Dear All,

when using the 3D optoelectronic systems body segments angles are usually reconstructed via certain number of markers attached to segments which, among other things, usually requires imaging and 3D position reconstruction of attached markers in context. Since camera is a device which projects points from 3D to 2D and subsequently depth information is lost, thus we need at least two cameras for 3D reconstruction (or camera accompanied with some other source of light such as laser, video projector or even desktop lamp would do etc.).

During the course of my Ph.D I have been considering possibilities of calculating the body segments angle using only one and uncalibrated camera (precisely speaking, knowing only internal camera parameters) and in even cases where movement is not constrained to 2D, but subject can freely move in all directions. Needless to say I believe using only one camera for segments angle calculation, would have certain practical advantages.

The idea has computer vision origin and I would like very much your opinion about its feasibilty for biomechanical applications, perhaps someone has been doing similar things.In brief, every image point can be back projected in space as a line coming through camera optical center and image point itself. When having two such image points, there is relatively simple formula to calculate angle alfa between such two lines:

cos(alfa)=(x1'*omega*x2/sqrt(x1'*omega*x1)/sqrt(x2'*omega*x2); where x1 and x2 are homogenous coordinates of two image points and matrix omega is image of the so-called image of absolute conic (IAC) which is readily calculated if matrix of camera internal parameters are known (possibly from some other calibration or attainable from some cameras settings, data sheet etc.)

Now, let's suppose that we have two body segments and on each segment three collinear markers attached on it (the more markers are welcome, but three are sufficient and condition of collinearity is probably one of the most critical parts for method to work) and markers distances are known/unchangeable in all frames (for instance from anthropometric measurements or three markers are fastened on a stick attached to body segment; I have seen some papers where people put various sticks to easy up angle calculations). Every segment, i.e. line formed by its collinear markers is characterized by the so-called point at infinity and its image (vanishing point) generally speaking is detectable on cameras image (2x2 mapping H between points in space and image can be found and vanishing point calculated as H*[1 0]'). It can be shown that angle spanned by two back projected lines from two detected vanishing points is equal to angle formed by segments, i.e. its correspodent lines in space.

Therefore, using the above formula if we can represent each segment with three collinear points and found its vanishing points on image we would be able to calculate angle between them, using single camera knowing only its internal parameters (maybe not so obvious, but in other words IAC depends only on cameras internal parameters, external ones are irrelevant).

I tried to be concise as possible and any comment would be appreciated, summary will follow up.

Best, Tomislav.

Tomislav Pribanic, M.Sc., EE

Department for Electronic Systems and Information Processing

Faculty of Electrical Engineering and Computing

3 Unska, 10000 Zagreb, Croatia

tel. ..385 1 612 98 67, fax. ..385 1 612 96 52

E-mail : tomislav.pribanic@fer.hr

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# body segments angle calculation using single and partiallycalibrated camera

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