1. Gradient Domain High Dynamic Range Compression
Raanan Fattal Dani Lischinski Michael Werman
The Hebrew University of Jerusalem

2. The Method in 1D log derivative attenuate exp integrate 2500:1 7.5:1
The method is really quite simple to explain in 1D.
Suppose we are given the 1D HDR image shown in this plot. The very first thing we do is to switch to the log-luminance domain. There are good reasons for that, which are explained in the paper.
Next, we compute the image gradients, which in 1D are simply the first derivatives. Now we can identify the strong luminance changes in the image as high spikes in the derivative.
The next step is to shrink these high spikes, while keeping the low derivatives unchanged. At this stage we have a new attenuated image derivative, and in order to get an image back, all we need to do is to integrate this derivative. Now we must undo the logarithm we took in the beginning, and we end up with a new 1D image.
Comparing the result with the original image we can see that the dynamic range was reduced from 2500:1 to only 7.5:1. Also, in the reduced
dynamic range the fine details are much more significant than they were before, and they no longer vanish when the entire range is displayed. *
The plot you see on the slide is a 1D HDR image.
First of all we do all the processing in the log domain. There are actually two good reasons for that:
The first reason is that our visual system’s response to brightness is roughly logarithmic;
The second reason is that gradients in the log domain correspond to local contrasts in the original image.

3. The Method in 2D Given: a log-luminance image H(x,y)
Compute an attenuation map
Compute an attenuated gradient field G:
Problem: G is not integrable!
Now let’s review the process in 2D.
Given a 2D log-luminance image H, we compute an attenuation map phi based on the magnitudes of the gradients in H. We’ll see how to construct this map later on.
Then we attenuate the gradient field by multiplying it with the map phi, and we get a new vector field G.
So now, just like in 1D we would like to reconstruct an image by integrating G. But, in the 2D case there is a problem: the modified vector field G is generally not integrable!

4. Solution I minimizes the integral:
Look for image I with gradient closest to G in the least squares sense.
I minimizes the integral:
The way we chose to overcome this problem is by searching for an image I, whose gradient field is closest to our field G, in the least squares sense.
This is done by minimizing the following integral taken over the entire image. The functional F inside the integral simply measures the L2 distance between the gradient field of I and the vector field G.
Poisson
equation

5. Solve
.. 1 … …

6. Attenuation log(Luminance) Gradient magnitude Attenuation map
We do this by computing a (Gaussian) pyramid of our luminance image.
For each level in the pyramid we compute its own gradient magnitude map.
And based on the gradient magnitudes we compute an attenuation map for each level. These map contain a factor between 0 and 1 at each pixel. Small gradients correspond to factors around 1, and high gradients correspond to smaller factors: the higher the gradient, the smaller its factor.
Now we have one such attenuation map for each pyramid level, but we would like to perform the attenuation at the original image resolution, so we will now combine them into a single map at the finest resolution level.
log(Luminance)
Gradient magnitude
Attenuation map

7. Multiscale Gradient Attenuation
Interpolate X =
Interpolate X =
This is done in a coarse-to-fine manner: we interpolate each attenuation map to the next level (click),
There we multiply it with the map already computed for that level to get an accumulated attenuation map (click),
And then we repeat this again and again until we reach the full resolution level.

8. Final Gradient Attenuation Map
So we have just spread the attenuation values from the coarser levels to the finer ones, and accumulate them.
And here’s the final attenuation map for our HDR scene.

9. Performance Measured on 1.8 GHz Pentium 4:
512 x 384: 1.1 sec
1024 x 768: 4.5 sec
Can be accelerated using processor-optimized libraries.
The entire process is fairly fast, as you can see from the timings on this slide.
Note that computation time grows linearly with the number of pixels in the image
Most of the work done by our method can be formulated as image convolutions, so we believe that the performance can be significantly accelerated by making use of the vector processing capabilities of today’s CPUs.

10. Informal comparison Gradient domain [Fattal et al.]
Bilateral [Durand et al.]
Photographic [Reinhard et al.]

11. Informal comparison Gradient domain [Fattal et al.]
Bilateral [Durand et al.]
Photographic [Reinhard et al.]

12. Formal validation
tonemapping
result
HDR display
tonemapping
result