Fourier Transform Tufts University傅里变换塔夫斯大学.pptVIP

Fourier Transform * A Fourier Transform is an integral transform that re-expresses a function in terms of different sine waves of varying amplitudes, wavelengths, and phases. So what does this mean exactly? Can be represented by: When you let these three waves interfere with each other you get your original wave function! Let’s start with an example…in 1-D Notice that it is symmetric around the central point and that the amount of points radiating outward correspond to the distinct frequencies used in creating the image. Increasing Frequency Increasing Frequency Since this object can be made up of 3 fundamental frequencies an ideal Fourier Transform would look something like this: Let’s Try it with Two-Dimensions! This image exclusively has 32 cycles in the vertical direction. This image exclusively has 8 cycles in the horizontal direction. You will notice that the second example is a little more smeared out. This is because the lines are more blurred so more sine waves are required to build it. The transform is weighted so brighter spots indicate sine waves more frequently used. So what is going on here? The u axis runs from left to right and it represents the horizontal component of the frequency. The v axis runs up and down and it corresponds to vertical components of the frequency. The central dot is an average of all the sine waves so it is usually the brightest dot and used as a point of reference for the rest of the points. Since this is inverse space, dots close to the origin will be further apart in real space than dots that are far apart on the Fourier Transform. (Again keeping in mind that these dots refer to the frequency of a component wave.) x-y coordinate system u-v coordinate system Fourier Transform Fourier Transform Images are from: /~brayer/vision/fourier.html Let’s Bring it Up a Few Notches This image exclusively has 4 cycles horizontally and 16 cycles vertically This image exclusively has 32 cycles horizontally and 2 cycles vertically An or