## 1 Introduction  
Ultra-precision diamond cutting, combining a single point diamond tool with an ultra-precision lathe, has been widely employed for the fabrication of microstructure elements, such as microlens arrays [1], compound eye freeform surfaces [2], and sinusoidal grids [3; 4]. The achievable machining accuracy of the ultra-precision diamond cutting is significantly affected by the geometry of the diamond tool, including cutting edge contour, cutting edge radius, and tool faces [5; 6; 7]. In order to achieve a nanometric machining accuracy, it is essential to conduct a quantitative evaluation of the geometry of the diamond tool, especially cutting edge radius, which determines the minimum depth of cut and the surface finish of the machined microstructures [8; 9]. Cutting edge radius of a diamond tool is usually required to be within 10 to 100 nm for ultra-precision diamond cutting [10]. Therefore, the methods for cutting edge radius measurement should have a nanometric measurement accuracy in both lateral and vertical directions [11]. Optical methods can conduct fast and non-contact measurements [12; 13; 14]. However, a lateral resolution that is larger than 10 nm is limited by the optical diffraction phenomenon which occurs around the cutting edge when an optical method is used to measure the geometry of the diamond tool. Scanning electron microscopes (SEMs) have a nanometric lateral resolution and a wide view field [15]. However, because SEMs have a limitation of two-dimensional (2D) projection, they can only be used for the qualitative evaluation of diamond tool cutting edge radius and cannot be used for the quantitative evaluation. In addition, the material of diamond tools also influences the measurement accuracy of the diamond tool cutting edge radius using SEMs [16]. Atomic force microscopes (AFMs), which can provide a three-dimensional (3D) measurement with a nanometric accuracy in both lateral and vertical directions, can directly measure the diamond tool cutting edge radius by accurately aligning the AFM cantilever tip with the diamond tool cutting edge within the measurement range of the AFM [17; 18; 19]. However, the process for an accurate alignment is extremely time-consuming and the AFM cantilever tip can also be easily damaged due to an inaccurate alignment operation. An indirect measurement method based on an AFM has been proposed to address the above problems using an AFM to measure cutting edge radius of a diamond tool [20]. A profile of a diamond tool cutting edge is copied on a copper workpiece by indenting the diamond tool into the workpiece on an ultra-precision lathe. Then, the copied profile is scanned by an AFM, from which the diamond tool cutting edge radius can be obtained. Since this indirect measurement method can protect the diamond tool and the AFM cantilever tip from damage, and also shorten the measurement time, it is attractive for practical applications. However, the elastic recovery of the copied profile can affect the measurement accuracy, which is not considered in the indirect measurement method based on the AFM. More importantly, since both the direct and indirect measurement methods are based on an AFM, the obtained profile is a geometric convolution of the AFM tip and the actual and copied diamond tool cutting edges. It is essential to eliminate the convolution effect of the AFM tip whose size is comparable to that of the diamond tool cutting edge, for achieving a high-precision diamond tool cutting edge radius measurement. There are two approaches for reducing the convolution effect of the AFM tip. One approach is to directly characterize the AFM tip with a form surface measuring instrument. The other is to make an error separation operation on the AFM images for removing the influence of the AFM tip profile. Compared with the direct characterization approach, the error separation approach is more effective because no additional form surface measuring instruments are necessary [21].