Dr. RENUKA Devi Sm

IMAGE MOSAICING USING PHASE CORRELATION AND FEATURE BASED APPROACH: A REVIEW

Keywords: Image Mosaicing, phase correlation, feature based approach

Mosaicing is a computer-vision-based approach to attain high field of view without compromising the image quality. Image Mosaicing technique enables to combine together many small images into a single large image, which represents more information. In this paper, images with overlapping regions are mosaiced using the phase correlation and feature based approaches. Phase correlation involves featureless registration, followed by mosaicing of images, and blending of image mosaic in order to reduce seams that may arise due to different environmental conditions such as lighting, camera focusing etc. In the efficient feature based approach involves in taking scale invariant features (using SIFT) in both the images. Later to find by using the perfectly matched corner pairs, homography is done to do mosaicing. A weighted average blending algorithm is used to seamlessly blend the two images.

INTRODUCTION

Mosaicing is basically a technique of image processing where in tiling of digital images is done to obtain large covering in an image. Mosaicing is a research area in computer vision, to understand and interpret the information represented by the images. Image mosaicing has many applications in the fields as diverse as forensic image restoration, computer generated special effects, digital video compression and many more [START_REF] Harris | A combined corner and edge detector[END_REF]. Information from many images can be combined into one image to aid for better understanding of what they represent. The amount of data which can be collected by the camera is small compared to the human eye. Though there exist complex lens to increase the field of view, it accounts to be costly. The other way of increasing the field of view is to capture the image from a long distance. However, very often, that is neither possible nor desirable as it may degrade the picture quality or resolution. This is where image mosaicing comes in.There are a number of methods to build image mosaics. One is to record an image onto a long filmstrip using the panorama camera. Another is to use a lens with a very large field of view, such as fish-eye lens, and a parabolic mirror. Although these methods directly capture the panoramic image, they are rather hardware intensive and expensive. On the other hand, image mosaicing algorithm which is a less hardware-intensive and very popular to obtain a wide field of view (FOV) image of a scene. It takes many regular pictures in order to cover the whole viewing space, and then stitches together the images to construct a single image with a larger field of view. Image registration, or alignment of images, is core technique of image mosaicing; it is the process of establishing point-to-point correspondence including translations, rotation and scaling between two images of the same scene. The Types of Image Mosaicing Techniques are can be broadly classified into direct or intensity based methods and feature based methods. Direct methods are found to be useful for mosaicing images having large overlapping regions with translations and small rotations. Feature based methods can usually handle small overlapping regions and in general tend to be more accurate. Image mosaicing using phase correlation is one of the earliest forms of mosaicing technique. It is an intensity based mosaicing technique. This is a traditional approach, which uses correlation, suffers from computational inefficiency and is sensitive to variations in the image intensity. To improve the efficiency of image mosaics, feature-based approach is used.

IMAGE MOSACING TYPES

Direct Based Method: Intensity-based methods compare intensity patterns in images via correlation metrics. It is based on the Fourier domain to match images that are translated and rotated with respect to one another. They register entire images or sub images. If sub images are registered, centers of corresponding sub images are treated as corresponding feature points. Feature based method [START_REF] Mallick | Feature based image mosaicking[END_REF]: It is based on accurate detection of image features. Correspondences between features lead to computation of the camera motion which can be tested for alignment. Feature-based methods establish a correspondence between a number of especially distinct points in images. In the absence of distinctive features, this kind of approach is likely to fail.Image mosaicing involves the individual images to be aligned to some reference image. Any Image mosaicing algorithm consists of three basic stages: a) Image Registration: Finding the homographies between the given images with respect to some reference image. b)Image Stitching/Alignment: Pasting the registered images so that overlapping regions between them are merged. c)Image Blending: Smooth out or rectify the intensity differences between the two stitched images. Forms of Mosaicing [START_REF] Mallick | Feature based image mosaicking[END_REF]:The most commonly used forms are planar and cylindrical mosaicing. i) Planar:When it is known that the image sequences are related by a (planar) homography, all the images can be warped onto a plane and the mosaic can be constructed. For every image, the transformation required to map it to the reference plane is estimated. After the registration is done, images are blended into a seam-less mosaic. ii) Cylindrical:The translational motion across images is estimated in terms of cylindrical coordinates. In other words, the input images are warped onto a cylinder mapping the world point P from Cartesian coordinates [X Y Z] to 2D cylindrical coordinates (θ, r). In Spherical mosaicing, the images are mapped to a sphere rather than cylinder.

IMAGE MOSAICING USING PHASE CORRELATION 3.1. Image Registration:

The registration method presented here uses the property of phase correlation which gives the translation and rotation parameters between two images. Suppose the two images I1 and I2 to be registered involve both translation and rotation with an angle of rotation being θ′ between them. When I2 is rotated by θ, there will be only translation left between the images and the phase correlation with I1 should give maximum peak. So by rotating I2 by one degree each time and computing the correlation peak for that angle, we reach a stage where there is only translation left between the images, which are characterized by the highest peak for the phase correlation. That angle becomes the angle of rotation .If there is no other transformation between the images other than translation, this shows a distinct peak at the point of the displacement. With this as the basis, translation is also found.Output: Registration parameters (tx, ty, θ) where tx and ty are translation in x and y directions respectively and θ is the rotation parameter. 3.2. Image Stitching:Image stitching is the next step following the registration. At this stage, the reference image is overlaid on the source image by pasting its pixels on a canvas at the appropriate location using the transformation parameters obtained in the registration process.

Image Blending:

The last step in mosaicing is Image Blending, which modifies the image gray levels in the vicinity of common boundary to obtain a smooth transition between images by removing the seams. Creating a blended image requires determining how pixels in an overlapping area should be presented. The advantage of Phase correlation technique is that it can be applied to best only for images having large overlap region with translation and smaller rotations.

FEATURE BASED IMAGE MOSAICING

In practice, the images generally have certain homographic transformation between them which cannot be solved using the phase correlation technique. Therefore, feature based method is used for mosaicing the images having homography between them. Concerning the purposes of image mosaicing, feature based methods aim to determine a relationship between the images through distinct features extracted from the images. Compared to direct methods which utilize comparisons of all the pixel intensities in each image while determining such a relationship, one large advantage of feature based methods is a major reduction in computational complexity. An additional, though not so decisive, advantage for feature based methods over direct methods is that nearly all feature based mosaicing systems have the property of strong structural similarity. The processing involved with feature based mosaicing systems can generally be broken down into five distinct stages. At each stage different algorithms can be implemented to compute results and then be used by thenext stage of mosaicing process. This is demonstrated in Figure 2.

Feature Matching:

In the second stage of processing, features are matched between each pair of overlapping images. These matches should be as accurate as possible, yet it is nearly impossible for a single algorithm in this stage to compute only the correct matches. For this reason, this processing stage produces putative feature correspondences. A good proportion of these correspondences should be correct, but need not be perfect matching, since RANSAC will later be used to eliminate the mismatches. These are used as seed correspondences. These putative correspondences are obtained matching the corner points using a combination of proximity and similarity of intensity neighbourhoods. There are two different methods to find the correspondences, first is using sum of squared differences (SSD) and the second is using normalized cross correlation (NCC).

Elimination of Outliers:

In the third stage of mosaicing for feature based systems, false matches are detected through an estimation process. This estimation process fits feature matches to a probabilistic model. The parameters of the model which best accommodate the feature set are then selected which in the case of mosaicing are the elements of a homography matrix.

