set of ridges such that time - frequency analysis can be performed on trajectories of maps or of flows . in this section, we illustrate the time - frequency analysis on the two models : the well - known standard map and the hydrogen atom in crossed magnetic and elliptically polarized microwave field .+ the standard map is a two - dimensional area - preserving map  : this example provides a benchmark for the time - frequency analysis since the phase space structures are well - known in this example . + the hydrogen atom driven by a elliptically polarized microwave field ( polarized in the orbital plane ) and a constant magnetic field perpendicular to the orbital plane is described by the following hamiltonian  : where is the contribution of the elliptically polarized microwave field and is the one of the magnetic field : where is the frequency of the microwave field and is the ellipticity degree . for , the microwave field is linearly polarized ( along the direction ) , and for , it is circularly polarized . in the circularly polarized case ,the problem has two effective degrees of freedom . for ,1[ ] . in this figure , the bold curve represents the main ridge or the main frequency ( i.e. where the normalized scalogram is maximum , see sec .[ sec24 ] ) .+ in this time interval , several transitions occur .for instance around , the main frequency jumps discontinuously from the ridge located around to the ridge located around .one could wrongly conclude that a sharp resonance transition occurred in the system . by looking at secondary ridges ,i.e. the ridges with smaller amplitudes than the main frequency , one can see that both ridges existed before and after ; nothing discontinuous happens in the system . from the ridge plot, the mechanism of resonance transition appears to be the following one : a second ridge at increases in amplitude from to while the first ridge at decreases in amplitude . on some time interval ( from to ) , several ridges of the same order in amplitude appear . at ,the second ridge becomes the main frequency .this situation is typical from other ridge plots of trajectories showing transitions from ridges and hence resonance transitions ( see fig .[ fig12 ] ) .thus we have a new characterization of resonance transition that applies in more general situations .resonance transitions are a manifestation of the fulfillment of the chirikov resonance overlap criterion  .ridge plot of the component of the trajectory of the standard map with and initial conditions and on the time interval  ] ( with a ridge detection threshold ) .   ridge plot of a chaotic orbit of hamiltonian ( [ hamrot ] ) for and in the maximum configuration obtained for initial conditions : . the upper part of the figure represents the poincar section of the trajectory . ]  figure [ fig9 ] depicts an enlargement of the ridge plot of fig .[ fig8 ] on the time interval  ] .the upper part of the ridge plot is obtained by analyzing the coordinate and the lower part is the one for the coordinate . ]three different type of regimes are observed on these figures : trapping regions where the frequency is constant , weakly chaotic regions where the frequency is changing smoothly ( for instance , from to ) , and strongly chaotic regions characterized by many short ridges ( for instance between and ) .+ the two main trapping regions in these figures are obtained from to where the trajectory is trapped around the elliptic island surrounding the periodic orbit with frequency , and from to where the trajectory is trapped around the elliptic island with frequency . + the part of phase space visited by the trajectory is compact ( this is due to the magnetic field ) so we do not expect great variations of the main frequency .+ in fig .[ fig10 ] , we depict a resonance transition that occurs in the system for and ( still in the maximum configuration ) for initial conditions in the poincar section .this figure displays the ridges of the coordinates and with a ridge detection threshold .ridge plot of the chaotic orbit of hamiltonian ( [ hamrot ] ) in the maximum configuration with the parameters and obtained for initial conditions in the poincar section .the upper part of the ridge plot is obtained by analyzing the coordinate and the lower part is obtained from the coordinate . ]there are two trapping regions : from to and from to .the first trapping region is characterized by two main frequencies and , and the second trapping region has three main frequencies , and .both regions are separated by a strongly chaotic region characterized by many short ridges ( resonance transition region ) .figure [ fig11 ] displays poincar section of the trajectory from to ( upper left panel ) , the section of this trajectory which is trapped between and ( upper right panel ) , and the section of the trajectory trapped from to ( lower panel ) . in the trapped regions ,the trajectory nearly fills a one - dimensional part of the poincar surface .poincar section of the trajectory of fig .