Jocelyne Troccaz

Michael Baumann

Peter Berkelman

Philippe Cinquin

Vincent Daanen

Antoine Leroy

Maud Marchal

Yohan Payan

Emmanuel Promayon

Sandrine Voros

S Bart

M Bolla

E Chartier-Kastler

J L Descotes

A Dusserre

J Y Giraud

J A Long

R Moalic

P Mozer

Medical image computing and computer-aided medical interventions applied to soft tissues. Work in progress in urology

Keywords: Computer-aided surgery, medical robotics, medical image registration, urology

Introduction

A short introduction to urology

Urology concerns the exploration, diagnostic and medical or surgical treatment of both the urinary apparatus of men and women and the genital apparatus of men. The organs of interest are the bladder, kidney, ureter, urethra and, for men, the prostate, penis and testicles (see Fig. 1). Pathologies include among others: lithiases (stones), cancers, traumas, stenoses, incontinence, infectious diseases, malformations and sterility. Urologic surgery also includes kidney transplantation. The major targets for robot or image-guided assistance are the prostate and the kidneys as detailed below.

Prostate cancer is one the most common malignancy among men. [START_REF] Parkin | Global Cancer Statistics in the year 2000[END_REF] reports year 2000 cancer statistics: 543000 cases and 204000 deaths were attributed to prostate cancer worldwide. Its detection is based on digital rectal examination (DRE) and Prostate Specific Antigen (PSA) rating and is confirmed through the anatomo-pathologic analysis of biopsies.

Treatments include watchful waiting, surgery (laparoscopic or conventional radical prostatectomy), chemotherapy, and destruction of the tumour using different physical agents including radiotherapy (radiation by external beams), brachytherapy (radiation by implanted radioactive seeds), HIghly Focused Ultrasound, radiofrequency, and cryoablation. Because of the immediate anatomical environment of the prostate, in particular the bladder and rectum, and because of the role of this gland in the sexual life of patients, special attention is paid to minimally invasive techniques. One objective is to minimize induced morbidity. However, from the clinician standpoint the earlier detection of cancers from PSA screening and the development of laparoscopic techniques, by targeting smaller area via smaller entry ports, yield to increasing difficulties. Thus, computer or robot assistance may be needed. Percutaneous access to the kidneys is also a challenging issue and concerns many patients. This technique can be used for any introduction of a needle in the kidney for diagnostic (biopsies) or therapeutic actions (radiofrequency cancer ablation or stone destruction for instance). The destruction of stones is a major clinical application: 5% of the occidental population is concerned. Traditionally, percutaneous access is controlled from real-time imaging (ultrasounds or fluoroscopy) whose drawbacks are respectively poor visibility and irradiation. There also, assistance would be welcome.

Dealing with soft tissues

Because urology deals with soft tissues, it is a perfect illustration of the difficulty to directly apply the computer-assistance know-how from bony structures to mobile and deformable tissues. Mobility and deformations have different origins:

-Intrinsic origin: some organs intrinsically move or are deformed to perform natural physiological activity such as breathing or cardiac rhythm. A foetus organ also has an intrinsic mobility due to foetal motion. Heart beating is quite predictable whilst foetal motion is not.

-Anatomical environment: other organs move or are deformed because of their anatomical environment. This is typically the case for the kidney that is moved up and down according to the diaphragmatic activity during breathing; this is also true for the prostate which position and orientation depend on the bladder and rectal filling, and in a lesser way, breathing.

-Patient position: the position and shape of some structures depend on the patient posture or position relatively to gravity. For example, the prostate position partly depends on the flexion of patient's legs.

-External action: finally, the therapeutic (needle insertion for instance) or diagnostic (e.g. ultrasound examination) action may move and deform the organ of interest. This is the case for the kidney and even more for the prostate, especially when an endorectal ultrasonic exam is performed.

Very often, the motion and deformation of an organ has multiple sources. For instance, the prostate moves and/or is deformed due to: patient breathing, patient posture, bladder and rectum natural or artificial filling, insertion of a needle, oedema from multiple needle insertions.

In the case of the prostate, several groups worldwide paid special attention in the mid-nineties to motion of the gland in the context of radiotherapy; since most intra-treatment localization approaches were based on X-Ray data where the prostate is not directly visible, it was important to quantify prostate motion with respect to bony structures. [START_REF] Van Herk | Quantification of organ motion during conformal radiotherapy of the prostate by three dimensional image registration[END_REF] performed separate CT/MR bone and prostate registrations to determine prostate mobility; rigid chamfer matching on segmented surfaces was used; [START_REF] Balter | Measurement of prostate movement over the course of routine radiotherapy using implanted markers[END_REF] used implanted prostate fiducials and X-ray images to perform a similar study. More recently deformation was studied specially in the context of imaging involving intrarectal probes or coils (see for instance [START_REF] Hirose | Quantitative MR imaging assessment of prostate gland deformation before and during MR Imaging-guided brachytherapy[END_REF]).

In order to be able to handle these anatomic changes several issues must be solved: models must be designed when the motion and/or deformation are predictable and repeatable; tracking capabilities must be developed based on intra-operative sensing (images, signals such as ECG); real-time re-planning may be necessary for the guiding system (for instance a robot) to adapt to these changes; finally robots should be synchronized to those motions and deformations in a discrete or continuous way. This raises very challenging robustness and safety issues. One important characteristic of those anatomic changes is their time scale with respect to the duration of the action to be performed. Consequently, different strategies may be selected: localization just before the action or tracking during the whole action. In the following sections, the way those questions have been solved in the case of urological targets will be analyzed.

Robotics and urology

Historically, urology was one of the first clinical domains where a robot was used for patients.

At the time -the late eighties -where most people dealt with neurosurgery or orthopaedics applications of robotics, the London Clinic and the Imperial College of London developed PROBOT [START_REF] Davies | Development of a surgeon robot for prostatectomies[END_REF]: a robot for the transurethral resection of the adenomatous prostate, i.e. the removal from the inside of the gland of extra-tissues compressing the urethra. The first test on a patient started in April 1991. After a feasibility study on 5 patients, a pre-clinical series with 40 patients was undertaken. Several versions of this system where developed; the first prototype was based on a PUMA 560 (from Unimation Inc.) connected to a passive frame. This frame is an elegant solution to safety issues since it constrains the tool movement inside a cone related to the task to be executed. The current system consists of a passive robot positioning a motorized frame with 3 degrees of freedom (dof) -conical motion plus translation of the resectoscope. [START_REF] Shah | Urobotics": robots in urology[END_REF] reports the difficult task of automatically controlling this robot for resection monitoring from the real-time intra-operative ultrasound images. Indeed, because soft tissues move and deform, two types of strategies may be used in robot control. The ideal approach would be to continuously and automatically close the robot control loop using intra-operative information about the organ motion. To our knowledge, such a solution has not yet been developed for urology. However in radiotherapy, where the tool is outside the body and the planning is rather simple (beam orientation with respect to.

the patient and duration of radiation), organ tracking ability was introduced. In [START_REF] Coste-Maniere | Robotic whole body stereotactic radiosurgery[END_REF] the motions of intra-body implanted fiducials are correlated to the motions of infra-red on-body markers for tracking breathing movements this process is however rather invasive.

[Sawada04] proposes a non invasive solution based on real-time image correlation for the detection of a pre-defined stage in the breathing cycle (full expiration for instance); this information is used for respiratory-gated radiotherapy treatment. The other and much simpler approach is to tele-operate robots: in that case the user closes the loop between robot motion and real-time image information. Such an approach is particularly interesting when operative planning is too complex to be explicitly defined. Intermediate solutions consist in adding motion tracking abilities to tele-operated robots (see [START_REF] Ginhoux | Active Filtering of Physiological Motion in Robotized Surgery Using Predictive Control[END_REF]) or to close the loop from imaging data in a more discrete way for simple tasks (see section 2.2).

Tele-operated robots

Endoscope holders

The first FDA 1 approved medical robot, AESOP (from Computer Motion Inc.) [START_REF] Sackier | Robotically assisted laparoscopic surgery: from concept to development[END_REF] had a significant clinical and industrial success. Two thousand AESOP were sold to around five hundred hospitals between years 1994 and 2000. AESOP has a SCARA architecture with 4 active and 2 passive (pivot rotation) dof; this tele-manipulator is voice controlled. Many other robotic endoscope holders have been developed in the academic and industrial tracks.

One of them designed at TIMC [START_REF] Berkelman | LER: The Light Endoscope Robot[END_REF] has the interesting property of being directly put on the patient abdomen skin (see Fig. 2). Because the robot is placed on the endoscope entry point, 3 dof (2 rotations and 1 translation) are sufficient to handle the endoscope motions.

Fig. 2. The LER (Light Endoscopic Robot, TIMC, Grenoble Hospital and School of Medicine): on a phantom (left), urological intervention on cadaver (right).

As compared to AESOP and to most of the other systems which are positioned on the operating room (OR) table, floor or ceiling, this very compact system follows the patient motions and is very easy to install. It weights 625g; it is voice controlled and completely sterilizable. Interesting evolutions of robotic endoscope holders deal with automatically control of robots from image information in order to track organs or instruments during the surgery (see [START_REF] Voros | Automatic Detection of Instruments in Laparoscopic Images: a First Step towards High Level Command of Robotized Endoscopic Holders[END_REF] for instance).

Tele-surgery robots

Based on the robotic endoscope holders experience, instrument holders have naturally been designed resulting in the so-called tele-surgery robots. ZEUS, an evolution of the AESOP, is composed of 3 separated 4 dof arms (one endoscope holder and two instrument holders).

