Theory and application of shapelets to the analysis of surface self-assembly imaging

A method for quantitative analysis of local pattern strength and defects in surface self-assembly imaging is presented and applied to images of stripe and hexagonal ordered domains. The presented method uses “shapelet” functions which were originally developed for quantitative analysis of images of galaxies $(\propto {10}^{20}\mathrm{m})$. In this work, they are used instead to quantify the presence of translational order in surface self-assembled films $(\propto {10}^{-9}\mathrm{m})$ through reformulation into “steerable” filters. The resulting method is computationally efficient (with respect to the number of filter evaluations), robust to variation in pattern feature shape, and, unlike previous approaches, is applicable to a wide variety of pattern types. An application of the method is presented which uses a nearest-neighbor analysis to distinguish between uniform (defect-free) and nonuniform (strained, defect-containing) regions within imaged self-assembled domains, both with striped and hexagonal patterns.

Figure 1

(a) Example of a (simulated) hexagonal self-assembled film from past work [9] where the field shows the surface coverage of a species ranging from $0\to 1 (\mathrm{black}\to \mathrm{white})$; (b) local hexagonal pattern strength or order resulting from applying the bond orientational order method to (a) where pattern strength ranges from no hexagonal order (black) to perfect hexagonal order (white). The length scale (both images) is 256 nm.

Figure 2

(a) Two-dimensional spectral density plot from the DFT of Fig. 1 with the origin at the center of the image; (b) radially averaged spectral density from (a).

Figure 3

Surface plots of the real (top row) and imaginary (bottom row) components of polar shapelet functions [Eq. (2)] with $n=0$ and $m=1\to 8$. Image length scale is normalized to 1 and $\beta =0.55$.

Figure 4

Plots of one-mode approximations of a (a) stripe $(a_1=1)$ and (b) hexagonal $(a_1=a_2=a_3=1)$ pattern using Eqs. (7) and (8); (c) schematic of rotations of a shapelet applied to a nonuniform stripe pattern.

Figure 5

Shapelet filter response $w_{n,m}^{\mathrm{opt}}$ versus pattern length-scale parameter $\lambda$ (a) with $\beta =\lambda$ and (b) $\beta =C\lambda$ (Table 1). Responses shown are correlations with the uniform hexagonal pattern given by Eqs. (7) and (8) with $a_1=a_2=a_3=1$ and $\frac{\lambda }{\Delta x}=20$.

Figure 6

Responses of steerable polar shapelet filters applied to uniform one-mode approximations of a (a) stripe and (b) hexagonal pattern from Eqs. (7) and (8). Optimal filter responses (above, gray-scale intensity) and orientations (below, see the colorbar) are shown for each pattern with steerable shapelets with $n=0$ and $m=1\to 6$ (left to right).

Figure 7

Examples of hexagonal surface self-assembly with features of varying character: (a) sharp interface, (b) semidiffuse interface, and (c) diffuse interface. Figures taken from Ref. [9].

Figure 8

Examples of nonuniform (a) stripe and (d) hexagonal patterns from simulations of surface self-assembly (taken from Ref. [9]). Local pattern response distance resulting from applying the guided machine learning algorithm to the (b) stripe and (e) hexagonal patterns where the response distance [Eq. (13)] was normalized to range from 0 (black) to 1 (white) and the user-specified set of pixels are highlighted in (a) and (d). [(c) and (f)] Overlay of the pattern response distance onto the original patterns with normalized pattern response distance varying from 0 (red) to 1 (green).