[ fig10 ] .the upper left panel is the trajectory from to , the upper right panel is the trajectory from to , and the lower panel is the trajectory from to . ]  from figs .[ fig9 ] and [ fig10 ] , we notice that the ridge plots are the same for the coordinate of the coordinate .the only difference is that a given ridge has different amplitudes according to whether it has been computed using or .one exception is fig .[ fig10 ] from to : only the ridges of sufficient amplitude are detected ( with a ridge detection threshold ) and the ridge around which is the main ridge for the coordinate has a low amplitude for the coordinate .this ridge appears if we include low amplitude ridges in the detection .+ the noticeable differences between the ridge plots for and the ones for are in the strong chaotic region where there are no elliptic islands and thus no trappings . in this regime which is the same as the one in fig .[ fig4 ] for the standard map , the significance of a frequency or a set of frequencies as an indicator is not clear .it means that the basis is not adapted to the strong chaotic regime .however , computing the ridge curves is useful since it provides a way to locate these strongly chaotic regions .it also provides a quantitative distinction between weak and strong chaos by looking at the number of short ridges .+ we highlight that computing a single instantaneous frequency from the maximum wavelet coefficients can provide erroneous information .figure [ fig12 ] depicts the ridges in a strongly chaotic zone of phase space ( the same trajectory as for fig .[ fig8 ] on the time interval  ] .the bold curve represents the main frequency curve . ]the time - frequency analysis can be carried out for higher dimensional systems .we consider the elliptically polarized case with represented by hamiltonian ( [ hamrot ] ) in the rotating frame . in this case, is no longer a conserved quantity so hamiltonian ( [ hamrot ] ) has three degrees of freedom .figure [ fig13 ] depicts the ridge plot of the coordinate with a ridge detection threshold , and figure [ fig14 ] depicts the ridge plot of the coordinate with a ridge detection threshold .ridge plot of the coordinate of the chaotic orbit of hamiltonian ( [ hamrot ] ) with with the parameters and obtained for initial conditions and solution of eq .( [ eqn : pw ] ) ; and are solution of and .the upper part of the ridge plot is for the time interval  ] .the ridge detection threshold is . ]ridge plot of the coordinate of the same trajectory as fig .[ fig13 ] with a ridge detection threshold . ]we notice that from this figure the trajectory looks more chaotic than in the circularly polarized case which is in agreement with refs . ( see the differences between fig .[ fig8 ] and fig .[ fig13 ] ) . from figs .[ fig13 ] and [ fig14 ] , we detect resonance trappings and weakly chaotic regions .for instance , from to the trajectory is trapped in a region where the main frequencies are ( which is the main frequency for ) and ( which is the main frequency for ) .the same trapping is observed on a longer time interval from to .this is confirmed by a fourier analysis of the segments of the trajectory : figure [ fig15 ] shows the power spectrum of two segments of the trajectory : from to which is expected to be close to quasiperiodic , and from to which is expected to be strongly chaotic with a broad - band spectrum .power spectrum of two segments of the trajectory of fig [ fig13 ] ( coordinate ) : from to ( upper panel ) , and from to ( lower panel ) . ]  by increasing the ridge detection threshold ( see fig . [ fig14 ] ) , we clearly see the trapping regions ( with one main flat ridge ) and the strongly chaotic regions .we notice that there are regions where there is one main flat ridge with additional short ridges ( for instance from to or from to ) .a fourier analysis of these segments of trajectory shows a main frequency and a broad - band component .this is a nearly trapped region near some elliptic island . from fig .