Estimation of Homography Using RANSAC[(Robust Homography Estimation for a Pair of Images):

In this stage, the final mapping is computed which will relate coordinates of two overlapping images captured of a common scene. The input to this stage is estimated feature match inliers between two images. A homography is the name of a matrix capable of projectively mapping points in one image to those in another. The main advantage of RANSAC [11] is its low cardinality when compared to that of other estimation models like Least squares where the parameters are estimated using all the data available. Cardinality gives the minimum number of data points required to determine the model parameters. After we get n putative correspondences, the RANSAC robust estimation is used in computing 2D homography. Estimate H from all the inliers using the Direct Linear Transform(DLT) algorithm [START_REF] Brown | A survey of image registration techniques[END_REF].

Perspective image transformation (Image Stitching):

To complete the process of mosaicing computed homographies are used to transform the set of individual images captured of a common scene, projecting them as one, final complete image. This single image containing all of the imaged portions of a single scene is called the mosaic, and this stage of processing is called perspective projection. 4.6 Blending: As a part of image mosaicing, separately recorded photographs are aligned and combined to cover the entire desired region. Since the parts may have been recorded under different conditions, including weather, lighting, film processing and noise, they may have different gray level characteristics. This may cause seems to be apparent between different parts. Therefore, blending techniques are employed as the "final stage" to eliminate the natural seams that appear at the region of transition requirements is to use this document as a template and simply type your text into it.

EXPERIMENTAL RESULTS

Results of

Figure 1 :

 1 (a) Input image1 (b)Input image 2 (c) Peak values of Phase correlation (d)Phase correlation plot after rotating image 2 Phase correlation method to find rotation and translation parameters.[4]

Figure 2 :

 2 Figure 2: Stages in feature based mosaicing

  Phase correlation: Phase correlation results are shown below on input images shown in figure (a) and (b).

5. 2

 2 Results for Feature Based Image Mosaicing: Here for each pair of images that are to be mosaiced, initial input images , images with corners, image depicting the putative correspondences, image indicating the matching of the inliers obtained from RANSAC, warped image and the final mosaiced image are presented along with the estimated homography (H).Fig 4 and Fig 5 shows the different stages in the mosaicing of Atlas map image and Lab images respectively.

Figure 3 :

 3 Figure 3: Results of Phase Correlation based mosaicking

Figure 4 :Figure 5 :

 45 Figure 4: Mosaicing process of Altas Map Images


Drawing on Deleuze's work about cinema (the ‗movement-image' and the ‗timeimage'), this paper explores formal and aesthetic resonances with sound-based music, distinguishing between aesthetics of energy, articulation and montage, and aesthetics of contemplation, space and virtual relations. A second perspective is given, focusing on how listening behaviors may impose a ‗movement-image' or a ‗time-image' lens through which we could experience and remember a work's form. This is exemplified with a short analysis of the first section of Chat Noir (1998)(1999)(2000) by Elizabeth Anderson.

Gilles Deleuze has been one of the most prominent philosophers of the 20 th century. In Milles plateaux (1980), he introduced with Félix Guattari the notion of ‗rhizome' to designate that multifaceted, complex and nonhierarchical system of relations, constituting a wider perspective than that of the genealogical, binary, causal and hierarchical tree, which constitutes a state of being rather than becoming. Whereas psychoanalysis explains the human mind by reference to the patient's past traumas, the notion of rhizome tends to emphasize the systemic nature of the mind, its plural relations to past, present and future. In another domain, the rhizome is apt in describing pack animals' movements and other group behaviors (Sutton & Martin-Jones 2008: 3-6).

What one should understand is that tree and rhizome do not form a binary category, but rather different perspectives on the same phenomena: when looking at the tree, one could forget its roots and consider it to be going upwards only; when considering the rhizome, one takes into account the tree's roots and their relations with other roots which may emerge as other trees or other forms of life. Instead of analyzing, for instance, how the form and deviation of the tree's trunk led to its being somehow crooked, we may look at how each branch's movement relates to other branches' movements, more or less independently of the trunk; we may be interested in how this tree's roots are actually longer than its emerging part, and how they are intertwined with what surrounds them, etc. With this in mind, we may ponder about how the way we look at things actually determines a lot more than we would like to admit.

The main question leading this paper will be: what if music weren't an art of time? Gisèle Brelet asserted that music was the art of time par excellence, consisting essentially in ‗the discovery of the temporal structures of consciousness and their translation in the only sense that could translate them: sound ' (1949: 632). I have often heard this point of view, as a student in music and musicology, modulated according to context, but it all boils down to the following: unlike sculpture or architecture, music isn't constituted of physical objects, so it could not be an art of space; furthermore, we do not have any control over its temporal unfolding (we cannot go back in time), and therefore, music must be an art of time. This is an ontological perspective on music and the arts in general. What I would like to propose here is not meant to replace this perspective, but to complement it with another possibility focusing on human processes rather than ontology.

My approach will be an intermedial one, using Deleuze's concepts about cinema to shed some light on how we could apprehend sound-based music (or rather musical form) in several contrasting ways. The aim is not to study how cinema techniques were recuperated by electroacoustic music (for a discussion of this, see Langlois 2012 and Brunson forthcoming), nor is it to explore the sociopolitical issues related to the use of certain kinds of images (for a discussion about this, see [START_REF] Sutton | Deleuze Reframed[END_REF]. Rather, after having presented the ‗movement-image' and the ‗time-image', their formal and aesthetic correspondences with music will be explored, in order to pave the way for a discussion about how the ‗time-image' might be discovered, poietically and esthesically, in acousmatic music.

Deleuze and cinema

Whereas he didn't write extensive work about music, Deleuze dedicated two books to the study of cinema, distinguishing between cinema before and after World War 2 (L'image-mouvement, 1983; L'image-temps, 1985). This does not mean that any and all movies from 1945 onwards respond to the characteristics of the ‗time-image', but that a new kind of relation to time and space in movies appeared at that time, after a period of transition.

The movement-image

Before World War 2, cinema relied nearly exclusively on ‗movement-images'. [START_REF] Deleuze | L'image-mouvement (Cinéma 1)[END_REF] defines the movement-image as being focused on characters and their reaction to the situations they're exposed tothis is what he calls the ‗sensorimotor link'. This means that the camera followed characters wherever they went, and that every sequence served the narrative thread. This also means that time was represented linearly, and that ellipses served the sole purpose of allowing the representation of durations greater than the duration of the movie itself (see also Sutton & Martin-Jones 2008: 92-93). This was the case with most western movies, among other genres.

Deleuze distinguished three main kinds of movement-images, related to perception, affection and action. The ‗perception-image' represents the perception of the situation by the characternot perception for its own sake, but perception of a situation that will affect the character -this will be represented in the ‗affection-image'. In turn, the character will react upon the situation in order to modify it -which will imply the ‗action-image'. A hypothetical example may help clarify this: watching a western movie, we may see a wide shot of a ghost town, centered on a character looking around in the street (perception-image). As a door creaks on the right, the character turns her head towards it, and we see a close shot of her eyes fixed on the source of the noise (affection-image). The camera goes to a medium shot, allowing us to see both the character and the house in front of her. She points her gun at the wall next to the door and shoots to kill (action-image). Now it really is a ghost town, and the camera may go back to a wide shot, or pursue with the character searching for the body of the person she just killed (new perception-image) to confirm their death and eventually show contentment (new affection-image) and leave town (new action-image). The whole situation is depicted linearly, the camera follows the character in the house, and the whole point of the sequence is to depict what has been described here, then follow up with the narrative.