Another system, the DaVinci (from Intuitive Surgical Inc.), is composed of 3 or 4 arms mounted on a single basis. Articulated instruments provide extra intra-body dof (see Fig. 3).

Both systems are based on master-slave architectures; the arms are tele-operated2 by the surgeon from endoscopic images. DaVinci proposes a "head-in" stereoscopic display (see Fig. 3) whilst Zeus includes a "head-mounted" stereoscopic display or a traditional screen. Intrabody dof are a major advantage of the DaVinci, increasing the surgeon's possibilities near open surgery conditions. Both systems are quite cumbersome and expensive; none of them include force feedback on the master workstation which may be a serious limitation for anastomoses for instance. The DaVinci has been extensively evaluated for urological applications. First robot-assisted laparoscopic radical prostatectomies were reported in [START_REF] Abbou | Remote laparoscopic prostatectomy carried out with a robot. Report of a case[END_REF], [START_REF] Binder | Robotically assisted radical prostatectomy[END_REF]. Very large series of patients have since been operated: the Vattikuti Institute in the Henry Ford Hospital of Detroit, USA, published in [START_REF] Menon | Vattikuti Institute prostatectomy, a technique of robotic radical prostatectomy for management of localized carcinoma of the prostate: experience of over 1100 cases[END_REF] a study concerning more than 1100 cases. In this centre, laparoscopic prostatectomies started in October 2000 and the DaVinci assistance was introduced in March 2001. A study comparing conventional/laparoscopic/robot-assisted laparoscopic procedures showed clear advantages of the robotic series on many points including shorter hospital stays, reduced pain, reduced blood loss, better PSA control, reduced positive margins, better continence, and less impotence. Another advantage of robot assistance is a reduction of the learning curve for laparoscopic procedures; [START_REF] Alhering | Successful transfer of open surgical skills to a laparoscopic environment using a robotic interface: initial experience with laparoscopic radical prostatectomy[END_REF] reports an improvement factor of about 10. Laparoscopic radical prostatectomy is probably one of the domains were the robotic clinical added-value was so clearly demonstrated. Other applications of such robots to urology are reported in full details in [UroCLin04]. Each time complex dissections, microsurgery or intracorporal suturing are necessary, the robot may be a precious assistant.

Several research projects in the world aim at developing competitive smaller and/or cheaper solutions with articulated intra-body instruments and endoscopes and master station offering force feedback. Planning tools are also developed in order to optimize the entry ports positioning, enabling both target access and collision-free motion of the robots (see for instance [START_REF] Coste-Manière | Optimal planning of robotically assisted heart surgery: Transfer precision in the operating room[END_REF]).

Image-guided robots

Many gestures in urology are carried under interventional radiology: the diagnostic or therapeutic tool is moved under control of an imaging modality. Ultrasounds or fluoroscopy enable continuous control: the operator can see in real-time the tool position and the anatomy;

CT or MRI allow asynchronous control: for instance, a needle is positioned, a control image is taken and the needle position is corrected if necessary, and so on. This idea has been exploited to control from medical images robots performing simple tasks such as a linear tool insertion.

Prostate biopsies and brachytherapies

From a technical viewpoint prostate biopsies and brachytherapies (see figure 4) are rather similar; they both consist in inserting needles in the prostate, either for tissue sampling or for radioactive seed placement, through transperineal or transrectal access, under imaging control -most often TransRectal UltraSound imaging (TRUS).

US-guided transrectal biopsy (left) and transperineal brachytherapy (right) operating principles.

From http://www.uropage.com/index.htm However, each biopsy makes use of a single ultrasonic (US) image in which the needle is visible whilst brachytherapy is based on a volume of images: often parallel axial US images acquired every 5mm. Brachytherapy is based on a careful patient-specific dose planning whilst biopsies are generally performed following a predefined global scheme (for instance sextant or 11-core protocols). Needle insertion is slow and manual during brachytherapies whilst biopsies are very rapidly performed using a biospy gun. As demonstrated by [START_REF] Hervely | Trajectory optimization for dynamic needle insertion[END_REF] in a different medical context, increasing needle velocity results in minimizing the displacement and deformation of the tissue. Thus automation may have a positive impact in terms of gesture accuracy. Moreover, prostate brachytherapy is based on the use of a template (a stereotactic grid) rigidly connected to the US probe. This restrains needle trajectories to lines parallel to the probe axis and results in potential collisions of the needles with the pelvic bone. Again, using a robot may enable various trajectory directions. [START_REF] Wei | Robot-assisted 3D-TRUS guided prostate brachytherapy: system integration and validation[END_REF] proposes to use a general purpose 6 dof robot for needle positioning and insertion. [START_REF] Davies | Brachytherapy -an example of a urological minimally invasive robotic procedure[END_REF] develops a special-purpose robot mimicking the conventional procedure (trajectories parallel to the US probe axis); a rotational dof for the needle is added to reduce the needle flexion during tissue penetration. Those systems are still laboratory test beds. [START_REF] Phee | Ultrasound guided robotic system for transperineal biopsy of the prostate[END_REF] describes a pre-clinical evaluation of a specific 9 dof system (positioning platform plus biopsy robot) for transperineal prostate biopsies. A 3D prostate geometric model of the prostate is approximated from series of close parallel US images enabling planning of the biopsies.

2.5mm accuracy is reported; those performances require very careful patient preparation and US probe handling. None of these systems really considers prostate motion and deformation during the procedure.

Another approach consists in performing transrectal prostate biopsies or brachytherapies with an intra-rectal robot under MRI control [START_REF] Susil | System for MR image-guided prostate interventions: canine study[END_REF]. Although conventional MR imaging (1.5T with endorectal antenna and T2 sequence) enables physicians to see precisely the prostate anatomy, using such a modality for biopsies is probably restricted to the few cases where USguided biopsies are not possible or not successful. Let us remind that in the United States (resp. in France) about 10 6 (resp. 10 5 ) series of diagnostic biopsies are performed each year).

In [START_REF] Susil | System for MR image-guided prostate interventions: canine study[END_REF], conventional MRI is used. The robot is inserted in the patient rectum and has three dof to reach the target defined on the MRI data: translation in the rectum, rotation around its main axis and progression of the needle. Thanks to its design, the robot does not disturb the magnetic field and includes two coils; one used as part of the imaging sensor and the other one as a position sensor. After validation on dogs, the robot is being clinically tested for transrectal biopsies and brachytherapies [Fichtinger04]; 2mm accuracy is reported; this remaining error is probably mainly due to the prostate motion and deformation during needle insertion. [START_REF] Chinzei | Surgical Assist Robot for the Active Navigation in the Intraoperative MRI: Hardware Design Issues[END_REF] proposes another robot for prostate biopsy or brachytherapy inside an open interventional MR system. Interventional MRI (0.5T) requires an additional conventional MRI exam which makes the procedure even more complex (see 3.2.1).

Percutaneous renal access

The purpose is to assist percutaneous access to the kidney. Since 1996 a robot named PAKY (Percutaneous Access of the KidneY) is developed by the Johns Hopkins groups in Baltimore (MD, USA). The robot has seven passive dof used to position a 3 active dof structure (2 for orientation and 1 for translation of the needle). Fluoroscopy is used for needle alignment and control during insertion. During the procedure, the patient is in apnoea in order to keep the kidney in a constant position. [START_REF] Cadeddu | Stereotactic mechanical percutaneous renal access[END_REF] reports in vitro and in vivo experiments. In [START_REF] Su | Robotic percutaneous access to the kidney : comparison with standard manual access[END_REF], for 23 patients, no significant difference is reported between the manual and robotic procedure in terms of precision, rapidity, number of attempts, complications. One advantage of the robotic procedure lies in the absence of irradiation of the human operator. [START_REF] Bascle | Needle placement under X-ray fluoroscopy using perspective invariants[END_REF] proposes a visual servoing approach from two fluoroscopic views enabling the automatic placement of the needle to a given target and entry point. This system was also applied to CTguided transperineal prostate biopsies through a single entry point.

Image-based urology

Image processing

Many papers propose tools to assist the segmentation of urologic images especially for the prostate where TRUS images have been paid close attention. Segmentation may be 2D, 2.5D (the segmentation of a given slice is used to help the segmentation of the following parallel one) or 3D. Most successful approaches make use of active contours and/or statistical models. However, for 2D images close to the prostate extremities, existing tools may not be robust enough due to the poor quality of those images. Other works concern the automatic segmentation of CT and MRI images of urological targets (kidney in particular). Because this problem is very vast and not typical of interventional systems, no details are given here.

[Shao03] and [START_REF] Zhu | Computer technology in detection and staging of prostate carcinoma: a review[END_REF] present good reviews of work concerning the image processing of prostate TRUS images.

Image fusion

MR and US imaging are probably the most used imaging modalities for prostate diagnosis and therapy. The interventional nature of US is counterbalanced by their traditional drawbacks: patient dependence, intra-and inter-operator variability, medium quality due to speckle, artefacts, etc. Conventional MRI using external or transrectal coils clearly show the prostate zonal anatomy which is useful for biopsy planning, whilst open MRI (also called iMRI for interventional MRI) enables near real-time control. This is why several research groups implemented fusion algorithm to benefit from complementary advantages of these modalities. Other imaging modalities are used such as CT imaging or histology sections; here also image fusion may be very useful. This paragraph describes different studies on multimodality fusion dedicated to prostate imaging.