[ fig14 ] , we notice that the trajectory is trapped in this island from to , and the main frequencies are and .in summary , we showed how instantaneous frequencies based on a ridge extraction of a wavelet decomposition of a single trajectory coordinate reveal the phase space structures ( resonance transitions , trappings , etc . )of chaotic systems .our method also gives a quantitative characterization of weak and strong chaos through trappings and through the number of short ridges .we have showed that this time - frequency analysis can be carried out on trajectories of maps and flows .note that nothing in our analysis depends on the dimensionality of the system : given a time series ( which can be e.g. , one of the coordinates of the system whatever the number of such coordinates may be , or indeed data sets ) , we can compute the ridges of the time - frequency landscape , leading to the instantaneous frequencies .moreover , the computational time is independent of the dimensionality of the system .the time - frequency method based on a single trajectory analysis is therefore very well suited for exploring phase space structures of various systems independently of the number of degrees of freedom .cc acknowledges useful discussions with n. garnier and j. laskar .this work was supported by the national science foundation and by onr grant no .n00014 - 01 - 1 - 0769 .given the following quasiperiodic function where , we derive an approximation of the ridges of the time - frequency landscape obtained using the windowed fourier transform and the continuous wavelet transform .the windowed fourier transform of trajectory ( [ eqn : qps ] ) is ( up to an overall constant ) : we want to study the behavior of around .we expand up to second order in the variable  : where e^{i\omega_k u }  e^{-\sigma^2(\omega_k-\omega_1)^2/2 } \right\}.\end{aligned}\ ] ] the ridge around is the local maximum of the absolute value of for each .the expression of the spectrogram is x^2 + o(x^3).\end{aligned}\ ] ] the ridge is approximately located at when the terms are small ( compared to 1 ) which is the case when is small or is sufficiently far from , the expression of the ridge becomes :  e^{-\sigma^2(\omega_k-\omega_1)^2/2},\ ] ] where .the instantaneous frequency curve is again a quasiperiodic function with frequencies .it oscillates around the main frequency with amplitude proportional to .these oscillations can be reduced by increasing the length of the window of the transform .the wavelet transform of trajectory ( [ eqn : qps ] ) is ( up to an overall constant ) : we want to study the behavior of the normalized scalogram around .we expand up to second order in the variable  : where e^{i\omega_k u }  e^{-\sigma^2\eta^2(\omega_k/\omega_1 - 1)^2/2 } \right\}. \end{aligned}\ ] ] the ridge around is given by the local maximum of for each .the expression of the normalized scalogram x^2 + o(x^3).\end{aligned}\ ] ] the ridge is approximately located at when the terms are small ( compared to 1 ) which is the case when is small or is sufficiently far from , the expression of the ridge becomes :   e^{-\sigma^2\eta^2(\omega_k/\omega_1 - 1)^2/2}.\ ] ]  as for the windowed fourier transform , the instantaneous frequency is a quasiperiodic function with frequencies .it oscillates around the main frequency .the amplitude of the oscillations is proportional to .they can be reduced by increasing the parameters of the wavelet and .in this section , we investigate the effect of the finite time on the computation of the instantaneous frequencies .we consider the signal on  ] , the function is equal to .thus we introduce the error function :           \\ & & + e^{i\omega t }    \left(\frac{\sigma^2\pi}{4}\right)^{1/4 }  e^{i(\omega-\xi)u}e^{-\sigma^2(\omega-\xi)^2/2 } \\ & & \qquad \times \left [  \mbox{erf}\left(\frac{t+u+i\sigma^2(\omega-\xi)}{\sigma\sqrt{2}}\right ) + \mbox{erf}\left(\frac{-u - i\sigma^2(\omega-\xi)}{\sigma\sqrt{2}}\right ) \right]\end{aligned}\ ] ] we assume that is large compared to and .we introduce the complex error function as  ] where the length of this interval is independent of the total time length and is proportional to .if we increase , the difference between the ridge and the frequency decreases as but the time interval where this effect is visible increases as .figure [ fig16 ] displays the ridge plots in three cases : the first panel is for where a flat ridge is expected , the second panel is for and , and the third panel is for and . + for , the extremum is a minimum for small and becomes a maximum for .thus the typical time of the effect of the discontinuity is again .for , there are two ridges approximately located at : these two ridges are symmetric with respect to .figure [ fig17 ] displays the two ridges in that case .the asymmetry on this figure is due to the correction of higher order terms in in the expansion of the spectrogram .the effect of a discontinuity at due to a periodisation is approximately the same as for the windowed fourier transform .the computations are very similar .the wavelet transform of is given by again two terms are important in the sum and . next we assume that is large and we express the wavelet transform using the complex error function : ,\ ] ] where and .the main difference with the windowed fourier case is that now depends on the frequency ( through the scale ) . for large and sufficiently close to , we get an approximation of and : where .thus the results of the windowed fourier transform can be transposed for the wavelet transform by replacing by .