The time-image

The years surrounding World War 2 saw the advent of a new kind of temporality in cinema, which Deleuze called the ‗time-image' (1985see also Sutton & Martin-Jones 2008: 93-96). Deleuze credits Hitchcock with the overthrowing of the movement-image and the use of ‗mental images', implying new relations of the camera to the spaces, characters and images of the movie. This is the time that filmmakers began using flashbacks in their movies, not only to explain the action taking place in the present, but also to put forward the fundamental relativity of memory (as is the case in Citizen Kane in 1941). This is also the time that trivial scenes, that would have been cut out from ‗action' movies, began to be integrated (in 1952, in Umberto D, de Sica films the maid doing her housework, which brings no new information regarding the narrative but anchors it in everyday reality). In such contexts, wide shots of landscape and close shots of faces do not imply an incoming action. Rather, they are there to be read, to be compared, to be understood, to reveal time in itself: this face for five seconds, this landscape for twelve and a half seconds, this long take of people picking apples far away for some time, etc. Deleuze distinguished between two main kinds of time-images, according to whether the relations to be read are actual or virtual. ‗Optical-images' imply actual optic relationships. The viewer is given some time to read these relationships. Perception is replaced by existence, as in the beginning of 2001: A Space Odyssey (1968), with its static shots of landscapes, or in the unfolding of Ozu's Tokyo Story (1953) in which the camera often stays put for some time while characters move around, enter and leave the room. Affection may be replaced by close shot portraits, as in Robbe-Grillet's L'homme qui ment (1968) in which portraits are inserted at several points without any narrative reason for it, or Ozu's Tokyo Story again, in which the characters who are talking are often filmed looking straight into the camera, mostly without any context. Action is replaced by the moving around of figures in space, those figures being related to each other by measures of distance or closeness, between them and to the camera, as we often see in the unfolding of 2001: A Space Odyssey, for instance when the character goes out in space to repair the ship. ‗Crystal-images', instead of readable optical relations, imply virtual relations. These can be related to memory (like in the uncertain, multifold past of Citizen Kane), lying (in L'homme qui ment, at the beginning, the narrator says the bar was empty while the image is showing him entering a bar full of people), mise en abyme (in 2001: A Space Odyssey, the camera actually appears on screen in the last section, filming the protagonist), etc. All of these relations imply reflection on the part of the viewer, in one way or another. The crystal shows several coexisting realities (present or past); time is given in itself and turned on itself; the viewer is left to link everything together and draw conclusions.

Deleuze interpreted by Bonnet

Recently, Antoine Bonnet, a French composer, wrote that Deleuze's thought about the arts is not supposed to shed light on the one art it actually talks about, but to inform readers on much more general ‗modes of thought' that could be exploited for a number of situations and objects (2015: 82). Borrowing Deleuze's concepts about cinema, Bonnet describes the evolution of music at the turn of the 20 th century. Interestingly enough, ‗atonality' is not the topic of his talk, thematicism is.

The parallel between a musical theme in classical-tonal music and what Deleuze describes as a character reacting to a situation in a movement-image is quite clear: at the heart of the sensorymotor scheme, the theme both implies and reveals a given situation (-perception-sound‖), forms its expression in the foreground (-affection-sound‖), modifies and measures its formal evolution (-action-sound‖). Conversely, in modern-atonal music, the theme, precisely because it isn't a theme per se anymore, becomes incapable of reflection, expression or inflexion on any situation, because it appears in any situation as a constitutive element amongst other elements, or even as a simple marking. (ibid.: 83) The theme and its development are not the topic of music anymore. ‗Themes' appear and disappear without getting more important or more salient than other constitutive materials. Bonnet asserts that Wagner, with his leitmotifs, initiated the ‗crisis' by proposing themes whose development doesn't concern themselves, but their context, their relations with what surrounds them (ibid.: 84). Then Debussy, under the influence of Wagner, ruptured the sensorimotor link of music, preferring colors and floating time over melodic articulation. Composers in the 20 th century then developed music's ‗depth of field', using superimposition, juxtaposition, hierarchies and distributions of planes: ‗on the scale of the century's modernity, this resulted in the generalization of complementary and superimposed strata in composition, at the expense of causal and linear successions' (ibid.: 85).

Preliminary discussion

Although this perspective is interesting, its limits should not be overlooked. The main bias to be found here is the fundamental arbitrariness of the correspondences, which appears in the one-to-one relationship established between the three movement-images and the roles of themes in thematic music. Perception-images, affection-images and action-images seem extremely specific to cinema and cinematic narrative. Whereas Bonnet's perspective is that of an artist, a composer (studying how one could conceive one's work, one's materials), forcing such a correspondence might not be the ideal way to explore Deleuze's notions musicologically.

At a higher level of abstraction, the sensorimotor link of the movement-image finds its correspondence in music with theme development and energetic articulation: wherever the main material goes, the ‗camera' follows; an energetic climax in a sound is an opportunity for an energetic climax in the whole piece; the whole form is to be regarded as directional, teleological, linear; ‗musical time' emerges from the articulation and development of materials in time and establishes a direction, a speed, a nature of time passing different from that of our everyday experience. The time-image finds its correspondence in music with fragmented forms and extended static situations left to be readthis will be explored further.

Besides the movement-image and the time-image, Sutton and Martin-Jones define another kind of image in more recent cinema, which they call the ‗hybrid-image ' (2008: 99 sq.). This name points out the coexistence of both movement-images and time-images in many movies and music videos nowadays. They show how Singh's The Cell (2000) can be described as a movement-image overall, since there is a clearly delineated plot and clear-cut characters with a linear development (a killer in a coma, his last victim about to die somewhere, the FBI and a psychologist working together to find the victim and save her), as well as scenes of another nature, where camera cuts seem irrational and arbitrary, where images are of a more symbolic nature, where chronology and the sensorimotor link seem abolished altogether -this second kind of scenes taking place in the killer's psychotic mind. Although Sutton and Martin-Jones refer to The Cell as a hybrid-image, they concede that this movie, just like many other movies made nowadaysthe Wachowskis' The Matrix (1999), Jonze's Being John Malkovitch (1999), Nolan's Memento (2000), etc.is mostly a movement-image with a clear narrative thread, where the time-image serves the purpose of representing sick minds, dreams, drug-induced states, or other ‗abnormal' characters, situations or phenomena.

Instead of using the time-image for its own sake, with its own dialectics and its own potential developments, moviemakers nowadays seem to be motivated by the contrast it offers with the normal, with the movementimage. Instead of offering people necessary reflexivity in movies, we give them an all-made explanation: this weird thing happening does not need to be understood for itself, to be reflected upon, it is simply the manifestation of an abnormal state that may help explain the story.

We may find this same relation in music, in the articulation of tension and release, dynamic and static sections, etc. The following section will underline different natures of space and time in music, as well as the musical entities that may emerge from these natures. This will allow us to see how the time-image, rather than the movement-image or the hegemonic hybrid-image, may be used as an aesthetic descriptor, an analytical model and a listening behavior to shed some new light on acousmatic music.

Space, time and entities in music

Among musicological works, we may find some aesthetic distinctions which may immediately find some resonance with movement-images and time-images.

