GEOMETRIC PROPERTIES OF THE 3D SPINE CURVE J.M. Sotoca1, M. Buendía2, J.M. Iñesta3 and F.J. Ferri4 1 Dpto. Lenguajes y Sistemas Informáticos. Universidad Jaume I. 2 Dpto. Fisiología. Universidad de Valencia. 3 Dpto. Lenguajes y Sistemas Informáticos. Universidad de Alicante. 4 Dpto. Informática. Universidad de Valencia.

A light source with a known pattern is utilised instead of a camera. STRUCTURED LIGHT: Range retrieval method alternative to stereo imaging. A light source with a known pattern is utilised instead of a camera. A set of landmarks are created on the objects by the light pattern. The 3D positions of those landmarks are computed. Pros: This method allows the surface reconstruction in objects without texture. Makes it easier to solve the stereo correspondence problem. Cons: Only valid in controlled environments. Sensitive to light condition changes and kinds of surfaces.

THE INDEXATION PROBLEM It’s the problem in structured light dual to the correspondence problem in stereovision. It represents the labelling of the landmarks artificially created by the pattern when it is projected over the scene. Once solved, the range data can be retrieved. Different approaches to help the solution: colour codes, binary patterns, constraints. We have introduced a mark in the pattern that sets a reference for landmark indexation.

EXPERIMENTAL SETTING: Simplification by means of a front plane. IMAGE 1: IMAGE 2: BACK PLANE CAMERA PROJECTOR FRONT PLANE CAMERA PROJECTOR VALID CALIBRATED SPACE

EXPERIMENTAL SETTINGS: Arbitrary direction of the optical axis. Relation of deep between distances with origin in the point Oc2p. D/d2 is connected with the angular aperture of beam of light between the front and the back planes. Oc2 Camera Projector Oc1 e12 P2 Back plane P1 d1 d2 Pp z P D Front plane Oc2p R2 Oc1p Rp R This way z is computed as a function only of distances between pixels, the distance between both calibration planes, D and the distance of the camera d2.

SURFACES RECONSTRUCTION: Application over back humans. Elements of the reconstruction: Back grid image. Front grid image. Object image with landmarks. Object grid image. Phases of the reconstruction: Mask region. Skeletonized and the node-seeking algorithms. Indexation of images. Topography map.

MORPHOLOGY OF THE SPINE: Medical problem. Serious deformities in the human spine are present in the 0.3 % of the population. Study of the thoracic and lumbar regions, analysing these pathologies that suppose bigger deformity: Scoliosis, kyphosis and lordosis. The detection is thought a clinic visualization of the cosmetic deformity. Frequent x-ray examinations are necessary. The habitual prognosis is realised measuring the Cobb angle and the projection of the vertebral pedicles. 45

MORPHOLOGY OF THE SPINE: Scoliosis. Characteristics: A lateral bend of the spine. Rotation of the vertebrae bodies. Prominence of the ribs and the disfiguring hump. The deterioration of the spine occurs quickly, so a prevention of the illness is necessary. Nomenclature of Ponsetti and Friedman: Cervical-thoracic. Thoracic. Thoraco-lumbar. Double major. Lumbar.

right thoracic-left lumbar MORPHOLOGY OF THE SPINE: The Ponsetti-Friedman classification. Cervical-thoracic Thoracic Thoraco-lumbar Double major right thoracic-left lumbar Lumbar

STUDY OF THE FRONT AND SAGITTAL PLANES: Thoracic scoliosis STUDY OF THE FRONT AND SAGITTAL PLANES: Thoracic scoliosis. The Cobb angle is 45.0 in the thoracic region. Projection of the spine curve for front X-ray image over back surface. The lateral asymmetry in the front plane is 41.7 in the thoracic region and 19.4 in the lumbar region. The kyphosis angle is 53.9 and the lordosis angle is 47.5.

STUDY OF THE FRONT AND SAGITTAL PLANES: Thoracic scoliosis STUDY OF THE FRONT AND SAGITTAL PLANES: Thoracic scoliosis. The Cobb angle is 45.0 in the thoracic region. Projection of the spine curve obtain with landmarks over the back surface. The lateral asymmetry in the front plane is 47.7 in the thoracic region and 32.1 in the lumbar region. The kyphosis angle is 50.4 and the lordosis angle is 51.9.

STUDY OF THE SPINE CURVE IN 3D: Curvature and torsion. C(u) : [pi, pi+1] 3 is a parameterisation of the spine curve by mean of a polynomial fitting. Px and Pz are the coefficients of the polynomial using a threshold in the corresponding correlation index, and nx y nz are the degrees of the polynomial, The curvature  and the torsion  can be calculated from an arbitrary parametric curve through the following expressions:

STUDY OF THE SPINE CURVE IN 3D: The frenet frame. For each point of the curve, a natural local reference system called Frenet frame can be defined by the following expressions: where t is the tangent vector of the curve, b is the binormal vector and n is the normal vector. If we consider  and  as the angle variations of the vectors t and b, respectively, can arrive to the following relations for the curvature and the torsion: where s is the arc length of the curve. Thus,  and  are the angular velocities of t and b. The curvature gives information about the changes in the orientation of the curve and torsion provides information about its rotation.

EXPERIMENTS AND RESULTS. A sample of 76 patients (42 female and 36 male). A group of 12 patients, aged from 11 and 18 years, had an idiopathic scoliosis process. The Ponsetti-Friedman classification: 4 thoracic, 2 thoraco-lumbar, 1 lumbar and 5 double major curves. The correlation index obtained between the lateral asymmetry in the spine curve obtain with landmarks and the Cobb angle obtain by means of front X-ray image was r = 0.89. The values for the kyphosis and lordosis angles for a group of 30 normal subjects were 44.511.8 and 34.110.0 degrees for male and 46.111.6 and 39.112.6 degrees for female.

STUDY OF THE SPINE CURVE IN 3D: A normal spine curve. Left: A representation of the curvature and the torsion . Right: The Frenet frame of normal spine curve.  ..............  x z y

STUDY OF THE SPINE CURVE IN 3D. Front plane Sagittal plane Thoracic scoliosis. The Cobb angle is 45.0 in the thoracic region. Left: A representation of the curvature and the torsion. Right: The Frenet frame of Front and Sagittal planes.

STUDY OF THE SPINE CURVE IN 3D. Left: A patient with a double major scoliosis with thoracic Cobb angle of 30 and lumbar Cobb angle of 30. Right: A patient with a thoraco-lumbar scoliosis with thoracic Cobb angle of 24 and lumbar Cobb angle of 12.

CONCLUDING REMARKS. A reconstruction of the back human surface has been developed using a structured light sheme. We compare the spine curve obtain with landmarks over the back surface with the projection of the spine curve in front X-ray image and have obtained a good correlations. We get a description of different types of deformities in the spine as a function of the curvature and torsion. Also, the Frenet frame is represented along the spine curve.