MRI/iMRI fusion

The Surgical Planning Laboratory (SPL) and Harvard Medical School have developed a navigation system for transperineal prostate biopsies under iMRI. Because of a lower intensity magnetic field, iMRI does not clearly show the prostate anatomy. This is why a preoperative MRI acquisition is performed (external pubic antenna, 1.5T, T2 FSE sequence) on which surgical planning is possible. Intra-operatively, iMRI data are collected (external pubic antenna, 0.5T, T2 FSE sequence) enabling volume reconstruction. A localized stereotactic grid similar to the one used for brachytherapy is calibrated with respect to the MRI system and is used as a guiding device. MRI and iMRI are registered using an intensity-based elastic registration enabling transfer of the planning data to the iMRI conditions. During the biopsy procedure, real-time FGR iMRI slices (acquisition time: 8s per slice) are obtained for biopsy tracking. The "3D Slicer" developed by SPL enables computing from the iMRI T2-volume the slice corresponding to the FGR one with planning information added. [START_REF] Hata | MR Imaging-guided Prostate Biopsy with Surgical Navigation Software: Device Validation and Feasibility[END_REF] reports two successful clinical cases. As mentioned previously, such an approach is especially interesting for patients for which ultrasounds have been unsuccessful (negative repeated USguided biopsies with increasing PSA) or impossible. [START_REF] Bharata | Evaluation of three-dimensional finite-element-based deformable registration of pre-and intraoperative prostate imaging[END_REF] describes another solution for MRI/iMRI registration based on elastic deformation of segmented data; a finite-element model of the pre-operative prostate is deformed using isotropic linear deformations to match the intra-operative prostate. Two regions of the prostate are considered with different elasticity parameters: the central gland and peripheral zone.

MRI/TRUS fusion

In many cases, bringing the MRI information to the US-guided procedures can be an interesting alternative. [START_REF] Kaplan | Real-time MRI-ultrasound image guided stereotactic biopsy[END_REF] describes experiments concerning the fusion of preoperative MRI data to TRUS images for transperinal biopsies. Six fiducial points are matched between the two modalities using a rigid registration. From this transform, a composite image is produced combining TRUS data and re-sliced matched MRI data. No attention is paid to motion or deformation that could occur between the two acquisitions or during the procedure.

In one of the two main types of brachytherapy protocols the radiation plan is prepared in the OR and is based on intra-operative TRUS acquisition and segmentation. One difficulty lies in this initial stage. [START_REF] Reynier | MRI/TRUS data fusion for prostate brachytherapy[END_REF] reports the evaluation of elastic surface registration between TRUS and MRI data for prostate brachytherapy. Pre-operatively, three orthogonal T2 TSE volumes are acquired using an endorectal coil and the prostate contours are segmented jointly on the three volumes. Intra-operatively, TRUS data are collected (axial and pseudo-sagittal slices -see Fig. 5-left). TRUS images are manually segmented by the urologist as in the conventional procedure and this results in a sparse intra-operative 3D prostate model (Fig. 5middle); the axial contours are needed for simulating and planning the dose. A preregistration consisting in superimposing the ultrasound and MRI data centres of gravity initializes the unknown transform before a second step of minimization using the Levenberg-Marquardt algorithm. Elastic registration allows for rotation and translation between data sets as well as local deformations or distortions. The method is derived from the octree-spline elastic registration described in [START_REF] Szeliski | Matching 3-D anatomical surfaces with non-rigid deformations using octree-splines[END_REF]. It makes use of an adaptive, hierarchical and regularized free-form deformation of one volume to the other coordinate system. The result is a 3-D function transforming any ultrasound data point to the corresponding MRI point. After MRI/TRUS surface-based elastic registration, composite MRI/TRUS images are computed; the operator can then visualize combined TRUS and MRI data of a same region (cf. Fig. 5right). This enables refining the TRUS segmentations of the axial slices especially near the prostate extremities (base and apex). Elastic registration accounts for prostate motion and deformation between the two acquisitions. Processing of large intra-operative modifications due to needle insertions is limited to manual re-planning made by the radiophysicist during the procedure. [START_REF] Reynier | MRI/TRUS data fusion for prostate brachytherapy[END_REF] reports millimetre registration accuracy and shows that such a modified segmentation may result in a significantly different radiation plan.

CT/MR fusion

For radiation therapy (external radiation therapy or brachytherapy), CT imaging plays a central role in treatment planning or post-operative dose evaluation due to the need for tissue density information. Meanwhile, in prostate treatments, MR is still the main diagnostic modality. [START_REF] Amdur | Prostate seed implant quality assessment using MR and CT image fusion[END_REF] presents CT/MR data fusion based on landmarks for seed implant quality assessment; the seeds are visible on post-operative CT images whilst the prostate is visible on pre-operative MR data. Rigid registration of CT and MR data is based on the use of a urinary catheter going through the prostate and placed before both MR and CT acquisitions making it visible on both modalities. After registration of the catheter, seeds visible on the CT data may be visualized with respect to the MR prostate anatomy. The authors report an average residual error measured on each slice at the centre of the urethra of 1.2mm for 11 patients and a maximum residual error of 2.5mm for one patient. [START_REF] Lian | Mapping of the prostate in endorectal coil-based MRI/MRSI and CT: A deformable registration and validation study[END_REF] proposes to use a Thin Plate Spline (TPS) function to map segmented prostate contours from MRI or MRSI (Magnetic Resonance Spectroscopic Imaging) to CT data. In this study, endorectal coils are used for MR imaging which may deform the prostate as compared to the CT acquisition situation; rigid registration may thus not be accurate enough. Obtained correspondence index of 93.1±5% and centroid position of 0.56±0.09mm of the registered surface slices quantify the registration accuracy and demonstrate the superiority of TPS registration over rigid one.

Histology/other modality fusion

Histology is the gold standard of cancer diagnosis. Therefore, registering histological data to other modalities may be very fruitful. After preparation and fixation in formalin of cancerous prostates surgically removed from patients, 3 to 5mm blocks are generally cut in a pseudoaxial direction; then a slice of 30 (to 50) microns thickness is obtained from each block and analyzed by a pathologist with a microscope for detection and characterization of cancerous cells. As proposed in [START_REF] Egevad | A new method for handling radical prostatectomies enabling fresh tissue harvesting, whole mount sections, and landmarks for alignment of sections[END_REF] artificial markers are generally used in order to build a 3D reconstruction from those sparse data.

[Shen01] has fused histological data from hundreds of patients using elastic surface registration in order to build a statistical atlas of cancer occurrence in the different prostate zones and to optimize biopsy schemes based on that atlas. [START_REF] Bart | MRI-histology registration in prostate cancer[END_REF] has proposed to fuse MRI to histological data of patients with an elastic surface registration as a tool to study and improve the understanding of MR imaging of cancerous zones (see Fig. 6). [START_REF] Taylor | Three-dimensional registration of prostate images from histology and ultrasound[END_REF] correlates histological data to US images and sonoelastometry. A first scaling factor is applied to account for prostate retraction during fixation; then, a rigid surface registration is applied.

The purpose is here to evaluate the predictive value of sonoelastometry for tumour detection.

Surgical navigation

Surgical navigation was introduced in neurosurgery in the eighties. It consists in tracking the surgical instruments relatively to poorly visible organs and/or to a pre-operative planning and to present this information visually to the surgeon. Those systems generally require trackers (optical, magnetic, US, mechanical) and involve data fusion and/or registering to enable transfer of the planning to the intra-operative conditions. Few urological applications exist.

Renal punctures

[Leroy04a,b, Mozer05] propose a system for navigating percutaneous access to the kidney (see Fig. 7). Pre-operatively two injected CT (early injection and late injection) are acquired under apnoea. Early injection allows visualization of the kidney envelope whilst late injection makes the internal structures of the kidney visible. Mono-modal rigid registration based on a chamfer matching is performed using the Analyze® software (from AnalyzeDirect Inc.).

Planning is based on those registered data. Intra-operatively a localized ultrasonic probe enables pseudo-3D echographic acquisition of the kidney under apnoea; the patient is anesthetized. After a first version using surface-based registration and requiring segmentation of the data -CT was segmented with the 3D watershed of NablaVision software (from GenericVision, ApS) -an intensity-based rigid registration was implemented and tested. The similarity measurement is the correlation ratio.

After registration, a graphical user interface guides the surgeon towards the planned trajectory and target position; the patient is again under apnoea. The system requires to the patient to be in the same apnoea configuration (full expiration for instance) during US acquisition and tool guidance. Experiments on volunteers have demonstrated a good repeatability of the kidney positioning under repeated apnoeas; this confirms literature results. Validation of the developed system on real data and on cadavers is promising. Real-time tracking of the kidney, for instance using magnetic localization, by avoiding the need for apnoea would greatly simplify the protocol; nevertheless the visual guidance may not be easy with a mobile target.

Robotic alternatives may thus have to be considered for instance by synchronizing the robot to a given stage of the respiratory cycle. The motion of the kidney due to the needle insertion

has not yet been studied. Here again, robotic access may minimize induced motion.

The same framework was applied to the guidance of stimulation electrode placement into the sacrum (S3 root) for incontinence treatment. CT/US surface-based registration of the pelvis originally developed for pelvic surgery (see [START_REF] Tonetti | Clinical results of percutaneous perlvic surgery. Computer assisted surgery using ultrasound compared to standard fluoroscopy[END_REF][START_REF] Daanen | A fully automated method for the delineation of osseous interface in the ultrasound images[END_REF]) was successfully applied to this application and validated though cadaver experiments [START_REF] Leroy | Méthodes de recalage scanner/échographie. Application à la navigation rénale des ponctions rénales percutanées[END_REF]. In this application the bone position is tracked thanks to an implanted rigid body.

Prostate biopsies

The precise realization of prostate biopsy schemes (for instance twelve biopsies regularly distributed on the prostate gland) faces the difficulty of using 2D images to guide a 3D action.

