Integer Wavelet with Only Positive coefficients

Question

Is there any Integer Wavelet Transform which produce only positive coefficients?
Currently i'm using Haar wavelet but it produces both positive and negative coefficients.

Answer 1

If you are talking about the wavelet by itself, no, because it is a zero-mean function (by construction), hence there are positive and negative values.

For wavelet coefficients from data, no, because of linearity: if for some $x$, $W(x)$ is positive, then $W(-x)=-W(x)$ is negative.

Note that for some signal models, with certain multiscale schemes, you could achieve your goal. You can find hereafter two references:

• Lossless data hiding based on integer wavelet transform

In data hiding, pieces of information represented by the data are hidden in the cover media. In some applications, people do care about the cover media. That is, the hidden data and the cover media may be closely related. For this type of data embedding, in addition to perceptual transparency, for some applications such as medical diagnosis and law enforcement, it is desired to invert the marked media back to the original cover media after the hidden data have been retrieved. The marking techniques satisfying this requirement are referred to as lossless, distortion-free or invertible data hiding techniques. From this point of view, it is observed that most of the current digital watermarking algorithms are not lossless.

• Distortionless data hiding based on integer wavelet transform

A novel distortionless image data hiding algorithm based on integer wavelet transform that can invert the stego-image into the original image without any distortion after the hidden data are extracted is proposed. This algorithm hides data into one (or more) middle bit-plane(s) of the integer wavelet transform coefficients in the middle and high frequency subbands. It can embed much more data compared with the existing distortionless data hiding techniques and satisfy the imperceptibility requirement. The image histogram modification is used to prevent greyscales from possible overflowing. Experimental results have demonstrated the validity of the algorithm.