Energies and entities in music

Bériachvili defines, among other musical spaces, form-space as an ‗imaginary vision of quasi-geometrical relations between sound elements' in spatialized time, and matter-space as ‗sound matter (...) putting forward associations with colors, lighting effects, haptic feelings, physical materiality, etc. ' (2010 :12). He distinguishes between two kinds of articulation of these two spaces, delineating two aesthetic stances. In expressionist space (espace expressif), form-space is foremost, matter being only a more or less auxiliary attribute (typical examples include expressionist atonal music, Stockhausen's serialism, a large part of Boulez's serialism, etc.): this is the expressive mode, ‗action time' (ibid.: 107). On the other hand, in impressionist space (espace impressif), ‗any movement (…) is perceived as an attribute of sound-matter (best examples may be Xenakis and a lot of Varèse and Debussy)' (ibid.: 54). Here, ‗perception equals contemplation of musical space, while the time of music unfolds -objectively‖ without merging with our interiority' (ibid.: 107). This distinction is also found in the opposition of musique concrete and musique acousmatique as seen by Schaeffer. Battier reports that to Schaeffer, the montage of sound-objects gives rise to time in musique concrete, while in musique acousmatique, time gives life to sound-objects, allows them to develop (2013: 708). With the former, objects are articulated expressively in time, bending time to their inner development; with the latter, objects are born, live on for some time, revealing progressively their multifaceted nature while keeping a sense of identity. [START_REF] Khosravi | Spectral Spatiality and the Acousmatic Listening Experience: The Birth of Autonomous Spectromorphological Entities[END_REF] talks of ‗autonomous spectromorphological entities' to designate those sounds, living through time in an acousmatic environment and which may acquire a form of autonomy, of life-like, zoomorphic existence. At that point, musical entities are spatial rather than temporal, and another kind of relation to music may be instated: the ‗space-form'.

Space-form and momentform

Denis [START_REF] Smalley | ‗Space-form and the Acoustic Image[END_REF] defined the ‗space-form' as this musical form where time serves space rather than being articulated by itthe notion of coexistence thus being more relevant than that of succession. Describing his listening experience of a soundscape, he writes that Possibly the most important strategy in arriving at an holistic view of the space-form of this experience is that I disregard temporal evolution: I can collapse the whole experience into a present moment, and that is largely how it rests in my memory. The temporal disposition of, and relations among, sounds serve to articulate and shape spectral and perspectival space, but even though my perception of sound is the product of time, I ultimately sideline time's formative role. (2007: 38) Time does not disappear, however: the unfolding of the space-form implies expectancies about continuity, change, ruptures, etc. (ibid.: 54). One piece of music could be a single space, or it could be developed as a succession of spaces, establishing virtual relations between spatial forms, movements, actual relations, potential relations and the interweaving of time and space. We may find this again in the concept of momentform, introduced by Stockhausen to designate those contemporary musical forms in which the whole is contained in each of its partsimplying the absence of any ontological beginning and end of the work, by contrast with ‗dramatic forms' with their ‗necessary' successive episodes, their ‗conclusive force', the ‗emotion' induced by tension, etc. (1988: 102). Such works, in fact, ‗have always already begun and can go on without limit' (ibid.: 110).

Kramer, in his book about the time of music, interpreted this notion of moment as delineating a ‗moment time', where moments happen one after the other without their order necessarily being relevant, although they are organized in such a way that they form an entire piece (1988: 50). In this perspective, each moment is a spaceform, and the grouping of all those moments reveals a momentform. In its relation to time, momentform is thus quite close to the crystal-image's interest towards fragmentation: time is given directly without any notion of progression, causality or directionality, as a segmented space-form. Time is not the medium of discourse anymore, it is either its topic or its container.

Relevant features in acousmatic music scores

To illustrate the inclusion of temporal parameters in a space-form, we may consider a crossroads in a city, such as the one used by Leigh Landy to exemplify the cyclic nature and divisibility of soundscapes: theoretically, we should never hear pedestrians' footsteps at the same time as the sound of cars moving. This soundscape is regulated by the cycle of traffic lights (2012: 44-45). We may go even further, studying Landy's transcriptions of yet another soundscape: his sequential score (Figure 1, on the right) reveals the systematic simultaneity of children playing and the sounds of the factory, as well as the regularity of church bells and trains, while cars and birdsongs seem to be linked somehow. But are those elements really of interest to us? Wouldn't we rather know, when listening to a soundscape, where each sound event, each sound entity is situated in the spatial setting? This is what is indicated in the spatial score (Figure 1, on the left): the church and factory are quite far away, cars are close to the factory, the train goes left to right and closer to the front, the school is rather distant on the right, overflown by birds, and there seems to be another element, a pond with a swan in it.

Figure 1. Two proposals for the transcription of a soundscape (Landy 2012: 36-37).

On the right, we have no indication about space. On the left, we have no indication about chronology. In this depicted space-form, relationships between musical entities are systemic, hierarchical and spatial rather than energetic, chronological and causal.

The time-image in acousmatic music

The distinction between movement-image and time-image, established by Deleuze about cinema, thus seems to find its parallel in music. Bériachvili distinguished between espaces expressifs and espaces impressifs. The first of these concerns movements (space), following one another ruled by sensorimotor causality; they happen through time and allow time's structuration. We could think of expressive, articulated montage in musique concrète, where the sensorimotor link still rules. On the other hand, espaces impressifs concern time and duration; sequences follow one another in a rather arbitrary manner, each representing a sustainable, lasting space. Time unfolds in space and allows its structuringmovements, coexistences, alternations, all become proprieties of space; causality and succession lose their relevance. We could talk about acousmatic 'montrage', which is to say that we are ‗shown' a space, an ‗image-de-son', to be observed and read rather than to be experienced empathically. Figure 2 illustrates this duality, where movement may be articulated through or contained in time.

The time-image as a form or an aesthetic stance

It is quite clear already that some musical works, due to their structures and languages, may fall in the expressionist categorykeeping the dialectics between high-energy and low-energy sections, tension and release and a somewhat narrative thread (this is what appears in many acousmatic works nowadays)while other works may fall in the impressionist category (putting aside the energetic dialectics to focus on something else) and be considered ‗time-images'.

The ‗time-image', in this sense, can be related to two musical forms: it is a momentform, whose beginning and end are arbitrary and whose plans are individual and articulated in a somewhat arbitrary manner; it is also a space-form, in which time serves the unfolding of space, whose main point is to establish spatial relations between entities rather than causalities and successions, and which should be, in that regard, listened to as we would a soundscape.

Porträten (2014) and Porträten Tvō (2015) by Cara Arndt 1 are exemplary from this point of view, each offering six portraits of a specified arbitrary duration (as well as a ‗clotted blood insert'), the topic of the work being the identity of each portrait and the putting together of contrasting portraits, none of which have any teleological development. We could call each portrait an ‗optic-image', while the whole form would be a ‗crystal-image'.

The crystal-image may be explored further, in its relation to reality, memory and time. How could time be folded on itself in electroacoustic music? Trevor Wishart's Journey into Space (1972) is founded on the ambiguity between reality and daydream, between trivial reality and the everyday routine of the average worker. This is manifest, in the work, in the transition between apparently straight recording of everyday scenes and more abstract material; this appears again at one of the climaxes of the work, when the ‗waking up' sequence is played and repeated several times to figure routine. 2 Furthering our exploration of the crystal-image, Luc Ferrari's Presque rien n°2, ainsi continue la nuit dans ma tête multiple (1977), besides its anecdotal nature (which makes it eligible to be described as a time-image already), includes lies from Ferrari about his whereabouts and about the reality his voice is describing in the piece. In fact, the voice has been added afterwards to make it sound like Ferrari is exploring the soundscape, microphone in hand, and discovering it at the same time as we do, whereas in fact everything was constructed in a studio. The piece also features ambiguities about reality, as the anecdotal soundscape is progressively invaded by more electronic sounds, allegedly coming from within Ferrari's ‗multiple head', a process used in three out of the four acousmatic Presque Rien. 3 These three examples show how we could analyze specific pieces or groups of pieces in order to assess whether they present characteristics of the time-image in their conception, their construction, their sound content, etc. This forms the poietics of analysis. Let us now turn towards the aesthesics of the time-image.

