Fig 1.
On the left: Growing syndesmophytes in the spine of a 30-year old male with ankylosing spondylitis. On the right: Multiple syndesmophytes, bridging adjacent vertebral bodies in the spine of a 37-year male with ankylosing spondylitis, responsible for the so called bamboo spine image on X-ray.
Fig 2.
The shape language primitives.
Fig 3.
The example of a contour representation.
Primitives are separated by dots, both black and white. Sinquads are separated by red dots.
Fig 4.
On the left the healthy cervical vertebrae, on the right the cervical vertebrae with a new bone formation.
Fig 5.
The contours divided into the sinquads.
Switches, marked in red, are the points between sinquads that were received as a result of description by primitives pij. Below each contour, there is a key, which denotes a sequence of transitions between sinquads. After each biquad, there are, additionally, values 0 or 1 which sequence inform about convexity of a given contour.
Table 1.
The received keys and equivalent classes in the analyzed data.
Fig 6.
The contours of healthy vertebrae.
The contours have been received by using SDA algorithm. In the first row, there are K0, K1 and K2 described by string 341.411. In the second row, there are K3, K4 with the description 341.411 and K5 described by string 341.430.341.411.
Fig 7.
The contours of the vertebrae with early pathological changes.
The contours of the vertebrae K2, K3 and K4 from Fig 4 with pathological changes received by SDA algorithm. The first and third ones are described by string 341.430.341.411. The second one is described by string 341.411.
Fig 8.
The contours of the vertebrae with serious pathological changes.
Each vertebra is described by untypical keys—see Table 1.
Fig 9.
The diagram of the proposed hierarchical method of vertebrae contours analysis.
Fig 10.
The determining of the angle α.
The examples of vertebrae with an angle α between 4−sinquad and 1−sinquad are marked. Auxiliary straight lines help to indicate the angle. From the left: a healthy bone, a bone with a small lesion, a bone with the advanced lesion.
Fig 11.
The graphs of the membership functions.
The membership functions are defined for each specified class i.e. K0, K1, K2; K3, K4, K5. The angle α is the angle between 4−sinquad and 1−sinquad.
Table 2.
The results of the classification.
Fig 12.
Three examples of vertebrae from K1 and the values of the membership functions to the class with syndesmophytes.
The values are calculated for vertebrae from the set K1 which are presented in Fig 10.