4.1 HDR
Figure 4.1: Image Acquisition Pipeline
From Figure 1: Image Acquisition Pipeline, we could see how the scene radiance(X) is nonlinearly mapped into its final digital values(Z). In other words, f(X)=Z, the composition of the characteristic curve of a device and all the nonlinear mapping, is a nonlinear function. Nonlinearity is mostly introduced by later processing steps such as analog to digital conversion and remapping. Therefore, our method to construct a High Dynamic Range(HDR) picture is to first recover the response curve, f(X)=Z and then by using the pixel values from the series of (LDR) pictures, we're able to get the scene radiance, which is like a reversed process. finally, our HDR picture will be constructed from these values of scene radiance.
Our general process will be:
1. We'll take pictures with different exposure times.
a) Knowing the exposure X and the exposure time delta t, we're able to recover irradiance through formula E = X/delta t.
b) Since f(X)=Z and function f could be reasonably conceived as increasing function, its inverse function is well defined. Then, we have
Figure 4.2
c) Using the formulas in a) and b), we have
Figure 4.3
where i is the index of sample (pixel) and j is the index of pictures with different exposure times. The formula also equals to
Figure 4.4
taking natural log on both sides, we will have
Figure 4.5
Making X=g(Z) the inverse function of Z=f(X), we get 
Figure 4.6
d) Thus, we transform our problem into a problem minimizing least-squared error between g(Z) and ln(E) and ln(delta t), which could be summarized by the following formula:
Figure 4.7
Note that the second term here is for smoothing g(Z). And lambda is a scaling factor subject to different pictures.
3. After we get the response curve by solving the equation above, when we use the pixel values Z of those LDR pictures, we'll be able to get the values of irradiance X at each point of the scene. In other words, we're trying to figure out in different areas, which group of pixel values from different LDR pictures with different exposure times is closest to the corresponding irradiance and best to reveal details of that area. Thus, it enables us to construct HDR picture in which pixel values in different areas are extracted from the LDR picture that captures its details best.
4. We finish our task to construct a HDR picture here. However, we still have to solve another problem, which is to display a HDR picture on a LDR display. And this leads to the second part of our implementation: Tone-mapping.
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1This content is available online at <http://cnx.org/content/m45500/1.1/>.