The time-image as a reception behavior

Smalley was abundantly clear on the fact that the listener may disregard temporal evolution and not retain it as a relevant feature for the formal imprint of the piece. This means that listeners may actually decide to listen to a piece of music as if it were a time-image, even if the piece does not present clear poietic or structural characteristics of a time-image.

The beginning of 2001: A Space Odyssey is a group of static wide shots, following one another, with certain durations and certain readable optic relations. Nothing in the movie indicates whether those shots are supposed to represent different moments presented chronologically, as well as different places, or if they are supposed to be the same moment in those different places. This ambiguity allows us to watch this prelude with no notion of a chronology of events, although it wasn't necessarily conceived that way. We may in the same way listen to some (or maybe to most) sound-based works without the need for successive sections to be formally understood as a linear succession. In that regard, even a work that was composed and conceived chronologically might sometimes be listened to in such a way that it becomes a time-image.

1 Both works may be found at https://soundcloud.com/caraarndt [05/07/2015].

2

For a deeper analysis of this piece, see Marty (2014).

3

For a deeper analysis of the series of Ferrari's Presque Rien, see [START_REF] Marty | ‗Presque Rien, de l'anecdote au surréalisme[END_REF]. This is linked with the notion of ‗reception behavior' as ‗this act, where objective, strategy, perceptual construction, symbolisations, and emotions are mutually dependent and progressively adapt to the object' (Delalande 1998: 23) 4 . A listening behavior imposes a framework for the reception of a piece of music or sound art by selecting pertinences, without predefining any content for this piece. The point, here, is to say that one work could be listened to, apprehended and remembered in very different ways according to whether listeners considered chronology as being relevant or not to the form of the work.

Chat Noir 5

As an example, we may take Chat Noir, by Elizabeth Anderson. Chat Noir, composed between 1998 and2000, belongs to the ‗Janus trilogy' (named that way in the recent online edition by empreintes DIGITALes) inspired by Robert Johnson's Owning Your Own Shadow (1991), a psychoanalytical account of the repression of our inner child because of society's norms, and the possibility to accept paradoxes in order to reconcile apparently opposite elements without them losing their force. In Anderson's work, opposites are found in the use of different timescales, as well as in the coexistence of very diffuse wefts and very clear, crispy sounds. Here, I will focus on the first two minutes of the work, which form, in my experience, a clearly delimited section divided in two subsections. As illustrated in Figure 3, the section begins with an articulation leading to a sustained modulated high B (1000 Hz). A whole-tone harmonic field appears in the background as a weft , followed by a low, noisy, granular sound , internally articulated by dystonic dry percussions7 . While the high B disappears, the harmonic field becomes inharmonic and widens toward low frequencies. The whole spectra densifies and widens in space until its culminating point around 1'00, with the emergence of a new noisy sound in the foreground. Energy is directed towards a rupture point at 1'07, with three punctuations opening up the second subsection. Everything is muted while a modulated C# axis appears far away. Around 1'25, a new articulation brings back the high B, the inharmonic weft, the dystonic dry percussions and the noisy sounds in the background. A thin shrill granulation appears briefly around 1'37, while a high vortex emerges around 1'45, closing up the section with a morendo between 1'55 and 2'05 on the high B from the beginning.

It is quite clear how this section, taken sequentially, reveals two goal-oriented processes: in the first minute, the spectrum widens towards low frequencies until rupture; in the second minute, the spectrum 4 For studies about listening behaviours in this framework (with the distinction between empathic, figurative and taxonomic behaviours), see [START_REF] Alcazar | ‗La escucha atenta como base para el análisis de las músicas electroacusticas[END_REF], [START_REF] Anderson | Materials, Meaning and Metaphor: Unveiling Spatio-Temporal Pertinences in Acousmatic Music[END_REF], [START_REF] Delalande | ‗La terrasse des audiences du clair de lune de Debussy : essai d'analyse esthésique[END_REF]Delalande ( , 2010Delalande ( , 2013)), Marty (2015a), [START_REF] Marty | ‗L'entretien d'explicitation pour analyser l'écoute des musiques acousmatiques[END_REF]. widens both way and closes up again. Some local internal energy appears in the modulated sounds and in the unfolding of wefts and granulations, participating in the general forward-oriented motion.

3.2.2. The time-image as model for listener-oriented analysis How could we analyze Chat Noir (or other acousmatic works) as a time-image, as a space-form? We would have several optic-images, separated by montage between sections (notably, strong surface contrasts that could not be missed) or ‗travelling' of the acousmatic camera. All of these optic-images could be collated together to make up a single space, a single world, of which we could explore mentally time and spaces, as well as potential relations, even though they might not have been presented in the piece.

It was noted by [START_REF] Cipollone | ‗Le temps représenté[END_REF] that the systematic two-dimensionality of musical scores influences how we listen to music (and how we conceive of musicin my experience, it is often hard for students to understand how they could realize a static score that does not include time indications). Landy's proposals (see above) are a good remedy to this situation, as are new transcription tools like Couprie's EAnalysis (see [START_REF] Couprie | ‗Le développement d'un outil d'aide à l'analyse musicale : bilan et perspectives musicologiques[END_REF], allowing free choice over dimensional mapping of acoustic and musical characteristics. Here, I will propose a ‗spatial' score, by which I mean that the space of the score will try to be analogous, up to a certain point, to the sonorous space that might be perceived, while time will be integrated in the score as little as possible, only to indicate montage points and transitions, while sections will be transcribed spatially.

It is necessary to define how components are to be analyzed, their characteristics and their functions. Since we're not interested in chronology and causality, we'll be most concerned with the spatial and systemic relations between entities (by contrast with the chronological outline or the unfolding of articulation, energy and causality). A classification of entities' functions is necessary in order to have a solid ground for analysis. With this aim in mind, it seems relevant to borrow Schafer's terminology about the study of soundscapes (1977), where chronology has very little importance compared with the actual roles of sound events and their systemic organization.

1. Background wefts are those sounds that are less salient, rather continuous and diffuse, unruptured, giving a general ‗color' to space. We may establish here a link with Schafer's ‗keynote', designating, in a soundscape, those sounds heard without being listened to, giving the soundscape (or time-image) its implicit identity (1977: 9). Background wefts will be transcribed as wide diffuse halos.

2. Soundmarks, as defined by Schafer, are the sonic equivalent of landmarks, salient by some characteristics and recognized by a community as representative of their space (1977: 10). This may mean that a sound should be internally stable, consistent in its temporal unfolding, relatively clear and contrasted with the background, recurrent enough that it would give the time-image its explicit identity. Soundmarks will be transcribed as determinate forms occupying a consequent amount of space.

3. Acousmatic animals are the ‗autonomous spectromorphological entities' described by Khosravi, whose clarity and behavior may facilitate their perception as living, self-conscious organisms. These entities are not sounds revealing the presence of animals, but become animals themselves (2012: 44-45). Animals could be solitary or swarming. According to their saliency and to their repetition, they may also play the role of soundmarks or background wefts in complex textures. Acousmatic swarms will be transcribed as groups of little forms occupying space; acousmatic solitary animals will be transcribed as determinate forms associated with dotted lines to indicate movement.

