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UNSW ENGINEERING @

UNSW

10Nicholas Webb (Author), David Taubman (Supervisor)15 October 2010

Reconstruction of Vertebra SliceGeometry from Orthogonal X-Rays

1. Introduction

1.1 ProblemTo reconstruct vertebra slice geometry from orthogonal X-rays (right, Kadoury 2007) of a spine. Below right is a diagram of two vertebrae.1.2 MotivationA solution to this problem would facilitate common 3D visualisation of the spine because biplanar X-rays are inexpensive, accurate and use relatively little radiation.

2.3 Accurate Contour Recovery

2. Method

2.1 Simulation of Data1) Simulate a vertebra slice, horizontal to the ground, like the simple concave shape on the right. The image is bi-level due to an assumption of constant bone and tissue densities.

4. Conclusion

This novel approach, with its novel contour detection algorithm, promises useful recovery of convex and simple concave bone slice shapes from orthogonal X-rays. 4.1 Further WorkA gradient descent-based system component should improve the contour estimate accuracy. Further algorithm development should work towards finding complex concave shapes such as a real lumbar vertebra slice.

2) Simulate x (above left) and y (above right) orthogonal X-ray projections of the simulated vertebra slice shape. 3) Construct a back projection image (right) from the simulated orthogonal X-ray projections. The algorithm searches this image for the vertebra slice shape contour.

2.2 Circular Shortest Path Contour Initialisation4) Select a Circular Search Space (CSS) subset of the back projection. The image on the right is an unwrapped and interpolated CSS (with radius the vertical axis and angle the horizontal axis).5) Calculate the back projection image gradient at the CSS points (shown on the right) with a tensor product spline image model.6) Calculate the initial contour as the circular shortest path in the back projection considering only image gradient. The red contour on the right demonstrates that this is the visual hull.

7) Measure the x (above left) and y (above right) projection errors of the contour estimate using the known orthogonal projections.8) Construct the image metric g(s) (right) which defines likely contour locations, being comprised of x and y projection errors, back projection gradient and a centre block which biases against too small contours.9) Calculate a more accurate contour (below left) with a novel method which minimises Equation 1. An existing algorithm (Appleton 2004) favours contour positions further from the image centre by finding the contour which minimises Equation 2. However, such an algorithm favours convex contours given a blurred metric like the output of step 8.

(1)

(2)

10) Smooth the contour from step 9 with a smoothing spline in accordance with an assumption that a real vertebra is relatively smooth (above right).11) Repeat steps 7-10 until the error is below 1%.

3. Results

The final contour (right) has both x and y projection errors below 1% of the true simulated values.

1.3 Novel ApproachExisting solutions rely on statistical models of the human spine. The novel approach here aims to reconstruct the shape of a horizontal vertebra slice from radiograph data only.