So, by doing it in two directions, you get down to one-quarter of the pixels. You can increase the blurring to create an extremely blurred version that needs relatively few pixels. Wavelet compression doesn't blur the whole picture. These are any region where you have a strong difference in intensity as you move in any direction from one pixel to another. If you keep the extremely blurred version of the smooth areas along with a few of the edges, you can get a beautiful rendering of your original image.
Compression is only the first half of the procedure.
After transmission, text, images, and data must undergo reconstruction, otherwise called decompression. Then, you keep multiplying the pixels by four, and hopefully put back the edges. Reconstruction is completely equivalent to compression, but much faster. You can get progressive renderings, with a handful of coefficients. Wavelet compression has yet to make a major impact on the transmission of images.