4. Lastly, signals are those salient entities with an articulatory role (Schafer 1977: 10). In the context of the time-image, signals will designate salient and clear montage points, articulating the sound flow by cutting from one sequence to the next. Signals will be transcribed as two vertical bars illustrating rupture, while continuous change will be transcribed as arrows.

The following is my interpretation of the first section of Chat Noir, inspired by the perspective developed thus far. Rather than inquiring with diverse listeners on their experience of the work when listening to it in this way, I preferred the realization of an analysis resulting from my own experience of it. This is because asking listeners would have implied a rather lengthy explanation about the framework, and it would have been extremely complicated, if not entirely impossible, to verify that they were, actually, listening to the work in the way described here.

Furthermore, as I said above, a listening behaviour does not impose any content on a piece, it defines the framework in which to listen to the piece. There could thus be other interpretations, maybe radically different ones, using the same framework, with different analysts. Therefore the fact that the analysis is presented the way it is here is not so much to describe an actual listening experience of the work, but rather to describe a potential listening experience, resting on the delineated perspective as a necessary condition.

Chat Noir as a time-image

In the first section, the high B is clearly the most salient sound, because of its characteristics as well as because of its repetition in the second subsection. I will consider it a soundmark, as it appears to me as the main marker of one identity of the work: a high-pitched harmonic sound, modulated and sustained. The articulation leading to it could be described as a signal, because of its trigger-like nature. The harmonic weft delineating the whole-tone field will be described as a background weft, a keynote, because of its unchanging nature, its diffuseness and its low intensity. The articulation gone, the high B and the weft constitute the only two entities to be heard for the first 30 seconds of the piece, illustrated in the first picture of Figure 4. The soundmark is generally centered, the weft is more diffuse.

While the weft's timbre darkens slowly , a new background weft is revealed , internally swarming and occupying all the width of our acousmatic screen. At a certain point, we don't see the soundmark anymore, only the darker, closer to earth, sounds. An animated entity emerges, moving through space as if on its own ---, which I'll describe as a solitary acousmatic animal, because of its saliency and its energetic movement. We have a new soundscape in front of us, illustrated in the second picture of Figure 4. Between the first two pictures, the arrow represents a gradual transition along 20 seconds, which we can see as a camera travelling, from looking up at the soundmark to looking down at the shadowy land which it dominates. No time has passed yet, no energy has been expended by any goal-oriented character.

We've been given this space to contemplate for some time, when the first montage cut happens. This is a signal, repeated three times, revealing at last the hand of the person who crafted this image. A new background weft appears , represented as a halo in the third picture of Figure 4, and its internal life is given for us to see for some 15 seconds. Its harmonicity and its relatively high position in spectral space links it to the first weft and to the soundmark. Maybe they are close to it somewhere. Indeed, after having been announced by its attached signal, the soundmark appears again, and the camera moves away to be able to see the whole of it, englobing both the soundmark and the keynotes. While the first animal can be heard far away, another animal ---, radically different, briefly appears on-screen. The internal life of the low weft, closer than it was the first time, may again be understood as a swarm of acousmatic rodents or insects, occupying space around the soundmark. On the whole, we could understand this whole section as framing different aspects of a single space. Following this perspective, the soundmark is the pillar around which little and big animals live, run, play with each other in tall grass constituted by the background wefts. We could even look at the whole of Chat Noir, and obtain Figure 5, which represents the unified space of the piece. Once represented and remembered in this way, the world of Chat Noir (and most certainly a lot of other pieces) can be explored mentally, establishing relations that weren't necessarily heard in the piece but that still exist as another way of framing this world. At this point, chronology, energy, flow, etc., might all be discarded as relevant elements to the formal experience of the piece. Music in the moment [START_REF] Levinson | Music in the Moment[END_REF][START_REF] Levinson | ‗Concaténationisme, architectonisme et compréhension musicale[END_REF]) is overcome, not to go towards an architectonic view of form, but rather towards a spatial and systemic one.

Concluding remarks

Whereas Deleuze talked about the time-image as a new aesthetics of cinema, emerging post-WW2, my perspective here has been slightly different: rather than an aesthetic concept, the time-image has been taken as an aesthesic concept, as a -model for listener-oriented analysis‖. Therefore, an analysis such as the one proposed here does not seek to represent the (communicative) will of the composer, it offers listener a specific perspective on the work.

As was the case with the notion of rhizome (as a counterpart to the genealogical tree), the idea of music as an art of space might actually need to be explained for listeners to be able to apprehend the work in that way: here space is not a new parameter of sound to be articulated through time, it is a new character of form, a new container for specific formal functions and listening behaviours.

Figure 2 .

 2 Figure 2. Movement-images over time and a time-image containing movement.

Figure 3 .

 3 Figure 3. Chat Noirlistening score for the first two minutes of the piece 6 .

5

  For a larger account about Elizabeth Anderson and the poietics of Chat Noir, see Marty (2015b)//www.electrocd.com/bio/anderson_el/discog/].

Figure 4 .

 4 Figure 4. Chat Noir as a time-imagespatial score for the first two minutes of the work.

Figure 5 .

 5 Figure 5. Chat Noir as a time-imageholistic spatial score for the whole work.

In this greyscale version of the score, as colors have been removed, some symbols have been changed to most aptly render the distinction between sounds. For a colored version of the score, see[START_REF] Marty | ‗L'entretien d'explicitation pour analyser l'écoute des musiques acousmatiques[END_REF]. To listen to this extract, refer to www.musimediane.com/delatour_marty , on the page of the chapter from Marty & Terrien (or contact the author).

-Dystonic‖ refers to a sound whose spectral content presents both nodes (noise bands) and inharmonic content. This is the translation coined by Lasse Thoresen for Schaeffer's -son cannelé‖.


Jovana Jovic

Sébastien Lengagne
email: lengagne@gmail.com

Philippe Fraisse

Christine Azevedo-Coste

Impact of Functional Electrical Stimulation of Lower Limbs during Sitting Pivot Transfer Motion for Paraplegic People Regular Paper

Keywords: Multi-contact Motion optimization, FES, Paraplegia, Sitting Pivot Transfer 1

Individuals with Spinal Cord Injury (SCI), perform Sitting Pivot Transfer (SPT) motion around fifteen times a day using upper extremities. It can lead to upper limbs pain and often shoulder complications. In this paper, we investigate the influence of Functional Electrical Stimulation (FES) on SPT motion of a paraplegic person. First, we proposed to develop a dynamic optimization method in order to predict SPT motion of an able-bodied subject. This approach has been validated by comparing the computed SPT trajectories with the ones measured during the experiment with an able-bodied subject. Then, we used the optimization tool to analyze the influence of FES on the SPT maneuver of paraplegic persons. Our results suggest that FES can decrease arm participation during the transfer motion of a paraplegic person.