The process thus strongly depends on the surgeon's ability to mentally integrate successive images and trajectories in a 3D space. Because US-guided biopsies only detect 75% of the cancers for the first series, assisting biopsies to guarantee that samples are correctly and regularly acquired in the targeted sites is an important objective. An exploratory study using a navigation system (see Fig. 8) was developed at TIMC. The US probe is localized using an optical system (Passive Polaris from Northern Digital Inc.) and the executed trajectories are recorded in a fixed reference system. Those recordings clearly demonstrate that the prostate moves very significantly with the US probe displacements; those displacements are necessary to orient the needle, which is rigidly attached to the probe, towards the targeted sites. Current work deals with US-based real-time registration enabling to represent the executed biopsies in a single prostate reference frame despite prostate intra-operative displacements.

Model-based urology

As seen briefly in this paper, shape and/or appearance models have been introduced as a priori information for image processing. Statistical models concerning the occurrence of cancers in the different zones of the prostate have also been constructed in order to optimize the biopsy strategies by maximizing the ability to detect an existing tumour. [Schostak05] reports a clinical evaluation of such an optimized scheme.

In the context of medical interventions dealing with soft tissues much attention is paid to biomechanical models; the purpose is in particular to better predict tissue motions and deformations and tool interaction with the tissues. In a first stage these models were mostly developed for training simulators especially in laparoscopic surgery. More recently, they are seen as a necessary input to the planning of a surgery performed on soft tissues. Some groups work on discrete models of the prostate complex environment; for instance Current limitations of planning approaches are due to the hypothesis of a homogeneous tissue and, for patient-specific planners, in the difficulty of determining in-vivo tissue physical parameters. Sonoelastometry, a very promising modality for the detection of cancers inside organs [START_REF] König | Initial experiences with real-time elastography guided biopsies of the prostate[END_REF], can also help for physical parameters identification.

Conclusion

As seen in this paper, many research projects are dedicated to urological applications.

Concerning target motion and deformation, some partial solutions have been proposed. Teleoperated robots enable adapting the surgical strategy to the anatomical state based on the surgeon know-how. Image-guided robots generally allow data acquisition just before the action to be executed; however none of them, in urology, is completely controlled by realtime data. Image fusion can consider changes occurring between the different acquisitions stages to be registered; however, no system yet enables real-time fusion of intra-operative data to pre-operative data for tracking purpose. Regarding the kidney, navigational assistance or robot actions are performed on a stabilized organ (breathing is temporarily held); no tracking is available yet. Biomechanical models are developed but none of them is yet used for guiding the intervention.

A lot has been done but significant research efforts must still be undertaken to fully consider a mobile and deformable target such as the kidney or the prostate. Intra-operative tracking based on intra-body markers (for instance magnetic markers or other [START_REF] Coste-Maniere | Robotic whole body stereotactic radiosurgery[END_REF][START_REF] Bricault | Computerassisted interventions on liver: feasibility of the "anchor needle" technique for real-time targeting of lesions with respiratory motion[END_REF]) or on real-time image processing is necessary. The development of autonomous robots servo-ed to real-time intra-operative data (see for instance [START_REF] Vitrani | Automatic guidance of a surgical instrument with ultrasound based visual servoing[END_REF]) raises very challenging robustness and safety issues since in this case the surgeon will no longer be able to directly supervise the robot actions. Finally, patient-specific modelling of organs mechanical properties is a key issue for predicting and recognizing anatomical changes and to allow precise planning. All those topics open very interesting and difficult scientific tracks for the coming years with a very strong clinical interest.

Fig. 1 :

 1 Fig. 1: Anatomy in urology : (left) Kidney and (right) prostate male anatomical environments (Respectively from http://www.bartleby.com/images and http://www.liv.ac.uk/researchintelligence/issue21/images/ )

Fig. 3 .

 3 Fig. 3. The DaVinci robot (Intuitive Surgical Device Inc., http://www.intuitivesurgical.com):the master-slave system (left and middle); and one of its articulated instruments showing intra-body dof (right)

Fig

  Fig. 4.

Fig. 5 .

 5 Fig. 5. IRM/TRUS fusion for brachytherapy (ProCur software, TIMC and Grenoble Hospital): (left) two of the axial and pseudo-sagittal TRUS images; (middle) 3D TRUS reconstruction; (right) composite image generated after MRI/TRUS elastic registration: for each axial TRUS image the corresponding MRI data are computed .

  Fig. 6. Histology/MRI fusion (ProPath software, TIMC and Pitié Salpétrière Hospital, Paris): Needles and their guide for landmark definition inside the prostate (a); macroscopic 3mm slices of the resected prostate (b); slice reconstruction (arrows highlight some landmarks) (c); volume reconstruction including capsule, peripheral zone, urethra and cancer segmentations (d); composite image after MRI/histology elastic registration (e).

Fig. 7 .

 7 Fig. 7. Navigated percutaneous access of the kidney (TIMC, Pitié-Salpétrière Hospital, Grenoble School of Medicine, PRAXIM): (from left to right) scanner volume et segmented kidney; pre-operative planning interface; US probe and puncture needle with their optical localizers (passive Polaris from NDI, address) ; CT/US fusion; puncture guidance.

Fig. 8 .

 8 Fig. 8. Prostate biopsy navigation (ProNavV1, TIMC, Grenoble Hospital) -Biopsy recording: (left) localized intra-rectal US probes (2D and 3D) -(right) visualization interface presenting US images, prostate model and recorded trajectories in a fixed reference system.

[ Kuroda03 ]

 Kuroda03 presents a simulator for rectal palpation of the prostate. The interactive nature of such models enabling intra-operative re-planning may be counterbalanced by difficult parameter identification. Because of their theoretical background and the ability to introduce rheological tissue parameters, finite-element models are the most commonly used. Based on DiMaio and Salcudean previous work (cf. [DiMaio03]), [Goksel04] proposes a brachytherapy simulator combining TRUS image generation to a needle tissue interaction model. [Mohammed02] combines statistical and biomechanical models for evaluation of intraoperative prostate deformation. Other groups work on steerable needle path planner taking benefit of needle flexion to generate paths avoiding anatomical obstacles [Alterovitz05].

Technically, nothing really constrains the surgeon on the master console to be close to the slave robot; in practice -except for some concept demonstrations such as[START_REF] Marescaux | Transatlantic robot-assisted telesurgery[END_REF] -the needs for reliability in data transmission and safety result in short-distance tele-surgery.

Acknowledgements

Research projects of TIMC described in this paper are/were supported by grants from the French Ministry of Health (PHRC 2003 Prostate-Echo, CIT PICAMI), the French Ministry of Research and ANVAR (MMM project in the RNTL Program), non-profit organizations (Association pour la Recherche contre le Cancer, Association Nationale de la Recherche Technique) and by Praxim-Medivision.


Herve Guillard

Jalal Lakhlili
email: jalal.lakhlili@ipp.mpg.de

Alexis Loyer

Ahmed Ratnani

Eric Sonnendrücker

Mesh generation for fusion applications

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mesh generation for fusion applications

Hervé Guillard 1 , Jalal Lakhlili 2 , Alexis Loyer 1 , Ahmed Ratnani 2 , Eric Sonnendrücker 2 .

1 Inria Sophia-Antipolis, Nice, France. 2 Max-Planck-Institut für Plasmaphysik, Garching, Germany.

Mesh generation for fusion applications

Hervé Guillard 1 , Jalal Lakhlili 2 , Alexis Loyer 1 , Ahmed Ratnani 2 , Eric Sonnendrücker 2 .

1 Inria Sophia-Antipolis, Nice, France. 2 Max-Planck-Institut für Plasmaphysik, Garching, Germany.

Context and motivations

-High anisotropy in magnetized fusion plasmas: ⇒ requires the use of flux aligned meshes.

-Complex and realistic geometries: ⇒ need other strategies (equidistribution, orthogonality).

-High order derivatives (in MHD for example): ⇒ require regular representation.

Sketch of a Tokamak reactor

Unified code for the generation of flux aligned mesh in the poloidal plane.

-Different codes and type of meshes.

-Different numerical methods:

-Semi-Lagrangian approaches.

-Finite difference, Finite volume -Finite Element (Spline or Hermite-Bezier on quadrangles; Powell-Sabin, Clough-Tocher on triangles, etc..) -Automatic decomposition of isolines including arbitrary number of X-Points.

Workflow diagram

-Topological set of the isolines consists of finite connected components and contains only closed orbits.

Examples of domain segementation for JET and WEST tokamaks:

Structured grid construction I. Isolines generation 1. Identify a mapping: (s, t)

∈ [0, 1] × [0, 1] → x(s, t) ∈ Ω 2.
Get the radial curve S (s), solution of the ODE:

     dx ds = ψ M -ψ 0 ∇ψ 2 ∇ψ x (0) = x 0
3. Compute the spline interpolation of the radial curve and generate a set of internal isolines.

II. Meshing of individual patch Ω k

1. Choose a finite number of isolines f -1 (c i ).

2. Approximate each level set by a spline:

f -1 (c i ) ∼ C i (t) = j C i j N j (t). N j : Bspline Basis. C i j : Control Points. 3. Construct a 2D tensor product mapping [0, 1] × [0, 1] → Ω k : S(s, t) = i j P i,j N i (s)N j (t), s.t. S(s i , t) = C i (t), ∀i.

