![]() Amodern interpretation of which states that "any well-behaved function can be represented by a superposition of sinusoidal waves". This was the discovery made by the mathematician Joseph Fourier. Eventually by adding enough waves you can exactly reproduce the original profile of the image. However, you can continue in this manner, adding more waves and adjusting them, so the resulting composite wave gets closer and closer to the actual profile of the original image. This ' wave superposition' (addition of waves) is much closer, but still does not exactly match the image pattern. See also Adding Biased Gradients for a alternative example to the above. Since this is a digital representation, the frequencies are multiples of a 'smallest' or unit frequency and the pixel coordinates represent the indices or integer multiples of this unit frequency.But how can an image be represented as a 'wave'? Images are WavesWell if we take a single row or column of pixel from any image, andgraph it (generated using "gnuplot" using the script " im_profile"), you will find that itlooks rather like a wave.Ĭonvert -size 1x150 gradient: -rotate 90 \ -function sinusoid 3.5,0.25.25 wave_1.png convert -size 1x150 gradient: -rotate 90 \ -function sinusoid 1.5,-90.13.15 wave_2.png convert -size 1x150 gradient: -rotate 90 \ -function sinusoid 0.6,-90.07.1 wave_3.png convert wave_1.png wave_2.png wave_3.png \ -background black -compose plus -flatten added_waves.png In this domain, each image channel is represented in terms of sinusoidal waves.In such a ' frequency domain', each channel has 'amplitude' values that are stored in locations based not on X,Y 'spatial' coordinates, but on X,Y 'frequencies'. This is just a fancy way of saying, the image is defined by the 'intensity values' it has at each 'location' or 'position in space'.But an image can also be represented in another way, known as the image's' frequency domain'. Thus each of the red, green and blue 'channels' contain a set of 'intensity' or 'grayscale' values.This is known as a raster image ' in the spatial domain'. But for our purposes here we will ignore transparency. The Fourier TransformAn image normally consists of an array of 'pixels' each of which are defined bya set of values: red, green, blue and sometimes transparency as well. ![]() For the more mathematically inclined, see: Fourier Transform, and 2D Fourier Transforms.Other mathematical references include Wikipedia pages on Fourier Transform, Discrete Fourier Transform and Fast FourierTransform as well as Complex Numbers. Other introductory discussions include Introduction To Fourier Transforms For Image Processing, Fourier Transforms, DFT and FFT, and Image Filtering in the Frequency Domain. If you find this too much, you can skip it and simply focus on the propertiesand examples, starting with FFT/IFT In ImageMagickFor those interested, another nice simple discussion, including analogies tooptics, can be found at An IntuitiveExplanation of Fourier Theory. It is the goal of this page to try to explain the background and simplifiedmathematics of the Fourier Transform and to give examples of the processingthat one can do by using the Fourier Transform. These include deconvolution (also known asdeblurring) of typical camera distortions such as motion blur and lens defocusand image matching using normalized cross correlation. But it can also provide new capabilities that one cannot do inthe normal image domain. Nevertheless, utilizing Fourier Transforms can provide new ways to do familiarprocessing such as enhancing brightness and contrast, blurring, sharpening andnoise removal. First, it is mathematicallyadvanced and second, the resulting images, which do not resemble the originalimage, are hard to interpret. IntroductionOne of the hardest concepts to comprehend in image processing is FourierTransforms.
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