Why is the high-pass filter result in a discrete wavelet transform (DWT) downsampled? From Wikipedia's description of the Discrete Wavelet Transform, a signal yields a set of:


approximation coefficients (low-pass: by averaging + downsampling)
detail coefficients (high-pass: convolving with wavelet + downsampling)




This process is cascaded over approximation coefficients:



Why is the high-frequency part of a signal downsampled and how does it retain information? Can it be reconstructed? Is information lost due to the Nyquist theorem?



Sources I looked at:


Mathworks: Wavelet decomposition algorithm
Guide for wavelet transforms for machine learning
Assume the original sampling rate is $F_s$ and that the filters are perfect brickwall Halfband filters. After the low pass filtering the frequency content is from $0 \rightarrow \frac{F_s}{4}$. Because of the new bandwidth, the sampling rate can be reduced to $F_s/2$.

For the high-pass filter, the frequency content lies in the range $\frac{F_s}{4} \rightarrow \frac{F_s}{2}$. There is no content in the range $0 \rightarrow \frac{F_s}{4}$. We can downsample by a factor of 2 in this case as well. The downsampling causes the frequency content in the $\frac{F_s}{4} \rightarrow \frac{F_s}{2}$ to be aliased into the range $0 \rightarrow \frac{F_s}{4}$, but because there is no frequency content in this range, due to the high-pass filter, we have not lost any information.