III. Gluing patches

Given 2 subdomains Ω 1 and Ω 2 described by:

S 1 (s, t) = i j P i,j N i (s)N j (t) S 2 (s, t) = i j Q i,j N i (s)N j (t)
with a common boundary for instance: S 1 (0, t) = S 2 (0, t)

1. C 0 continuity: P 0,j = Q 0,j , ∀j.

2. G 1 continuity: geometric condition on the control points, P 1,j , P 0,j = Q 0,j , Q 1,j have to be aligned.

  Input data: Flux field ψ from equilibrium code + the original mesh (+ boundary definition of the vacuum vessel) Data regularization Segmentation: Decomposition of the domain into several patches → 1 patch is homeomorphic to a logical square Splines approximation: Construction of mesh grids of individual patches Wall interaction treatment Gluing of diffrent patches to obtain the global mesh ⇒ export the description into HDF5 output Pre-processing of the data Isolines regularization If the equilibrium solver results are not smooth enough : low-order discretization, coarse resolution, .. ⇒ Replace the flux function by its Clough Tocher interpolant on the refined mesh (locally). Segmentation -Based on Morse theory and Reeb graph.


Joris Mithalal
email: joris.mithalal@univ-lyon1.fr

Nicolas Balacheff
email: nicolas.balacheff@imag.fr

THE INSTRUMENTAL DECONSTRUCTION AS A LINK BETWEEN DRAWING AND GEOMETRICAL FIGURE

Keywords: deductive geometry, 3D dynamic geometry, instrumental deconstruction, drawings

. We show that the Instrumental Deconstruction process is multifaceted and central in this evolution. We present a theoretical framework, in relation with an experiment based on a 3D Dynamic Geometry Environment. Based on a case study, we show that construction tasks with specific representations make the instrumental work play a key role in the learning of geometry.

INTRODUCTION

Drawings play a central role in the learning of geometry but they more often than not turn into an obstacle to the learning of mathematical proof: seeing results in a diagram leads student to question the need to prove "mathematically". For instance, Jahnke (2007, p. 79) noted that "in an empirical study covering ca. 3000 British high achieving students of grades 8 and 9, more than 50% chose a 'pragmatic argument' as warranty for the angle sum theorem for triangles".

We explored the reasons for this tension between "seeing on a drawing" and proving [START_REF] Mithalal | Déconstruction instrumentale et déconstruction dimensionnelle dans le contexte de la géométrie dynamique tridimensionnelle[END_REF]. One outcome of this research supports the claim that 3D Dynamic Geometry Environments (DGEs) could provide a way to reduce this tension: deductive geometry is needed to perform certain kind of constructions, hence mathematical proof and construction tasks turn to be connected.

In this article, we report how these tasks may help the students use of deductive geometry, by analyzing a case study involving 10 th grade students. We develop Duval's work on visualization and deconstructions to make it fully operational in this context, especially by refining the Instrumental Deconstruction notion. We use the cK¢ modeling framework [START_REF] Balacheff | cK¢, a model to reason on learners' conceptions[END_REF] to analyze how construction tasks are fundamental in this process. Our intention is to show that it is possible to create strong and productive links between manipulation and deduction by using construction tasks in 3D DGEs.

We first focus on the role of drawings in geometry and the skills students have to master when doing deductive geometry. In this way, 3D geometry tasks with DGEs are promising. The second section is dedicated to show that, contrary to some previous assumption [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF], the learning of these skills can be done continuously, and that Instrumental Deconstruction plays a pivotal role. Then, to understand this role, we refine this notion and show that two types of Instrumental Deconstruction are involved in the learning of geometry.

WORKING WITH DRAWINGS IN GEOMETRY

GEOMETRICAL OBJECTS, FIGURES AND DRAWINGS

The drawing problem is multifaceted:

"Diagrams in two-dimensional geometry play an ambiguous role: on the one hand, they refer to theoretical geometrical properties, while on the other, they offer spatio-graphical properties that can give rise to a student's perceptual activity. Students often conclude that it is possible to construct a geometrical diagram using only visual cues, or to deduce a property empirically by checking the diagram. When students are asked by a teacher to construct a diagram, the teacher expects them to work at the level of geometry using theoretical knowledge, whereas students often stay at a graphical level and try only to satisfy the visual constraints." (Laborde, 2004, p. 2).

However, a strict opposition between manipulation of drawings and abstract reasoning would be too simplistic. As [START_REF] Parzysz | Knowing" vs "Seeing", Problems of the plane representation of space geometry figures[END_REF] noted, students' activity in geometry involves "seeing" as well as "knowing"; it is a mix of observation and deduction.

Research on the learning of geometry raises the question of understanding the nature of geometrical objects and of the role they play in discussing and establishing the validity of a statement. At the core of this question is the "double nature" -figural and conceptual -of geometrical objects which led Fischbein (1993, pp. 148-149) to coin the notion of figural concept emphasizing the close link in geometry between ideal objects and their concrete representations. But, as Fischbein noted, the use of the word "figure" is ambiguous. He himself chose to denote by the expression "geometrical figure" the mental image "the properties of which are completely controlled by a definition" (ibid.). Because we will not explore mental images, we will adopt in this article the distinction made by Laborde and Capponi (1994, pp. 168-169), between drawings and figures which extends Fischbein's choice and will facilitate our analysis. Following Laborde and Capponi, a figure is the association of a theoretical referent defined by the conjunction of geometrical properties, the geometrical object, and a set of drawings, its representations. Hence a drawing can be considered for itself, as a graphical object, or as one of the possible representation of a geometrical object. Then, solving geometry problems is neither purely experimental nor purely deductive, but it involves going back and forth between both (Parzysz, 2006, p. 128).

Studying the role of drawings as constraints in the learning of geometry, [START_REF] Chaachoua | Fonctions du dessin dans l'enseignement de la géométrie dans l'espace[END_REF] distinguishes two functions:

1. An illustration function: Drawings can illustrate the problem's statements, eliciting hypothesis, or can point to elements which could be relevant for the resolution. (ibid.p. 32-34) 2. An experimentation function: drawings provide a field where a large part of the resolution is caried out, experimentally. Working on drawings involves identifying new sub-figures or relations, making or testing conjectures, etc. This use of drawings is specific to the heuristic phase of the problem solving process. (ibid p.35) Chaachoua (1997, pp.42-45) showed that to students' eyes, drawings could either represent nothing but themselves, represent another material object, or stand for a geometrical object (considered conceptually). In the latter case, they are only means supporting geometrical activities. [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] makes it more precise, explaining that there are two modalities of visualization which guide the exploration and the interpretation3 of a drawing: the iconic and the non-iconic.

Iconic Visualization is the most spontaneous approach to a drawing; it means that one recognizes an object because its shape is similar to an already known object. This is why some students identify a square as a rhombus, but not as a square (Fig. 1). They cannot, as well, modify or analyze a drawing: adding lines changes the appearance, and hence the nature of the related object. [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] shows that as this is the usual way of visualizing, pupils tend to keep working with Iconic Visualization. Chaachoua's work demonstrates that being able to use nothing but this kind of visualization is an obstacle to mobilizing deductive geometry, as it involves experimental work and analytic study of the drawings.

Figure 1: A square, a rhombus and "something from a square" (Iconic Visualization)

To Non-Iconic Visualization, a drawing is nothing but one of the representations of a geometrical object. Shape and appearance are no longer fundamental characteristics, as the drawing is seen as the assembly of lower dimension components, the figural units (points, lines, segments, circles). Therefore, there is no contradiction between modifying a drawing and considering that it remains a representation of the same object. But passing from Iconic Visualization to Non-Iconic Visualization is neither easy nor natural. Duval (2005, pp. 45-48) stresses that going from high dimension objects to lowers components is the most common spontaneous way of visualization, whereas identifying first inner components in order to recompose bigger ones is specific to mathematics and needs training. This dimensional hiatus is an important obstacle to the learning of deductive geometry. Duval (2005, pp. 16-17) explains that using drawings in order to solve geometry problems involves three distinct operations: Mereological Deconstruction, Instrumental Deconstruction and Dimensional Deconstruction. We will not use the first type which corresponds to "heuristic breaking down of the shapes" (Laborde, Mithalal, & Restrepo, 2009, p. 5), but focus on the two others which relate drawing and discourse:

THE ROLE OF VISUALIZATIONS AND DECONSTRUCTIONS

-Instrumental Deconstruction answers the question "How is it possible to produce this drawing with a given set of tools?". A drawing is considered as the result of a construction process associated with a geometrical object. For instance, an Instrumental Deconstruction of a rhombus could be: "construct a segment [AC], the perpendicular bisector of this segment, B and D belonging to this perpendicular bisector so that AB=AD". Properties of the object may appear more "mechanical" than "geometrical". -Dimensional Deconstruction is a discursive interpretation: a figure is considered as a set of figural units linked by a set of geometrical properties, and the drawing is analyzed accordingly. A rhombus is now "four distinct points A, B, C, D, with AB=BC=CD=DA". This does not refer to a process, but it is based on the properties of the geometrical object; discourse is the main representation of the geometrical object.

We can notice a connection with the "drawing"4 Vs "constructing" distinction [START_REF] Healy | Messing up[END_REF].

Iconic Visualization-based strategies focus on the shape and its perceived features; it guides the production of a drawing which "looks" right. Dimensional Deconstruction requires a theoretical approach and the use of geometrical properties; the interpretation it provides of the graphical objects may lead to a "constructing" characterization -no matter the size or the orientation, the important point is that the figure cannot be "messed up". Nevertheless, the deconstructions are not about effective processes students perform, but about the mechanism involved in their treatments of the drawings. Therefore, Instrumental Deconstruction can be either dedicated to reproducing a given shape -and in this case it is close to the drawing process -, or to using mechanical properties linked to instruments, in which case it is close to the constructing process.

