Re: Combining multiple photographic exposures into an HDR image
- From: Jeff Mather <jeff.mather@xxxxxxxxxxxxx>
- Date: Mon, 01 Oct 2007 17:05:02 -0400
aruzinsky wrote:
"Often that response curve is unknown, and you can get surprisingly good results with a simple average."
That should be a simple average before, not after, dividing by exposure
time. In an additive white noise model, the optimum weights for
minimizing the variance of a weighted average are inversely proportional
to the noise variances. In this model, the standard deviation of noise is
proportional to square root of exposure time whereas the signal is
proportional to the exposure time. Thus, a 2 second exposure would have
the same noise std. dev. as adding two 1 second exposures. After scaling
to equal signal strengths by dividing by exposure time, the noise
variances will be inversely proportional to the exposure times and an
average with weights proportional to exposure time will be optimum. This
is the same as using a simple average before dividing by exposure time.
"Underexposed pixels fall below a small threshold and only contribute
noise."
I am unconvinced that such a threshold exists.
Hey look, you're probably right; I'm just going by the information in Reinhard, et al., which I previously mentioned, and by my own experience writing algorithms to make HDR images from multiple exposures.
Here's what Reinhard, Ward, Pattanaik, and Debevec have to say about creating HDR images from multiple LDR exposures:
"By taking multiple exposures, each image in the sequence will have different pixels properly exposed, and other pixels under- or overexposed. However, each pixel will be properly exposed in one or more images in the sequence. It is therefore possible and desirable to ignore very dark and very bright pixels in the subsequent computations.
"Under the assumption that the capturing device is perfectly linear, each exposure may be brought into the same domain by dividing each pixel by the image's exposure time. From the recorded Radiance values L_e, this effectively recovers irradiance values E_e by factoring out the exposure duration.
"Once each image is in the same unit of measurement, corresponding pixels may be averaged across exposures -- excluding, of course, under- and overexposed pixels. The result is an HDR image.
". . . In practice this procedure needs to be refined to include camera response curves, image alignment techniques, and ghost and lens flare removal."
_High Dynamic Range Imaging: Acquisition, Display, and Image-based Lighting_, p. 117-118. (ISBN: 978-0-12-585263-0)
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