Wavelet Decomposition of Images

One type of wavelet which exists, and which has continued to be of some interest to computational signal processing, is the Haar Wavelet. It’s thought to have a low-pass and a high-pass version complementing each other. This would be the low-pass Haar Wavelet:

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And this would be the high-pass version:

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These wavelets are intrinsically flawed, in that if they are applied to audio signals, they will produce poor frequency response each. But they do have as an important Mathematical property, that from its low-pass and its high-pass component, the original signal can be reconstructed fully.

Now, there is also something called a wavelet transform, but I seldom see it used.

If we wanted to extend the Haar Wavelet to the 2D domain, then a first approach might be, to apply it twice, once, one-dimensionally, along each axis of an image. But in reality, this would give the following low-frequency component:

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And only, the following high-frequency component:

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This creates an issue with common sense, because in order to be able to reconstruct the original signal – in this case an image – we’d need to arrive at 4 reduced values, not 2, because the original signal had 4 distinct values.

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And so closer inspection should reveal, that the wavelet reduction of images has 3 distinct high-frequency components: ( :1 )

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