Compressive Acquisition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 38

Fig. 3.17 An example of relit images of a scene generated from a reflectance field captured using
only 1000 nonadaptive illumination patterns [110]. (a) The scene relit with a high-frequency
illumination condition. (b) The scene relit under a natural illumination condition. (c) A ground
truth photograph of the scene

If we denote multiple illumination conditions as a matrix L =[ l 0 ,···, l m ] and the

corresponding observations as a matrix C =[ c 0 ,···, c m ] then the acquisition stage

can be compactly written as

C = TL .

The observations at the i th pixel are given by the following equation

c i = t i L , (3.27)

where t i is a row of the transport matrix T .From( 3.27 ), one can clearly see the
connection to CS where L plays the role of sensing matrix.
In order to apply the theory of CS to light transport data, one has to find a
basis in which t i are sparse or compressible. It has been shown that certain types
of reflectance functions can be sparse or compressible in spherical harmonics or
wavelet basis. Assuming that we have such an orthogonal basis, B , one can write
the observation matrix as

C = TL

BB T = T ( ) L

TB T L

= , (3.28)

where T = TB . Using one of the CS measurement ensembles φ , one can define the

illumination patterns as L = B φ . Combining this with ( 3.28 ) we obtain

T

B T B C = ( ) φ (3.29)

T

=

φ .

(3.30)