Angular Fourier Mapping; Highlighting lattice structures without destroying original data
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Angular Fourier Mapping; Highlighting lattice structures without destroying original data Johannes H. Kindt*, James B. Thompson*, George T. Paloczi*, #, Martina Michenfelder§, Bettye L. Smith*, §, Galen Stucky¶, Daniel E. Morse§, and Paul K. Hansma* *Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA. # Now at Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA. § Department of Molecular, Cellular and Developmental Biology, University of California Santa Barbara, Santa Barbara, CA 93106, USA. ¶ Department of Chemistry, University of California Santa Barbara, Santa Barbara, CA 93106, USA. ABSTRACT A two-dimensional Fourier transformation, FT, is used to isolate two different lattice structures within one scanning probe microscope, SPM, image. The isolated structures are then used to create a two-color map that encodes the presence of these structures within the image. The color map is normalized in brightness and then used to color-code the original black and white SPM data. The distribution of different structures becomes obvious, while all original brightness information is preserved in this combined image. INTRODUCTION Lattice-scale SPM images are often obscured by noise. Digital filtering can be employed to remove that noise and make features such as step-edges more clearly visible. The most broadly used filtering technique involves a conversion of the image into the frequency domain (FT), selection of image frequencies and exclusion of noise frequencies, and an inverse transformation [1]. While this is a powerful method, many researchers avoid using it because its effects on the original data are hard to predict. In principle, a carefully tuned Fourier filter can turn white noise into any image desired. The algorithm proposed here employs Fourier filtering but it puts restrictions on the filtering parameters, making the result more predictable. Furthermore, the original black/white coded information is not changed, but only color-coded with the result of the digital filter. To return to the original image the color coding can be removed by anyone, e.g. using a Xerox copier with a flat spectral response. The main application for this method is to visually separate lattice structures of different periodicity and orientation. Algorithm The Angular Fourier Mapping algorithm can be divided into the following six consecutive steps: 1. Two-dimensional FT of the original SPM image Original Image: O(x,y) Fig. 1 Fig. 2 2D FT: F(x,y) = F(O(x,y))
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Figure 1. Original height image of a transition in lattice Figure 2. Two-dimensional FT of the original data. structure, observed by atomic force microscopy in liquid
2. User selection of two lattices by periodicity and orientation The FT translates lattice periodicity into radii from the center. Angles translate into angles. A lattice is fully defined by its period and orientation. A lattice in the original image appears as a bright maximum in the Fourier image. Two different lattices are now select
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