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

Debevec and Malik 1997 Image series Pixel Value Z = f(Exposure) • 1 • 1 • 1 • 1 • 1 • 2 • 2 • 2 • 2 • 2 • 3 • 3 • 3 • 3 • 3 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

For each pixel site i in each image j, want: Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 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

Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec -4.9875 -4.7250 -4.4628 -4.2095 -3.9744 … 2.0197 2.0654 2.1149 2.1686 2.2255 Example of what g(Zij) looks like 256 unknowns 10

Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec -4.9875 -4.7250 -4.4628 -4.2095 -3.9744 … 2.0197 2.0654 2.1149 2.1686 2.2255 Example of what g(Zij) looks like 256 unknowns 10

Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec -4.9875 -4.7250 -4.4628 -4.2095 -3.9744 … 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

Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec -4.9875 -4.7250 -4.4628 -4.2095 -3.9744 … 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

Image Sequence • 1 • 1 • 1 • 2 • 2 • 2 • 3 • 3 • 3 Dtj = 1/64 sec num_samples * num_images equations -4.9875 -4.7250 -4.4628 -4.2095 -3.9744 … 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

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 num_samples * num_images equations Weight each equation by trustworthiness 254 equations … Lock down one point in g() 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

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