Wavelets: Frequency Analysis

The following workspace allows you to visualize 2D magnitude spectra of wavelet analysis performed on an image. A one-level wavelet analysis decomposes an image into four regions. Beginning at the upper left corner of the image and in clockwise order we find 4 regions:
  1. low-pass decomposition,
  2. high-pass decomposition in the width direction,
  3. high-pass decomposition in the height direction, and
  4. high-pass decomposition in both width and height directions.
Of each decomposition we want to visualize the its frequency content, magnitude spectrum.

Shown below is the one-level forward wavelet transform based on Daubechies filters, and its log(mag+1) spectrum.


Forward wavelet transform and its log(mag+1) spectrum

Now extract the four decomposition regions, find their log(mag+1) spectrums and display the results. Shown below are the results in clockwise order beginning from the upper left-hand corner.


1st quadrant_____________________2nd quadrant


3rd quadrant____________________ 4th quadrant

Observe that all low frequencies are concentrated around the center of the image of the 1st quadrant which corresponds to the continuous term (DC). In the 2nd quadrant we encounter high frequencies along the width direction, and in the 3rd quadrant the high frequencies are along the height direction. In the last quadrant high frequencies exist along both width and height directions.

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