+ using the expansion of the error function , we obtain an approximation of the value of the maximum of the wavelet coefficients and hence an approximation of the ridge . for and for ,this value is : in the case of the wavelet ridge , we notice that the difference is proportional to the frequency , i.e. the the error in determining the frequency will be smaller for low frequencies than for large ones .+ for close to zero , we get an approximation of the ridge : the typical width in time of the error in determining the frequency is given by we notice that increasing the parameters of the wavelet decrease the amplitude of the error but increases the typical length time where this error is visible .+ the wavelet method can not determine frequencies such that , i.e.  frequencies such that in summary , the fact that the trajectory is known on a finite time interval creates in general a discontinuity on the boundaries of this interval .this distorts the ridge on a small time interval .it creates one or two ridges ; when there are two ridges , it means that the main original ridge is a minimum of the spectrogram .when there is one ridge , it can be below or above the main ridge depending on the value of the fractional part of .we have seen that the amplitude of this distortion is inversely proportional to the size of the window of the fourier transform and that the typical time interval during which this distortion is visible is proportional to this size of the window .+ figure [ fig18 ] represents the influence of the discontinuity in the time - frequency plane .the gray region is the portion of the time - frequency plane that has to be discarded due to the influence of the discontinuity on the boundaries . for the windowed fourier transform ,this error does not depend on the frequency whereas for the wavelet transfrom , the distorsion influences greatly the low frequencies .schematic representation of the finite time effect on the computation of the ridges : the gray region is the part of the time - frequency plane where the discontinuity of the signal between and influences the computation of the instantaneous frequencies for the wavelet transform and for the windowed fourier transform . ]j. von milczewski and t. uzer .mapping multidimensional intramolecular dynamics using frequency analysis . in d.l .thompson , editor , _ modern methods for multidimensional dynamics computations in chemistry _ , pages 190 200 , singapore , 1998 .world scientific .j. laskar .introduction to frequency map analysis . in c.sim , editor , _ hamiltonian systems with three or more degrees of freedom _ , nato asi series , pages 134 150 .kluwer academic publishers , dordrecht , 1999 .huang , z. shen , s.r .long , m.c .shih , q. zheng , n .- c .yen , c.c .tung , and h.h .liu . the empirical mode decomposition and the hilbert spectrum for nonlinear and non - stationay time series analysis ., 454:903 995 , 1998 .  `http://www-stat.stanford.edu/~wavelab ` .j.b . buckheit and d.l .wavelab and reproducible research . in a.antoniadis and g. oppenheim , editors , _ wavelets and statistics_. springer - verlag , berlin , 1995 .d. delande .chaos in atomic and molecular physics . in m.j .giannoni , a. voros , and j. zinn - justin , editors , _ chaos and quantum physics _ , les houches ,session lii , 1989 , pages 665 726 .elsevier , 1991 .

Abstract: we describe a method for analyzing the phase space structures of hamiltonian systems . this method is based on a time - frequency decomposition of a trajectory using wavelets . the ridges of the time - frequency landscape of a trajectory , also called instantaneous frequencies , enable us to analyze the phase space structures . in particular , this method detects resonance trappings and transitions and allows a characterization of the notion of weak and strong chaos . we illustrate the method with the trajectories of the standard map and the hydrogen atom in crossed magnetic and elliptically polarized microwave fields .   time - frequency analysis , wavelets , hamiltonian systems 05.45.-a , 32.80.rm