Visualization and deconstructions differ in nature but have strong links. Visualization deals with the perception of drawings. Deconstructions are processes involved in their analysis. Dimensional Deconstruction connects drawings to geometrical objects, hence bridging the gap between drawing and proving, while Instrumental Deconstruction lays the ground for the problem-solving process. Non-Iconic Visualization is a necessary condition for Dimensional Deconstruction to be operational, which itself is an unavoidable tool to engage in deductive geometry and the construction of intellectual proofs 5 . [START_REF] Duval | Les différents fonctionnements d'une figure dans une démarche géométrique[END_REF][START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] claims that the transition from the geometry of drawings to the geometry of figures, which is necessary to the learning of mathematical proof in geometry, requires the double transition from Iconic Visualization to Non-iconic Visualization, and from Instrumental Deconstruction to Dimensional deconstruction. [START_REF] Duval | Les différents fonctionnements d'une figure dans une démarche géométrique[END_REF][START_REF] Duval | Representation, vision and visualization: cognitive functions in mathematical thinking. Basic issues for learning[END_REF][START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] considers that the two deconstructions are distinct processes: each has to be taught separately with specialized activities. On the contrary, the case study we report in this paper will demonstrate that it is possible to bridge the gap between both, and to make Instrumental Deconstruction and Dimensional Deconstruction the two sides of a single problem-solving process. We will show that students can be led to using Non-Iconic Visualization because of the failure of Iconic Visualization, and that analyzing the role of the artifacts helps to understand why Instrumental Deconstruction and Dimensional Deconstruction can be strongly linked. In the remainder of this paper, section 2 will be dedicated to show the continuum between both deconstructions, and section 3 to refine the notion of Instrumental Deconstruction.

USING 3D-GEOMETRY: WHY, WHAT FOR, HOW?

Two dimensional (2D) representations of 3D geometrical objects raise several visualization issues. First, it is no longer possible to confuse the representation with the object itself. Secondly, one cannot rely confidently on the visual information, as perception can be misleading. Then, the analysis of the relation between the representation and the associated object necessarily involves reconstructing the latter using properties of low dimensions units (e.g. lines, points) of the former. These issues make Iconic Visualization insufficient for 3D geometrical problem solving, opening the possibility for Non-Iconic Visualization to emerge as a way of restoring the capability of perceiving reliable information. This emergence is very complicated in a paper-and-pencil environment. However, the access to 3D geometry can be facilitated by the use of DGEs which allow the representation and the direct manipulation of 3D geometrical objects (Chaachoua, 1997, p. 265). We have chosen Cabri 3D6 which has such characteristics, allowing the construction and the manipulation of drawings simulating 3D geometrical objects.

DRAWINGS AND TOOLS IN CABRI 3D

Cabri 3D drawings are images on a computer screen simulating 3D geometrical objects, which means that it is possible to manipulate what appears first as a 2D representation but which behaves when manipulated with properties evoking the third dimension. The user can continuously change the point of view of the objects by manipulating one of its component. The third dimension is simulated by a "fog effect" that makes objects progressively fade away while they are increasingly distant. As a screen is flat, moving a point in space has been split in two operations: in a plane parallel to (xOy) with constant z coordinate, or following a parallel to the (Oz) axis.

The combination of various views of the same object allows the user to detect misleading perceptions of the 2D representation and lowers the risk of attributing some wrong properties to the geometrical object (fig. 2). Following the design principles of DGEs, a drawing keeps the properties underlining the geometric properties used to produce it when any of its free point is moved on the surface of the screen.

CABRI 3D AS A MILIEU TO STIMULATE GEOMETRICAL REASONING

DGEs, as geometrical computer-based environments whose design principle is the direct manipulation of the representations of geometrical objects, share common characteristics but also differ in many ways either in by the choice of the functionalities or the look and feel of the interface. Hence we will use the concept of milieu7 of the Theory of didactical situations (Brousseau, 1997, pp. 226-249) in order to indicate our focus on one key feature that these environments have in common: the nature of the feedback they provide following the direct manipulation of base elements of a representation [START_REF] Laborde | The Computer as Part of the Learning Environment: The Case of Geometry[END_REF]. Every DGE allows dragging points and modifying a drawing with respect to the geometrical properties underpinning its construction; this construction is then pragmatically valid when it cannot be messed up [START_REF] Healy | Messing up[END_REF][START_REF] Laborde | The Computer as Part of the Learning Environment: The Case of Geometry[END_REF]. This characteristic makes DGEs relevant environments to set a milieu for teaching figures [START_REF] Laborde | Dynamic geometry as a source of rich learning contexts for the complex activity of proving[END_REF][START_REF] Laborde | Cabri-géomètre constituant d'un milieu pour l'apprentissage de la notion de figure géométrique[END_REF] and introducting the dialectic of conjecturing and proving [START_REF] Baccaglini-Frank | Generating Conjectures in Dynamic Geometry: The Maintaining Dragging Model[END_REF].

This study reported here has been carried out as part of the scientific programme of the Cabri-geometry research group; it used the Cabri3D DGE.

Cabri 3D makes it possible to design elementary construction tasks, not requiring at first high geometrical knowledge. However, any user soon realizes that mere perception is not sufficient in order to get reliable information. For instance, it is hardly possible to construct a cube-looking polyhedron only by adjusting the position of the points, whereas it is very easy to construct a square-like polygon using this strategy in a 2D DGE. Since visual information is not sufficient to ensure the correctness of constructions or to get information, geometrical knowledge and geometrical reasoning are necessary. The complexity and the fundamental character of the dragging process [START_REF] Hatterman | The dragging process in three dimensional dynamic geometry environments (DGE)[END_REF] should favour the move towards both Non-inconic visualization and Dimensional deconstruction, hence stimulating the transition to geometrical reasoning and intellectual proof. This is what we will illustrate and analyze in the remaining of this article.

BRIDGING THE GAP BETWEEN THE TWO DECONSTRUCTIONS: A CASE STUDY

The following case study aims at describing the continuum between Iconic and Non-Iconic Visualization. We examine the role of Instrumental Deconstruction in this change of the visualization process, and how it eventually provokes the use of Dimensional Deconstruction. We use the data gathered from an experiment detailed in Mithalal (2010)8 .

CONTEXT OF THE STUDY

The experiment was carried out with four classes of French 10 th grade students, working by pairs on Cabri 3D activities. These students had a limited experience with 3D geometry and elementary shapes. A first one hour session was dedicated to learning how to use Cabri 3D, and the second which we analyze here, was a construction activity. Our analysis is based on the records of each pair of students' voices and on their activity by video recording the screens, transcribed afterwards.

The task was the following:

Let ABCDEHGF 1 F 2 F 3 be a truncated cube (Fig. 3).Find as many ways as possible to construct the missing vertex 9

, and describe each method in a few words.

The point A can be manipulated. Even after the little training they had received, the constraint on whatever solution they came up with was clear to students: the cube they obtain by constructing the missing vertex must remain a cube whenever dragging A. With such a task, Iconic Visualization would lead to accept the construction of the missing part of the shape without considering geometrical properties, but it will be immediately messed up by dragging the point A. On the contrary Non-Iconic Visualization could lead to use the cube properties (such as symmetry) to construct F, hence meeting the DGE basic criterion which is to get a robust result when manipulating free points. On the basis of an a priori analysis, four distinct interpretations corresponding to specific resolution strategies were identified (Mithalal, 2010, p. 149):

1. To draw the missing part so that the result looks like a cube; 2. To draw the missing part so that the result looks like a cube and keeps the same measures as the original; 3. To draw a point using a construction procedure that fits with a cube; 4. To construct a point linked to the truncated cube with its geometrical properties.

CONNECTING ICONIC TO NON-ICONIC VISUALIZATION AND INSTRUMENTAL TO DIMENSIONAL DECONSTRUCTION

This subsection aims at showing the differences between visually controlled drawing (corresponding to Iconic Visualization), Instrumental and Dimensional Deconstructions, and to analyze their potential relations. For this purpose, we will use excerpts of full transcripts available in [START_REF] Mithalal | Déconstruction instrumentale et déconstruction dimensionnelle dans le contexte de la géométrie dynamique tridimensionnelle[END_REF]. We will give only a partial account of the students' exchanges and indicate their timing in order to alert us that these are not continuous dialogs, but illustrating sentences. This data has been analyzed in [START_REF] Mithalal | Déconstruction instrumentale et déconstruction dimensionnelle dans le contexte de la géométrie dynamique tridimensionnelle[END_REF], identifying precisely the pupil's strategies and the related conceptions. We will describe the evolution of these conceptions during the solving process by using the excerpts to illustrate the identified evolution.

FROM ICONIC TO NON-ICONIC VISUALIZATION

Even when a solution is based on Iconic Visualization, as it is illustrated by the case of Ludovic and Pierre quoted below, students may use Cabri3D geometrical tools in an attempt to lower the randomness of the action they performed (Mithalal, 2010, p. 163).

Pierre and Ludovic tried to construct the shape relying on a visual monitoring of their action on the screen. They spent about 25 minutes trying to locate the right place of a point. They concluded after unsuccessful trials that is was impossible, and decided that they needed another method: 16'06, Ludovic: Anyway, we can't see anything. Their aim remained that "it looks good", but they realized that the means for it had to be improved. They then tried various methods, including intersecting the three truncated edges (two would have been enough), constructing parallels, or using vectors (they failed).

The dominant approach, determined by Iconic Visualization, was to try first to complete the shape. But the failure of this approach pushed students to use the construction tools of Cabri 3D. To do so they needed to break the cube down in order to identify figural units, hence switching to Instrumental Deconstruction. Iconic Visualization and Instrumental Deconstruction worked together for two distinct but complementary purposes, the former providing perceptual assessment of the outcomes of the latter. As a consequence, the students moved step-by-step to Non-Iconic Visualization.

