![]() In this example, a card with a random arrangement of oval-shaped dots is held in front of the webcam. (E) The filtered FT is included underneath the main FT plot. (D) The right-hand column shows the Fourier transform (FT) of the acquired image that uses a similar grayscale color map as the original image. (C) The reconstructed image, obtained by inverse transformation after applying the filter, is displayed below the main image. (B) In the center column, the top figure displays the images captured by the webcam they are converted into grayscale by equally combining the red, green, and blue channels. (A) The user control panel includes buttons for starting and stopping the frame acquisition process. The spacing between Fourier peaks is inversely proportional to the spacing between lines in real space. Figure 1 shows an example of the reciprocal relationship between real and Fourier space, where a periodic grid pattern is translated into a series of peaks. ![]() Here, I present an interactive lesson plan that uses computer software to enable students to predict two-dimensional (2D) FTs of various patterns and test their predictions in real time. Specifically, it is important for instructors to address advanced learning goals by providing opportunities for students to generate, analyze, and evaluate predictions. Students should gain a conceptual understanding of how patterns in one domain are translated into its conjugate domain, as well as more practical knowledge, such as intuitively predicting the effect of a Fourier filter. Classroom activities are needed that provide students hands-on experience utilizing Fourier techniques. Modern pedagogical approaches are designed to develop competency across the entire cognitive spectrum: Remembering, Understanding, Applying, Analyzing, Evaluating, and Creating ( 14). Thus, it is challenging for students not equipped with strong mathematical skills to understand what a FT does and, more so, to acquire intuition for how a FT translates one function into its conjugate function in the Fourier domain. Although there are many excellent articles and textbooks on Fourier methods, pedagogical approaches can be highly mathematical ( 9– 11), introduced within the context of specific techniques ( 12), or be brief and oversimplified ( 13). Developing an intuitive understanding of FTs is therefore essential for undergraduate students to fully grasp the principles behind these techniques. Although certain experimental methods, such as infrared spectroscopy, are nearly universal in undergraduate teaching laboratories, FTs are automatically carried out by internal software libraries with preprogrammed settings that are typically hidden from the user ( 8). Modern molecular dynamics simulation packages implement fast FT algorithms to improve computational accuracy and efficiency ( 7). Spatial reconstruction algorithms based on FTs are at the core of modern biomedical imaging applications, such as magnetic resonance imaging ( 6). ![]() Biophysical characterization techniques, including nuclear magnetic resonance (NMR) spectroscopy ( 1), infrared spectroscopy ( 2), x-ray crystallography ( 3), mass spectrometry ( 4), and differential scanning calorimetry, rely on FTs for data processing or analysis ( 5). This interactive approach enables students with limited mathematical skills to achieve a certain level of intuition for how FTs translate patterns from real space into the corresponding Fourier space.įourier Transforms (FTs) are an essential mathematical tool for numerous experimental and theoretical methods. During the lesson, students are asked to predict the features observed in the FT and then place the patterns in front of the webcam to test their predictions. Several patterns are included to be printed on paper and held up to the webcam as input. The materials include a computer program to capture video from a webcam and display the original images side-by-side with the corresponding plot in the Fourier domain. Here, I introduce interactive teaching tools for upper-level undergraduate courses and describe a practical lesson plan for FTs. Despite FTs being a core component of multiple experimental techniques, undergraduate courses typically approach FTs from a mathematical perspective, leaving students with a lack of intuition on how an FT works. Fourier transforms (FT) are universal in chemistry, physics, and biology. ![]()
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