Application of 3 × 3 transformation matrix in the correction of three-dimensional AFM image tilts through coordinate transformation

In atomic force microscopy (AFM), it is difficult to maintain samples in a perfect horizontal position on the piezoelectric stage of an AFM to obtain accurate measurement of surface features and to enhance the visual effect of topographic images. Correction of the tilted image therefore requires the use of an appropriate 3 × 3 transformation matrix which will transform the coordinates to simultaneously correct the tilt in both the x-and y-axes due to the three-dimensional nature of AFM images. In this study, application of transformation matrix in three-dimensions for the simultaneous correction of image tilts in both the x- and y-axes by the method of coordinate transformation in three-dimensions is presented. Application of the matrix in simultaneously correcting tilts in the x- and y-axes of an AFM image is implemented by simulating an image assuming there are tilts in both the x- and y-axes. Results of features measurement after transforming the coordinates with the matrix show the efficacy of the matrix. This will conveniently replace the line-by-line analysis done to remove tilts in AFM, which has been the practice long before now. It also has the advantage of correcting images by simultaneously removing tilts in both the x- and y-axes as a veritable replacement of having to separately correcting tilts in images. This matrix can be used to transform coordinates in three-dimensions to obtain accurate step height and linewidth in AFM images.