Errors of tilt

Whilst parallax errors could be effectively ruled out as a result of the equipment used and the distance between lens and subject in the shots taken, the horizontal alignment of each tooth might be subject to observer error. Various methods can be used to ensure that the cervical plane of the crown is kept horizontal: a spirit level across the dental arch crown surface can be used in cases where teeth are all in an even plane, with crown surfaces parallel to the cervical plane. These cases are rare, however, because tooth wear is rarely along a horizontal plane, particularly along the buccolingual axis. Even if a spirit level small enough to place on each tooth individually could be found, it would still not help in the levelling of the tooth due to this differential wear. Another way to level the cervical plane of the tooth would be to immerse the tooth in sand that had itself been levelled, so that the cervical plane lies at the surface of the sand. However, when photographing very delicate fossil specimens, there is a very real danger of damaging the tooth, the dental arch or the cranial remains of a specimen during the process of immersing the specimen in sand or when trying to adjust the

80

specimen to make it level. To avoid damaging delicate specimens, a mount was ideally pre-prepared, using a cast of the specimen, embedded in a very malleable foam material (protected by a plastic film to avoid residues from the foam being left on the specimen). This kind of pre-preparation allowed for good pre-levelling of the cast’s occlusal surface and served to protect the actual specimen during handling. However, casts were not always available and such precautions and preparations were not always possible.

Ultimately, levelling along the mesiodistal axis can in principle be ensured by using a spirit level held or propped up in line with the cervical plane, but in practice, levelling across the buccolingual axis is more effectively achieved by careful alignment by eye from above the tooth through the viewfinder or via the live screen of the camera of both side walls of the tooth, so that they equally match each other for tilt.

In view of the potential for observer error of tilt (particularly buccolingually), it was decided to apply 3D imaging software to quantify this potential error. Coefficients of correlation between pairwise sets of landmark distances on a single 3D tooth image, tilted at regular intervals using Amira software with 2D snapshots taken at increasing degrees of tilt buccolingually were calculated. A particularly pristine (unworn) tooth was chosen for this exercise (in which case any change of tilt would be further

exaggerated by parallax errors due to the slope of the cusp peaks, so this would be equivalent to the “worst case scenario” for tilt error bars), and the same landmark arrangements were measured at 2 degree tilt intervals from -10˚to +10˚, with each set of data being compared to the 0˚ (fully horizontal) untilted view. These 10 sets of

correlation measurements were then compared to measurements taken from the same arrangement of landmark placements situated on a different tooth completely. The

81

results, presented in Figure 3.2, showed that tilt does not affect the outputs of a shape analysis very significantly at angles of tilt up to 4 degrees either way. When an unworn tooth is tilted at + 2 degrees or at – 2 degrees, the correlation coefficient of both sets of data against the horizontal was 0.99, indicating a strong correlation between the outputs at this degree of tilt against the horizontal. The correlation coefficient was still above 0.95 at the +/- 4˚ tilt level).

Figure 3.2 Landmark distances from the centre of 11 views of a pristine tooth at varying degrees of tilt compared to the same calculation for a different tooth entirely.

In this figure, “119” represents a 6 year-old human tooth from a collection of 3D tooth scans held at Toulouse University (with zero degrees of tilt being represented by the cobalt blue line right in the centre of the 119 series), and “TM1517” – the dark blue line with the very distinct contour, is a fossil tooth. Note that for the “119” series, the tan-coloured and the maroon-coloured lines (-2 degrees and +2 degrees respectively) on either side of the cobalt line (0 degrees) are highly correlated to the cobalt 0 degree line, so that up to 2 degrees of tilt would render very similar measurements in any statistical or geometric morphometric analysis. The dark blue line represented by tooth TM1517 at zero degrees of tilt is highly uncorrelated to the cobalt line (tooth 119 at zero degrees), the contour of its respective landmarks following a very distinct pattern.

(Acknowledgements: Professor José Braga – University of Toulouse, for access to the 3D images and to Jean Dumoncel – University of Toulouse, for training on the Amira software utilised in this exercise.)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 TM1517 119/-10 119/-8 119/-6 119/-4 119/-2 119/ 0 119/2 119/4 119/6 119/8 119/10

82

To estimate a possible “normal” degree of observer error that might be incurred by the same observer in levelling teeth successively on separate occasions, three different 2D photographs of the same tooth (the M1 of the holotype of Paranthropus robustus, TM

1517) were then embedded into the 3D image of the same tooth using Amira 3D imaging software, and their angles of tilt were compared. There was virtually no difference (0.014˚ along the x-axis, 0.107˚ along the y-axis and 0.098˚along the z-axis) between angle of tilt of two of the shots, both taken from directly above the tooth on two separate occasions, and there was a tilt of less than 2˚ in each direction in the case of a shot focussed on the adjacent tooth (i.e., when the focus was necessarily not directed toward that particular tooth). The coefficient of correlation (R2_{) for pairwise}

landmark measurements in the same tooth at a 2˚ tilt (against the level) was 0.99. Thus it can be reasonably assumed that provided that due care and attention has been taken in levelling each tooth before taking the photograph, measurement errors caused by tilt will in all likelihood be limited to less than 1% and would certainly be very unlikely to be as much as 5% (equivalent to more than 4˚ of tilt in either direction, which would seem improbable, since already at 4˚, the sidewall tilt is clearly visible from above, and is easily avoidable). In a worn tooth (as in the case of the majority of the fossil teeth analysed) these potential tilt error bars will be lower, since the flatter the surface, the smaller the additional parallax error as the tooth is tilted (landmarks on a flat surface remain more equidistant from each other as the tooth is tilted, whereas landmarks on steep cusp surfaces move differentially apart as the tooth is tilted).

The following figures (Figure 3.3 and Figure 3.4) illustrate the process used to calculate error bars, using Amira 3D software:

83

Zero degrees from horizontal – side wall (buccolingual) slopes observably even

+/- 2˚- acceptable tilt (1% error each way vs. the horizontal, on an unworn tooth)

+/- 4˚ - visible, avoidable tilt (<5% error vs. the horizontal, on an unworn

tooth)

+/- 6˚- unacceptable, fully avoidable tilt +/- 8˚- very unacceptable, fully avoidable tilt

+/- 10˚- extremely unacceptable, fully avoidable tilt

Figure 3.3 2D snapshots of 3D unworn tooth images for the calculation of tilt error bars.

84

Figure 3.4 illustrates how, by using 3D software, it is possible to calculate tilt differences between various existing 2D photographs.

A B C

Figure 3.4 Calculation of the angle of tilt for existing 2D photographs. Using Amira 3D imaging software, three similar 2D photographs (including the photograph figuring in A above) taken on separate occasions (two from directly vertically above the tooth and one from directly above the adjacent tooth) were individually inserted into slices of the 3D image of the same tooth (B – enamel in blue, dentine in red), so that the 3D image was perfectly superimposed over the 2D perimeter (C). The angle of tilt of the 3D image was then calculated for each image and compared for differences.