Algorithm Error Study for Photometric Measurement System Transformation

Author Name(s): S.V. Prytkov, A.A. Ashryatov
Author Email: ashryatov@rambler.ru

Abstract

The article studies the errors of conversion algorithms for photometric systems using the example of the Cγ → Bβ transformation. The considered algorithms lead to different types of interpolation at the final stage: on scattered data and bilinear interpolation. The authors study the way the error behaves with a different grid pitch of photometric angles and with a different exponent k, which determines the photometric body of a round-symmetric source 𝐼(𝛼) = 𝑐𝑜𝑠(𝛼).

Introduction

American [7] and Russian [2] standards have the formulas for the transition between photometric systems. But as they showed in [4] there is a number of inaccuracies in these documents that do not allow to use the proposed solutions in practice. It also shows how to combine photometric system correctly, and it is indicated that the data structure becomes irregular after transformation, therefore, it is additionally necessary to interpolate the values of the luminous intensity. It is impossible to generate photometric data files without this [6, 8]. Two options are possible here. If we combine the old system (the system with the original data) with the new one, then the interpolation of the luminous intensity values is carried out in the new system. In this case, the interpolation nodes form an irregular grid, so you need to use the appropriate interpolation (for example, using the Delaunay triangulation) [5]. If we combine the new system (the system in which we want to obtain the values of the luminous intensity) with the old one, then the interpolation of the luminous intensity values is carried out in the old system respectively. In this case, the interpolation nodes form a rectangular grid; therefore, the luminous intensity values can be found via bilinear interpolation [3]. This article studies the error distribution for these methods.

Conclusion

Figures 2–9 show that, the relative error increases for both methods as one approaches the poles of the Bβ system. And it can reach large values. And at first glance, the method of transformation using the interpolation on scattered data is significantly inferior to the method with bipolar interpolation. However, if you look at the DCLI, it becomes obvious that a large error is observed for small values of luminous intensity (<0.01 I0), which is not significant from the point of view of lighting practice. Which by the way is confirmed by table 1, which presents the luminous flux before and after the conversion of photometric systems. It also implies that with the increase of k, that is, when a photometric body is compressed, the error of luminous flux determination increases, but even for a concentrated DCLI it does not exceed 0.6%. It is also interesting to note that the errors of the light flux after the transformation of Cγ → Bβ using the interpolation on the scattered data are less than in the method with bilinear interpolation. This suggests that the first method behaves worse only when it approaches the poles of Bβ photometric system. The study showed that you can use both methods in practice.

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