CONNECTING INSTRUMENTAL TO DIMENSIONAL DECONSTRUCTION

The central place of construction tools and direct manipulation could orient the students activity towards Instrumental Deconstruction, thanks to what could be described as a mechanical conception of the DGE. However, as the tools made available by such a software can be read as reification of geometry primitives, Dimensional Deconstruction could prove useful for anticipating the strategies and their validity, and grounding argumentation. This is illustrated by the case of Marie, Paul and Julie. These students first decided that "reconstructing the cube" was not sufficient and that they had to draw the missing part.

13'19, Paul: Maybe we should try with a point, just a point here, at the right place.

13'24, Marie: But in this case you leave it to chance! 13'27, Paul: No! I mean, yes! Well, you make it… Their problem was to be sure that the construction was correct, and they eventually decided that it was "approximately good", which was enough for them. Marie's objection was about the ways of controlling the position of F, so they decided to use constructing functionalities of the software, such as edges intersections or symmetry. During half an hour, the correctness remained determined by the visual aspect of the shape (Iconic Visualization) and the robustness when moving free points (Instrumental Deconstruction).Then they succeeded in constructing F by intersecting 3 edges (Fig. 4, left picture).

Later, they tried another solution by constructing a figure symmetric to the truncated cube, and using it to construct lines (Fig. 4,2 nd &3 rd pictures). Deciding whether this last method was really or not, turned to be a Dimensional Deconstruction problem:

25'46, Julie: Look, you construct the same10 next to it. You'll see, the shape will do schlack, schlack, you see it will be a rhombus. Then, you use from this one to that one. To its symmetric. Then, you only have to do this,… Then you create the line, and you make it. The result was visually different from the former one, and so was the construction process although they used the same lines. The students decided that this solution was identical to the previous one (Fig. 4, left picture), not because of the shape or the construction process, but because they used the same lines. Thus, their interpretation of the strategies turned to be based on Dimensional Deconstruction. This shows how Instrumental Deconstruction and Dimensional Deconstruction can be strongly linked. Dimensional Deconstruction facilitates the design of the construction process and evaluating its validity. Instrumental Deconstruction problems make Dimensional Deconstruction (hence geometrical knowledge) play a greater role.

THE PIVOTAL ROLE OF INSTRUMENTAL DECONSTRUCTION

These examples illustrate the two main reasons for the evolution of the students' activity:

-First, students look for more reliable ways to act and for more efficient visual monitoring.

-Second, they need to better anticipate and manage their actions.

In the first case Instrumental Deconstruction renders reliable and efficient an approach initially based on Iconic Visualization. Then, Instrumental Deconstruction turns to be the operational side of the Dimensional Deconstruction, which itself provides the needed geometrical grounds required to justify the construction process. It .strongly modifies the way problems are addressed, becoming more theoretical, which makes deductive geometry meaningful.

Nevertheless, such evolutions are difficult to describe and to analyze because the data related to these two roles of the Instrumental Deconstructions are similar. The same processes might be used with different purposes, and a similar aim may generate different strategies. To get a more accurate view, we used the cK¢ model which accounts for the tight relationship between operators, instantiated by students' actions which are observed, and the associated controls which motivate or justify their use.

INSTRUMENTAL DECONSTRUCTIONS

In order to understand how small variations of the students' means may indicate greater evolution of their purpose, we will highlight here two kinds of Instrumental Deconstructions. For this we will use the cK¢ modeling framework to analyse data in a precise way, relating action and control along a problem solving process in relation to the students conceptions. The following section introduces shortly the cK¢ framework.

CONDITIONAL ACTION VERSUS CONTROL

The cK¢ modeling framework [START_REF] Balacheff | cK¢, a model to reason on learners' conceptions[END_REF] is based on the Theory of Conceptual Fields [START_REF] Vergnaud | The Theory of Conceptual Fields[END_REF] and on the Theory of Didactical Situations [START_REF] Brousseau | Theory of Didactical Situations[END_REF]. This model accepts the postulate shared by these theories stating that the first aspect characterizing a piece of mathematical knowledge is the set of problems it allows to solve. This set directs the students conceptions and grounds their legitimacy. This situated nature of ways of knowing in mathematics is recognized since the seminal work of [START_REF] Vinner | Concept definition, concept image and the notion of function[END_REF], and [START_REF] Confrey | A Review of the Research on Student Conceptions in Mathematics, Science, and Programming[END_REF]. Their critical analysis of the research on misconceptions led to the adoption of the generic term "conception" (ibid. p.29). In this context, the cK¢ model considers a conception not as a student's characteristic, but as a property of the interacting system formed by the student and a milieu with which he or she is acting and interacting (Balacheff, 2013, p. 6). It characterizes it by the conjunction of four elements: its domain of validity (a set of problems), the representation system it mobilized (possibly blending several semiotic systems), the set of operators which are available to contribute to solving problems, and the control structure which provide means to make choices, assess feedback, take decisions, and judge the advancement of a problem solving process.

We focus for this case-study on the interactions between operators and controls, considering that the dimension of the representation system, which consists of a combination of natural language and Cabri 3D representations and associated tools, is common to all observed pair of students. The identification of operators and controls is based on a close analysis of the students discourse, identifying the utterances and their evolution, and the related actions at the interface of the DGE 13 . An issue when performing this analysis is that some controls and operators may look 13 For a complete presentation of the cK¢ model, the reader is invited to refer to Mithalal (2010, Appendix pp. 11 -26).

the same (Pedemonte & Balacheff, 2016, p. 121). For instance, "the symmetrical image of a segment S 1 is a segment S 2 whose extremities are symmetrical to the extremities of the segment S 1 " could work either as an operator (To construct

[A'B']=Sym d ([AB]), link A'=Sym d (A)to B'=Sym d (B))
or as a control (The symmetrical image of a segment is another segment whose extremities are symmetrical to the extremities of the initial one). This correspond to the dual nature, operational versus predicative, of knowledge pointed by [START_REF] Vergnaud | The Theory of Conceptual Fields[END_REF]. In order to decide which of the two meanings to consider for a single utterance, we took the following criterion: if, based on the observed behavior and recorded discourse, an utterance could be represented by a formula of the form: [Statement 1 ⇒ Action] we classified it as an operator, but we classified it as a control if it could be represented by a formula of the form: [Statement 1 ⇒ Statement2]. Indeed, operators are easier to identify because of their observable materialization as behaviors often associated to discourse, and explicit interaction with the milieu whereas controls are often left implicit. However a well designed experiment involving social interaction facilitates their identification as they are more likely to be made explicit during the verbal communication between students as to their reasons for their choices and decisions.

Controls play different roles at different phases of the problem-solving process:

(1) In an initial phase, controls determine the type of problem, a set of possibly adequate operators (a selection function [START_REF] Vadcard | Étude de la notion d'angle sous le point de vue des conceptions[END_REF]), and a priori reasons for the relevance of a strategy.

(2) During the resolution phase of the problem, referent controls "regulate the anticipation of the action and the adequacy between the expectations and the results" (Gaudin, 2005, p. 153).

(3) In a final phase, controls provide a posteriori assessment of the validity of a solution.

The analysis of phase (2) requires a special attention in order to distinguish operators from controls. During this phase controls provide information about the coherence of an approach and its possible evolution. Phases (1) and

(3) are more about how geometry problems are considered by students, and the level of proving they engage in. The analytical tool provided by cK¢ facilitated eliciting the way Instrumental and Dimensional Deconstructions shape the problem solving process and evolve as levels in the transition from pragmatic proofs (based on drawings) to intellectual proof.

THE IMPORTANCE OF IDENTIFYING CONTROLS BEYOND THE OBSERVED BEHAVIORS

In this section we show that identifying the Instrumental Deconstruction is insufficient to interpret the strategies of the students.

Let us go back to the case of Pierre and Ludovic. They constructed the lines from the truncated edges, then their intersection14 , and filled in the cube so that "it looks like a cube". From a mathematical point of view, the three lines were useless but needed in order to complete the shape. Then Pierre and Ludovic added the three "missing" triangles, confirming their focus on the shape (Fig. 5). Marie intended to draw the missing corner so it looked like a cube. Her interpretation and final control was: σ visual =[The aim is to construct an object ⇒ The result has to look like the referent shape]. This control is related to a drawing process [START_REF] Healy | Messing up[END_REF] which is typical of Iconic Visualization, as was Pierre and Ludovic's. Both groups used similar operators: rline=[construct a line on a segment ⇒ select the 'straight line' tool and left-click on the segment] and r triangle a triangle ⇒ select the 'triangle' tool and left click on the three vertices]. These operators do not require theoretical control, and can be performed purely visually. The key difference between these groups is that, in order to control the construction process, Pierre and Ludovic only used σ visual Marie's work indicates that she could anticipate the validity of her construction. This anticipation was necessarily based on non-visual control backed by known geometrical properties, and then on Dimensional Deconstruction : σ DD =[The aim is to construct an object ⇒ The tools to be used are those corresponding to the objects' geometrical properties].

Even if the procedures and their outcomes are very similar, focusing on controls reveals significant differences: Marie, Paul and Julie considered that the exercise was about the shape, but they could use geometrical knowledge as tools for controlling their actions. On the other hand, Pierre and Ludovic failed when geometrical controls were needed. With the same aim to reproduce a shape, two Instrumental Deconstructions underpin processes differing in the nature of their associated controls, namely, (1) visual control (Iconic Visualization), (2) geometrical controls (Non-iconic Visualization).

EVOLUTION OF THE INSTRUMENTAL DECONSTRUCTION

As illustrated by the former cases, on the basis of the controls involved, it is possible to distinguish two types of Instrumental Deconstruction. The first one [Instrumental Deconstruction / Iconic Visualisation] involves shape-based controls, that is Iconic visualization, the other one [Instrumental Deconstruction / Non-Iconic Visualisation] is based on geometrical controls, that is Non-Iconic Visualization. Of course, these are types and for instance we may consider in the above examples, that Pierre and Ludovic used [Instrumental Deconstruction / Iconic Visualisation] but that Marie was midway between the two types.

