The Light Transport Matrix expresses the relationship between the light source (e.g., projector or display) and the camera in linear form. It can represent as follows:

c = T p

The column vector p is the projected pattern (pxq, 1). The column vector c is the image captured by the camera (mxn, 1). The matrix T is the light transport matrix (mxn, pxq). Where p and q are the sizes of the projected pattern, and m and n are the sizes of the image captured by the camera.

The light transport matrix is used for relighting [Debevec et al., 2000] and dual-photography [Sen et al., 2005].

This project uses multiplexed illumination [Schechner et al., 2003] to acquire the light transport matrix. Multiple sources are illuminated in a single capture in the measurement, and the matrix is estimated after capturing images. Here I use the unstructured light pattern [Couture et al., 2011] for projection. This method can be set to any spatial frequency to generate random patterns with multiple frequencies.

Suppose N images are taken (note: N must be greater than the number of light sources (N>=pxq)). Stack the vectors c and p to create the matrices cn (mxn, N) and pn (pxq, N). The relation to the light transport matrix T (mxn, pxq) is as follows:

cn = T pn

If there is a pseudo-inverse pn+ of matrix pn, then matrix T can be calculated as follows:

T = cn pn+

Here is the setup for the measurement. A display (EIZO, FlexScan EV2451) is used as a light source. The original resolution of the display is 1920x1080, but I set it to 48x27 (i.e. p=48, q=27) to reduce the measurement time. The image is taken with a color camera (FLIR, BFS-U3-23S3C-C) set to 480x300 (i.e. m=480, n=300). In the scene, a color checker and glass objects are placed.

Here are the results of relighting using the light transport matrix.

This repository provides the measured data and programs. So you can do relighting too.

You can download the measured data from here. This data is set to 16x9 display resolution (i.e. p=16, q=9) to reduce the size.

• NumPy
• OpenCV
• scikit-image
• structuredlight (https://github.com/elerac/structuredlight)

• 01_calculate_light_transport.py: Calculate the light transport matrix from measured data.
• 02_relighting.py: Relighting with the light transport matrix.
• 03_visualize.py: Visualize the distribution from light sources at specific camera pixel.

Screen shot of 03_visualize.py.

• Light transport matrices (Computational Photography, CMU)
• Debevec et al., "Acquiring the reflectance field of a human face." SIGGRAPH 2000.
• Sen et al., "Dual photography." SIGGRAPH 2005.
• Schechner et al., "A theory of multiplexed illumination." ICCV 2003.
• Couture et al., "Unstructured light scanning to overcome interreflections." ICCV 2011.