# A Novel 3D Topography Stitching Algorithm Based on Reflectance and Multimap
## 1 Introduction  
Many physical phenomena are linked to surface roughness, either in manufacturing, such as friction and visual aspects, or in human daily life, such as object gripping and walking. Therefore, roughness study is an excellent analysis tool to better understand physical phenomena. This requires good quality data at the scale at which the physical phenomena occur. Scale can be defined as a segment of spatial wavelengths (or, alternatively, frequencies) [1]. Multiscale analysis is an advanced technique which isolates physical phenomena according to their scale of application. This facilitates the understanding phase of each phenomenon. The observable scale range depends on the profilometry technique used for the measurement of a given morphology, the measurement apparatus and its equipment(objective lens for optical system or probe for scanning probe microscopy, also known as SPM). Especially important, optical profilometers (focus variation microscope, confocal microscope and interferometer) can only measure a fixed field of view at given vertical and lateral resolutions. Their values are linked to the equipped objective lens. Therefore, the scale range is also fixed and often limited. Higher magnification objective lenses measure smaller details [2], but this comes at the price of a smaller field of measurement [3]. It is then crucial for metrologists to estimate, by making assumptions, the most relevant scale range which can capture the morphology linked to the studied physical phenomena, hoping that each phenomenon is included in the scale range. This arbitrary aspect is problematic and the scales of interest are often lacking from the topography or not completely measured. Stitching tries to overcome this problem of limited scale range of topographic measures using 3D optical profilometers. This is an elementary map-merging technique which uses overlapping regions of two (or more) elementary maps to detect common structures and to perform data alignment, also called registration. Stitching increases the field of measurement while preserving the vertical and lateral resolutions. This then increases the scale range, which facilitates the detection of different physical phenomena during multiscale analysis, as more scales are explored [4; 5]. As an example, tribology studies often rely on surface topography as contact is highly dependent on the morphology of first bodies and third body. Also, topography is useful for observation of surface wear, and is a complement to image for the determination of surface components and their respective wear. Lubricization has been widely studied, and surface morphology influences the distribution of lubricants. Tribology of surfaces is complex and the multiscale nature of the involved physical phenomena, interacting with each other's, is difficult to study in the limited scale range of non-stitched topographies measured with optical profilometers. By increasing the scale range of the measured topography, stitching facilitates the simultaneous observation of physical phenomena involved in tribology, to which surface form, waviness and roughness all contribute. Using the same objective lens ensures that the same measurement gauge is used, which facilitates comparison of surfaces. Stitching also increases the number of pixels on each map (such as hundreds of millions or a few billions of pixels per map), which adds statistical stability to roughness parameters (ISO 25178-2 [6]). Therefore, stitching helps in the determination of the most relevant roughness parameters through statistical analysis [7], useful for the optimization of surface functionality (such as non-exhaustively: wettability, friction, or brightness). While the principle of stitching is easy to understand, stitching algorithms are actually complex, as the difficulty is in registering overlapping data which are similar but not identical.