1. Recovering HDR radiance maps for Project 5
James Hays
CS 129 Fall 2012

2. Debevec and Malik 1997 Image series Pixel Value Z = f(Exposure)
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Dt = 1/64 sec
Dt = 1/16 sec
Dt = 1/4 sec
Dt = 1 sec
Dt = 4 sec
Assumption – f is monotonic and invertible. Probably true.
Pixel Value Z = f(Exposure)
Pixel Value Z = f(Radiance * Dt)
f -1(Pixel Value Z) = Radiance * Dt
g(Pixel Value Z) = ln(Radiance) + ln(Dt) 10

3. For each pixel site i in each image j, want:
Image Sequence
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Dtj = 1/64 sec
Dtj = 1/16 sec
Dtj = 1/4 sec
For each pixel site i in each image j, want:
g() is the unknown discrete inverse response function.
Zij is the known (but noisy) pixel value from 0 to 255 for location i in image j.
Ei is the unknown radiance at location i.
Δtj is the known exposure time, e.g. 1/16, 1/125, etc. for image j
Equation 2 in Debevec 10

4. Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec …
2.0197
2.0654
2.1149
2.1686
2.2255
Example of what g(Zij) looks like
256 unknowns 10

5. Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec …
2.0197
2.0654
2.1149
2.1686
2.2255
Example of what g(Zij) looks like
256 unknowns 10

6. Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec …
2.0197
2.0654
2.1149
2.1686
2.2255
Example of what g(Zij) looks like
256 unknowns
We expect g() to vary smoothly 10

7. Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec …
2.0197
2.0654
2.1149
2.1686
2.2255
Add constraints on second derivative of g()
Example of what g(Zij) looks like
256 unknowns
We expect g() to vary smoothly 10

8. Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec
num_samples * num_images equations …
2.0197
2.0654
2.1149
2.1686
2.2255
Example form of g(Zij)
256 unknowns
254 equations …
We expect g() to vary smoothly
Lock down one point in g() 10

9. Image Sequence We can’t trust all of these values
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• 1
• 1
• 2
• 2
• 2
• 3
• 3
• 3
Dtj = 1/64 sec
Dtj = 1/16 sec
Dtj = 1/4 sec
num_samples * num_images equations
Weight each equation by trustworthiness
254 equations …
Lock down one point in g() 10

10. Image Sequence We can’t trust all of these values
• 1
• 1
• 1
• 2
• 2
• 2
• 3
• 3
• 3
Dtj = 1/64 sec
Dtj = 1/16 sec
Dtj = 1/4 sec
Weight each equation by trustworthiness … 10

11. x = A\b A x = b How do we convert this to Ax=b form for Matlab? …
256
num_samples * num_images equations
num_samples * num_images
non-zero entries
num_samples 1 …
254
254 equations 1
256 + num_samples 1
x = A\b
1 equation 10