Exploring High Dynamic Range Imaging: §5.2 Wavelets and HDR 1.2 2006/12/16 22:29:09 US/Central 2006/12/25 16:33:03.283 US/Central Tianhe Yang tian.yang@rice.edu Sarah Nicole McGee snmcgee@rice.edu Taylor T Johnson ttj@rice.edu Robert Lowell Ortman rlortman@rice.edu Tianhe Yang tian.yang@rice.edu Sarah Nicole McGee snmcgee@rice.edu Taylor T Johnson ttj@rice.edu Robert Lowell Ortman rlortman@rice.edu Mark A. Davenport md@rice.edu Richard G. Baraniuk richb@rice.edu After implementing numerous tone operators, we were curious as to whether or not there were any applications for wavelets in HDR. After researching, we found that while wavelets are more often used for data compression and noise reduction in HDR, wavelets can also be used to create an unsharp mask to adjust the sharpness of an image. Christian Buil, “Comet High Dynamic Range Imagery: Application to Total Eclipse Processing.” (12 Dec 2006) <http://www.astrosurf.com/buil/iris/tutorial19/doc41_us.htm>. The unsharp mask works by increasing the contrast at the edges. In photography, unsharp masks are created by making a blurred version of the original image in a wet darkroom and then overlays the two images. In most cases, the blurred version is created by applying a Gaussian blur to a copy of the image. The two matrices are then subtracted. If the value in the difference matrix is greater than a specified threshold, the darker colors are made darker and the lighter colors are made lighter in the sharpened image. Dale Cotton and Brian D. Buck, “Understanding the Digital Unsharp Mask,” (12 Dec 2006) <http://www.luminous-landscape.com/tutorials/understanding/series/understanding-usm.shtml>. In our unsharp mask, instead of using a Gaussian blur, we use wavelets to generate the blurred mask. While the Fourier representation only retains the frequency information, the wavelet representation contains both frequency and time information. Since it is impossible to compute all of the wavelet coefficients, we will use the discrete wavelet transform to analyze our signal. “Wavelet” Wikipedia (12 Dec 2006) <http://en.wikipedia.org/wiki/Wavelet>.The Haar DWT is generated using filter banks. The filter bank splits the image into various frequency bands. To create a level one 2D filter bank, make two copies of the image matrix. Run a low pass filter on one copy and a high pass filter on the other copy. Next downsample each copy of the image matrix by removing all of the even columns. Then recombine the two downsampled copies so that the new matrix has the same matrix dimensions as the original. As a result, one half of the new matrix has been low passed while the other half has been high passed. Now that you have filtered the column vectors, you need to do the same thing for the row vectors. The resulting matrix should have four sections: a low passed section that was low passed again (approximation matrix), a low passed section that was then high passed (horizontal details), a high passed section that was then low passed (vertical details), and a high passed section that was high passed again (diagonal details). The section that looks most like the original image is the one that was low passed twice. However, it is still blurrier than the original because the high frequencies were removed from the image. To create multi-level filter banks, continue to high pass, low pass, and then downsample the section of the matrix that is always low passed. To create a blurred image, zero out all of the matrices except for the approximation matrix, then apply the IDWT for the Haar wavelet. S. Allen Broughton, “Chapter 7: Filter Banks and the Discrete Wavelet Transform” 1998 (Terre Haute: Rose-Hulman Institute of Technology, 12 Dec 2006) <http://www.rose-hulman.edu/~brought/Epus/Imaging/waveimage.html> 2. We have found that the image is sharpened if you use a level 1 filter bank and even more so with a level 2 filter bank. However, there is visually no difference between sharpening with a level 2 filter bank and a level 3 filter bank. Also if the amount you sharpen by is too great, then the sharpened image will have halos. However, as opposed to diffuse halos that are produced by sharpening with a Gaussian blur, the filter banks produce a somewhat blockier halo effect. While no information is gained by applying an unsharp mask, it can be used to enhance the images produced by some of our own images.

Original Image

Spliting the image into 3 levels of filter banks

Zeroing out everything except the approximation matrix

Reconstructing the image using the zeroed-out filter banks to create a blurred image