Introduction

Among all transfer activities, SPT is reported to be one of the most commonly used by individuals with SCI [START_REF] Bromley | Tetraplegia and paraplegia: a guide for physiotherapists, 5th ededn[END_REF]. The performance of SPT may rank among the most demanding functional mobility activities for the upper limbs Gagnon, Nadeau, Desjardins & Noreau (2008). Transferring from a wheelchair to a treatment table, bed, tub/shower bench, toilet seat, car seat and vice versa represents typical SPT performed by individuals with SCI. When initiating SPT, paraplegic individuals move their wheelchair as close as possible to the target seat. They move the buttocks forward close to the front edge of the wheelchair seat and, with the help of their arms, place firmly their feet on the floor. Then, they place one hand, called the trailing hand, on a stable position on the wheelchair and the other hand, known as the leading hand, on the target surface far enough to leave sufficient space for the buttocks. From this starting position, subjects bend their trunk forward and sideways, meanwhile they lift up their body and sustain their weight with the arms. After that, with a very rapid twisting motion they place the buttocks on the target seat. The transfer is concluded when the subjects reach again a seated postural stability. The performance of SPT varies from one person to the other depending on the subject's features, like lower limb spasticity, and environmental factors, like wheelchair design Gagnon Usually SPT is divided into three distinct phases:

• Pre-lift is the preparatory phase that ends when the trunk bends forward, • Lift is the phase when most of the body weight is supported by the upper limbs, • Post-lift is the rebalancing phase, when the subject is sitting on the new seat.

The lift phase lasts forty percent of the entire transfer duration Gagnon, Nadeau, Noreau, Eng& Gravel (2008). A lot of scientific works have been focused on experimental studies in which the kinetics and the kinematics of the SPT motion has been analyzed The trajectory of Center of Mass (CoM) position estimated using experimental data were compared with the ones obtained using optimization process. Once the results were validated, the influence of FES stimulation on the corresponding motion in paraplegic subjects was investigated.

Method

Kinematic model

In this study, we used the biomechanical model of a human shown on

Dynamic modeling and balance

To study the contact forces of the hands, we start from the the dynamic model as presented:

T T q q q q q q q q                            (1)
1 for the right part of the model the values of y joint limits are opposite.

Where n q   is a vector containing the joint positions,

Computation of the contact forces

Equation ( 1) emphasizes the link between the joint trajectories   q t , the contact forces F and the joint torques  . From the joint trajectories, there is an infinity of couple forces-torques. Here, we present how to find a set of contact forces that compensates the dynamic effects, ensures desired torques and encourages the balance as much as possible.

Due to the non-planar contact points, we cannot use classical method such as the Zero Moment Point [START_REF] Vukobratović | Zero-moment point: Thirty five years of its life[END_REF] to characterize balance. In order to ensure the balance and desired torque values, the contact forces must counterpart the dynamic effects and fulfill the friction and unilateral constraint to avoid undesired sliding or taking off, such as:

2 2 D J F = 0 T  (2) 1, 1, [ ] D J F T e
e e e    

(3) To solve Equations ( 2) and ( 3), one can use the Moore-Penrose pseudo-inverse matrix of the Jacobian matrix.

2 2 2 0 {1,..., } n i f t n i i i F i N F F                (4)
However, the pseudo-inverse matrix minimizes the instantaneous norm of the contact forces, without any effect on the friction or sliding constraint. We rather consider the contact forces that are as close as possible to the normal direction to the contact surface, i.e. that are the solution to the following problem:

2 2 2 1 min β ( || || ) 2 [ ] [ ] 0 t n i i i i i i i i i i F F P A F D A                        (5)
 2 1 1 , 1, D W [ ] D [ ] i i i i i e i e e F P A A              (6)
with:

  1 , , W [ ] i i i i i i i e i i e i P A A P A A                             (7)
with   , , , ,

i i i i i i W diag      
(we assume that the z-axis is the normal direction of the contact forces) and     is a square matrix that can be easily inverted, by using, for instance, the Gauss-Jordan algorithm. From the computation of the contact forces, we can compute the joints torques using Equation (1).

Optimization process

In mathematics or computational science, optimization refers to the selection of the best element from a set of available alternatives. In our case, the optimization problem consists in minimizing a real function while ensuring some constraints, such as respecting joint limits, ensuring balance during the motion, etc. As mentioned in Introduction section, this approach was already used in the case of motion generation in the robotics field and field of human motion science Lee In this paper, we refer to the optimization process in order to generate SPT motion for the biomechanical model as presented in Section 2. This optimal motion, described through a parameter vector X, minimizes a cost function C and ensures a set of discrete , k z and continuous inequality i g and equality j h constraints. Thus the optimal motion is the solution of:

  argmin C X , [0, ] ( , ) 0 , [0, ] ( , ) 0 ( , ) 0 f i f j k k i t T g t j t T h t k z t           X X X ( 8
)
The parameter vector

  1, 1 1, , , , , ,
m K j i j i i q t b t p    (9)
As presented in Lengagne, Kheddar & Yoshida (2011), we used a time-interval discretization based on a polynomial approximation to consider a finite set of constraints that properly represents the continuous constraints. We used the optimization software IPOPT [START_REF] Wächter | On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming[END_REF] that is a free and effective solution with C++ interface. We consider cost function C as the weighted sum of joint torques and joint jerks, as presented:

... 2 2 0 0 ( )

T T i i i C q a dt b qi dt        (

Experimental Validation

As mentioned above, the SPT can be divided into three phases: pre-lift, lift and post-lift phases. For each phase an initial and final position can be defined:

• Pre-lift phase: the subject starts sitting on the initial chair with the feet on the floor, the right hand on the handle on the right side of the initial chair and left hand on the handle on the left side of the target chair; • Lift phase: the buttocks are moved from the initial to the target chair, while the positions of the feet and hands do not change; • Post-lift phase: the subject regains a sitting posture on the target chair, position of the feet and hands unchanged.

One able-bodied subject (male, age 29, height 1.75m, weight 72kg) participated in this study after signing an informed consent. The angle between the two chairs was 20 degrees. Kinematic variables were measured using a stereophotogrammetric system (8 Mx cameras, VICON®) at 100Hz sampling rate. In order to estimate accurately small postural modifications, 35 reflective markers have been located on the subject's body using Plug-In-Gait template http://www.vicon.com/ (n.d.). The experimental set-up is presented in Figure 2. The subject was instructed to assume comfortable feet placement, to place both hands on the handles and to keep the same position of the feet and hands during the experiment. The initial position of the subject is shown in Figure 2. The subject performed the SPT at his own speed and way five times.

The Vicon Nexus® and LIFEMOD® softwares have been used to estimate CoM positions from experimental data. Each trial was time-normalized to 100 samples. Root Mean Square error (RMS) and Pearson's correlation coefficients between experimental and optimized CoM positions have been calculated in order to compare those variables.

Results

The ability of optimization process to predict SPT trajectories

Optimal SPT trajectories were calculated using described cost function. In Fig. 3 Table 2 presents Root Mean Square error and Person's correlation coefficients calculated between CoM position computed using our optimization process and CoM position calculated using experimental data for all five trials. The RMS error is smaller than 10cm in A/P direction (Trial 5) and M/L direction (Trial 1). Observing Person's correlation coefficients, it can be noticed that our optimization process well reproduces the behavior of our subject in M/L and vertical direction.

Influence of FES assistance on hand forces during SPT motion

The optimal STP trajectories are calculated for nine different scenarios (Sc). The first scenario, Sc 1, represents the behavior of an able bodied subject, allowing the variations of knee joint torque. In the other scenarios, the value of the knee joint torque is constant during the motion. In the second scenario, Sc 2, the knee torque is 0 Nm and represents the behavior of a paraplegic subject performing the motion without FES assistance. Other scenarios, from the third (Sc 3) to the ninth (Sc 9), represent a paraplegic subject performing FES-assisted SPT task while the stimulation parameters were increased. The torque values are following. The following assumptions have been made:

• for Sc 2 -Sc 9, the voluntary control of the knee joints is not possible, • the virtually stimulated bi-articular muscles produce torque control only at the knee joints, • knee joint torques do not change during the lift phase of the transfer (it has zero value during pre and post lift phase) and no stimulation leads to a null knee joint torque.

