MATLAB Implementation of Fourier Transform

In the domain of signal and image processing, several fundamental techniques are employed to analyze and manipulate data. The Fourier transform serves as a cornerstone method, enabling the decomposition of signals into their constituent frequency components. In MATLAB, this can be implemented using the built-in `fft()` function for discrete signals or `fft2()` for 2D images, often accompanied by `fftshift()` to center the low-frequency components. Equidistant sampling involves capturing signal values at uniform intervals, which is critical for discrete signal analysis. MATLAB's `linspace()` function can generate equally spaced time vectors, while `resample()` or `interp1()` functions help maintain uniform sampling when working with non-uniform data. This approach ensures proper frequency representation in subsequent Fourier analysis. Equal-angle sampling refers to acquiring data points at regular angular intervals around a reference point, commonly used in circular or rotational analysis. This can be implemented using polar coordinate transformations with `pol2cart()` and trigonometric functions to generate uniformly distributed angular samples. Contour extraction techniques identify and extract object boundaries from digital images. MATLAB provides several approaches including edge detection algorithms like `edge()` with various operators (Sobel, Canny), or using `bwboundaries()` for binary images. The `contour()` function can also plot contour lines for 2D data sets, while `imcontour()` specializes in image contour visualization. Morphological processing involves shape-based operations to analyze and modify object structures. MATLAB's Image Processing Toolbox offers functions like `imdilate()` for dilation, `imerode()` for erosion, `imopen()` for opening operations, and `imclose()` for closing operations. These operations are particularly useful for noise removal, gap filling, and boundary smoothing in binary images.