The identification of these types of controls provides good indicators of the evolutions of the student's activity, and of the way they may move towards Dimensional Deconstruction. Let us go back to the case of Pierre and Ludovic, decomposing it into successive strategies.

Initial strategy

The first objective of Ludovic was to draw the missing vertex of the cube by creating a point and dragging it at the right position. To do so he could use the operator r=[create a point⇒ select the 'point' tool and left-click on the screen]. The point was then dragged at the right place, using visual control. This strategy was based on Iconic Visualization, and the a priori referent and the posteriori controls were identical (σ visual ). Ludovic used another control σ point of view =[The aim is to construct an object ⇒ The result has to look like the referent shape when changing the viewpoint]. This control is close to σ visual , and it is still based on Iconic Visualization, but it is a good way to get more reliable information on the results. This information made them realize that the shape was not correct ("It looks good, and when it moves, it looks like nothing…") and they had to find another strategy to draw the shape correctly.

Borrowed strategy

The new strategy was borrowed from other students (27'25: "We'll do the same as they do! Look at their solution!)". The successful construction they saw on the screen of the other students was enough for them to justify their new strategy, constructing lines and intersections (r line and rintersection=[construct an intersection point of two lines ⇒ select the 'point' tool and left-click on the lines intersection]). It was carried out step by step, using the Cabri3D tools so that the result looked like a cube; the initial and final controls were still visual.

Alternative strategies

In search for another strategy, they kept the objective of constructing an intersection point of lines based on the truncated edges, but with different Cabri-tools than for the borrowed strategy. To do so, they tried the parallel tool: rparallel=[construct line on a truncated edge, parallel to another edge ⇒ select the 'parallel' tool, left-click on the edge giving the direction, and on the truncated edge]. Even if visual control was still possible, this operator required a control based on geometrical properties of the cube: σ cube edges =[The aim is to construct a line on a truncated edge ⇒ This line is parallel to other edges]. This control needs the identification of figural units and geometrical properties, hence a move towards a Dimensional Deconstruction which required a Non-iconic Visualization.

To succeed in the task based on [Instrumental Deconstruction / Iconic Visualisation], visual control was fundamental for Pierre and Ludovic, and was supported by the use of Cabri 3D's tools. This control didn't allow them to succeed, and the DGE feedback gave evidence to their failure hence made them look for more reliable strategies. These strategies were based on geometrical tools which could be controlled with σ visual at the graphical interface of the DGE. Pierre and Ludovic tried to use the tetrahedron tool and, again, the feedback showed them that they were wrong. They failed again trying to use vectors, but were more successful using "line on segment" or "parallel" which were easier to control, possibly due to better knowledge and visual feedback. They gradually used more and more complex geometrical tools, and in this way the Instrumental Deconstruction evolved. At the beginning, it was based on experimental trials and error and required σ visual .

Step by step, they started to take into account relations between figural units (for instance with σ cube edges ), some of them being visible (intersections), others not (vectors). This was a first step towards Non-Iconic Visualization, and some of the referent controls were based on Dimensional Deconstruction. Nevertheless, as they couldn't use any other control but visualization, they failed when trying to use more elaborated strategies.

This evolution lasted one hour, and we cannot assume it to be a stable evolution. Nevertheless, this identification of operators and controls brings out the progression from [Instrumental Deconstruction / Iconic Visualisation] to [Instrumental Deconstruction / Non-Iconic Visualisation] which appears to be, at the same time, both complex and a key to students move from drawings to figures.

DISCUSSION

Our aim in this article was to clarify the relations between Iconic Visualization and Non-iconic Visualization, between Instrumental Deconstruction and Dimensional Deconstruction. We also tried to understand how constructions tasks in a 3D DGE, by creating an "intellectual need" (Harel, 2008, p. 488), may stimulate students' move from drawings to figures which is an essential step towards mathematical proving. To do so, we had to refine methodological and theoretical tools.

Following [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF] work, we recognize that perception plays a key role in the mathematical work of the students. However, in order to connect the working on the drawings to identifying or representing geometrical properties, the students must be able to perform specific operations, Instrumental Deconstructions and Dimensional Deconstructions, that require a change in perception and the use of Non-Iconic Visualization. Duval (2005, pp. 19-20) considered that these two deconstructions are too different to be connected. On the contrary, our case study shows that it is possible for students to move towards Dimensional Deconstruction, and at the same time towards Non-Iconic Visualization, by a specific use of Instrumental Deconstruction.

Based on a 3D DGE, the task we designed played a crucial role in this process. It rendered visual information less reliable, and then disqualified Iconic Visualization. The direct manipulation features of the DGE facilitates working on the drawings with empirical strategies, but the reproduction task design makes it inefficient. Finally, the DGE tools based on geometrical properties shed light on the figural units and their relations, so that the Instrumental Deconstruction comes to be strongly linked to the geometrical objects. This transition opens the possibility of a move from pragmatic proofs to intellectual proofs, which remains to be confirmed by specific research.

As a preliminary work, we had to characterize more precisely the way students' work changed. We used the cK¢ modeling framework to refine the analysis by distinguishing the evolution of the operators and controls involved.

Our case study showed that it is possible for the pupils to use more efficient operators, without a fundamental change of the control system. On the other hand, such a change has to be interpreted carefully, as very similar construction processes may mean very different things about the student's ability to engage in Dimensional Deconstruction. It therefore makes sense to distinguish two types of Instrumental Deconstruction, the first one more related to Iconic Visualization, the second one to Non-Iconic Visualization. This allows to theoretically consider the possibility of a continuous evolution between the two kinds of visualization.

In conclusion, this case study shows that it is possible to design specific tasks to provoke the need for intellectual proof. The construction tasks we used, based on a 3D DGE, provided opportunities to facilitate the gradual transition of students' focus from drawings to figures, which have direct consequences for the teaching of mathematical proof. As in this case Iconic Visualisation was no longer reliable, there was an intellectual need for ways of controling actions independently from sight, in order not to "leave it to chance" (Marie, p.10). The evolution from [Instrumental Deconstruction / Iconic Visualisation] to [Instrumental Deconstruction / Non-Iconic Visualisation] was a response to the problem raised by the milieu: how is it possible to be confident in a given construction process? A priori theoretical control of the constructions came to be needed.

Once they changed their construction process control, students also changed their interpretation of what "a new strategy" could be, now based on dimensional deconstruction (see p.8). There were no practical reasons to do so, as this could have been based on the construction process, but the situation raised an "intellectual need […] inextricably linked to another notion: epistemological justification" (Harel, 2008, p. 488). In this way, such a task raises the need for intellectual proof. It made its subsequent teaching and learning required and meaningful.

Figure 2 :

 2 Figure 2: Moving a drawing to get more reliable visual information

Figure 3 :

 3 Figure 3: The truncated cube students had to complete

  21'57, Pierre: It looks good, and when it moves, it looks like nothing… 26'22, Pierre: We have to find another solution.

  11 31'50, Marie: Wait, wait, because I just thought about it: the [second] cube is absolutely useless… 32'14, Pierre: Yes, in fact you just did the same! 12 32'15, Marie: Exactly. That's it: you just added a cube on the two sides.

Figure 4 :

 4 Figure 4: Construction with three lines or with the symmetrical of the cube.

Figure 5 :

 5 Figure 5: Trying to draw the missing part of the cube with 3 linesAnother group of students, Marie, Paul and Julie, followed the same construction process: construct the lines from the truncated edges, then the intersection. They engaged in this strategy at the very beginning:

In this case, seeing means both perceiving and interpreting what is perceived. We will keep this meaning for "visualization".

A reviewer drew our attention to the fact that "in the United States, with DGS, "draw" means to make a figure that "looks right"."Construct" means to make a figure that is right due to how it is constructed and hence cannot be "messed up"." In this article, we use "draw" to denote the production of a drawing in whatever way (sketching or constructing), and we use "construct" for the production of a drawing carefully using instruments and/or knowledge with the intention to be right (but sometimes being wrong).

By "intellectual proofs" we mean "[proofs] which use verbalizations of the properties of objects and of their relationships […]it requires a genuine construction of language means as an operative tool. The problem-solver must be able to use language and symbols as means to compute on statements and relations."(Balacheff, 1988, p. 285)

Cabri 3D is a Cabrilog software. More information available here: https://cabri.com/fr/cabri3d/

The concept of "milieu", from the Theory of Didactical Situation, refers to the "system opposing the taught system"(Brousseau, 1997, p. 57). In other words it is this part of the environment wich is meaningful from the perspective of the learning stake by the kind of feedback it provides to the student. "That is to say that the student's answer must not be motivated by obligations related to the didactical contract but by adidactical necessities of her relationships to the milieu" (ibid.).

The experiment was carried out in French classes, so the transcripts in this article are our own translation. For the French original transcripts, see[START_REF] Mithalal | Déconstruction instrumentale et déconstruction dimensionnelle dans le contexte de la géométrie dynamique tridimensionnelle[END_REF].

In French vertex is "sommet", which has not only a mathematical meaning but also a meaning in everyday language, namely, "summit".

The same truncated cube, symmetrical to the original one with respect to (ABC)

Julie described how to construct F as an intersection of two lines.

The same as the previous method, which was the intersection of two edges

Cabri provides the user with an "intersection" tool, that constructs the intersection of two objects. It is not obvious that it does not construct the intersection of the three lines, as it only displays a message saying "this intersection point".