After validating our approach as presented previously, we computed the contact forces on the hands using the biomechanical model presented in Section 2, and the cost function descibed in Section 2.4. The results of the computation for the nine scenarios are presented in Figure 4.

As expected, the contact forces under the hands are lower in the case of an able-bodied person ( Sc 1) than in the case of a paraplegic person performing the motion without assistance of FES (Sc 2). In addition, analyzing other scenarios, it appears that the stimulation of the knee joints has an impact during the lift phase during which the contact forces of the hands decrease when the equivalent torque produced by the FES of lower limbs muscles increases.

Conclusion

The results of this study indicate that it is possible to describe sitting pivot transfer of an able-bodied subject within the frame of optimization theory. Taking into account the body dynamics and kinetics, the proposed method was able to reproduce motion with the CoM trajectories quite closed to the one measured during experiments or estimated from the optimization process.

Using it, we have shown that functional electrical stimulation applied on paralyzed lower limbs has an impact on arm efforts during the SPT motion. Our study indicates that using FES decreases arm efforts during the SPT motion. The ability to achieve those transfers with minimal participation of the upper limbs would help to preserve long-term shoulder integrity and improve the life of paraplegic patients.

Besides modeling of dynamics and kinematics of human body, Future studies will include modelling of muscle properties and finding the optimal FES pattern in order to minimize the fatigue of stimulated lower limbs muscles during SPT motion.

  , Nadeau, Noreau, Eng & Gravel (2008) ; Koontz, Kankipati, Lin, Cooper & Boninger (2011). SCI persons perform around fifteen SPT motions a day Gagnon, Nadeau, Noreau, Dehail & Gravel (2008); Gagnon, Nadeau, Noreau, Eng & Gravel (2008).

Figure 1 .

 1 The model consists in nineteen segments and eighteen spherical joints, i.e. fifty four Degrees of Freedom (DoF), but we set a constant value to the six DoF of the neck. Despite the existence of accurate models for the knees and shoulders Gini et al. (2007); Murray & Larochelle (1998), we consider that spherical joints are good enough to describe the whole body motion. The spherical joints are considered as three rotations around X, Z and Y axis. The joint limits are the one presented in Table 1. The kinematic and dynamic properties of this model are scaled regarding the desired height and weight using the anthropometric Winter tables Winter (2009). 70 : 0.70 -0.09 : 0.09 Mid-spine -0.61 : 0.47 -0.34 : 0.34 -0.63 : 0.63 Lower neck -1.13 : 0.70 -0.61 : 0.61 -0.61 : 0.61 Upper neck -1.13 : 0.70 -0.61 : 0.61 -0.61 : 0.61 left clacivula 1 -0.01 : 0.01 -0.15 : 0.15 -0.15 : 0.15 left shoulder 1 -2.26 : 3.14 -2.26 : 0.00 -1.04 : 0.52 left elbow 1 -0.001 : 2.79 -1.57 : 1.57 -0.01 : 0.01 left wrist 1 -1.22 : 0.87 -0.01 : 0.01 -0.52 : 0.35 left hip 1 -0.87 : 1.65 -0.35 : 1.13 -0.61 : 0.61 left knee 1 -2.26 : 0.001 -0.01 : 0.01 -0.01 : 0.01 left ankle 1 -0.69 : 0.52 -0.34 : 0.34 -0.34 : 0.34

Figure 1 .

 1 Figure 1. Representation of the kinematic model and joint frames for zero joint positions.

n

   the vector of the joint torque 1 the dynamics effects (sum of inertia, Coriolis, centrifugal and gravity) due to the joint trajectories, F ,... the vector of the contact forces. Note that the generalized formulation of the inverse dynamic model usually contains the position and orientation of the waist that are computed from the joint angles and the position of the right foot knowing its constant position and orientation.

  (2) and (4) refer to the balance criteria presented in[START_REF] Harada | Motion planning for walking pattern generation of humanoid, Intelligent Robots and Systems[END_REF] which states that the contact wrench sum must remain in the contact wrench cone to ensure balance.

1 i

 1 value to modify the repartition of the different contact forces and i  gives more importance to the tangential components regarding to the normal one for each contact forces. The solution to problem (5) counterparts the dynamics effects and produces the desired joint torque e  while fulfilling as best as possible the friction and unilateral constraint (by setting  ). The solution of problem (5) is:

  et al. (2005); Lengagne, Kheddar & Yoshida (2011); Mombaur et al. (2005).

  parameters as presented in Section 2.3 and f T is the motion duration. As already used in Lee et al. (2005), we represent the joint trajectories   q t using B-Spline basis functions De Boor (1978):

   . The values a and b were set heuristically as explained in Lengagne, Kheddar, Druon & Yoshida (2011).

Figure 2 .

 2 Figure 2. Experimental setup and initial position of the subject.

  , trajectories of CoM position in Anterior-Posterior (A/P), vertical, and Medio-Lateral (M/L) directions are presented. It can be observed that differences between the optimized CoM positions and the ones calculated from measured data in anterior-posterior and vertical direction are not bigger than the variability among the different trials of the same subject. The difference between the optimized CoM position in M/L direction and the corresponding ones estimated from experimental data is slightly bigger than in the two other directions.

Figure 3 .

 3 Figure 3. Computed (red line) CoM position in A/P, vertical and M/L direction in able-bodied subject. The mean CoM trajectory, in A/P, vertical and M/L direction, calculated from experimental data (black line) and gray lines marking the plus/minus standard deviation interval are shown.

Figure 4 .

 4 Figure 4. Representation of the contact forces for the several cases. Number of Trial 1 2 3 4 5 CoM A/P RMS 0.1 0.0873 0.0741 0.0507 0.081 Correlation Coefficients 0.6732 0.2951 0.1069 0.7178 0.6402 CoM Vertical RMS 0.0581 0.0511 0.0409 0.0.056 0.0452 Correlation Coefficients 0.6552 0.6732 0.6706 0.7395 0.6908 CoM M/L RMS 0.0863 0.0716 0.0481 0.0.0687 0.1078 Correlation Coefficients 0.9017 0.9722 0.9939 0.9191 0.8643

  Sc 3: 5 Nm, Sc 4: 10 Nm, Sc 5: 15 Nm, Sc 6: 20 Nm, Sc 7: 30 Nm, Sc 8: 40 Nm and Sc 9: 50 Nm. The virtually stimulated muscles are quadriceps and biceps femoris.

Table 1 .

 1 Joints limits for the biomechanical model (in radian).

Table 2 .

 2 RMS error and Person's correlation coefficients calculated between CoM position computed using optimization process and CoM position calculated using experimental data.

Int J Adv Robotic Sy, 2013, Vol. 10, 25:2013 www.intechopen.com

Int J Adv Robotic Sy, 2013, Vol. 10, 25:2013 www.intechopen.com

Jovana Jovic, Sébastien Lengagne, Philippe Fraisse and Christine Azevedo-Coste: Impact of Functional Electrical Stimulation of Lower Limbs during Sitting Pivot Transfer Motion for Paraplegic People www.intechopen.com

Int J Adv Robotic Sy, 2013, Vol. 10, 25:2013 www.intechopen.com

Int J Adv Robotic Sy, 2013, Vol. 10, 25:2013 www.intechopen.com

Acknowledgments

The authors thank the subject for participating to this study and making it possible. This work is partially supported by grants from the German Research Foundation (DFG: Deutsche Forschungsgemeinschaft).