Compressive Sensing for Vision - Sparse Representations and Compressive Sensing for Imaging and Vision - page 57

a

Ground truth observation matrix C G

b

†

Estimated observation matrix CL L C C G

c

4 4

2 2

0 0

−2 −2

−4 −4

−6 −6

200 400 600 800 1000 200 400 600 800 1000

Frames
Ground truth state sequence

Frames

Estimated state sequence

Fig. 4.10 Ground truth and estimated parameters corresponding to Figure 4.9 . Shown are the top
10 columns of the observation matrix and state sequences. Matlab's “jet” colormap (red =+ large

and blue = − large) is used in (a) and (b) [127]

definitions, ( 4.28 ) can be written as

dx 0

dy 0

g 0 = = Gs . (4.29)

The curl of the gradient field can be defined as the loop integral around a box of four
pixels

curl (

y , x )= dx ( y + 1 , x ) − dx ( y , x )+ dy ( y , x ) − dy ( y , x + 1 ) ,

which in the matrix form can be written as follows

d = Cg , (4.30)

where d denotes the vector of stacked curl values and C denotes the curl operator
matrix.
Since the gradient field g 0 is integrable, Cg 0 = 0 . However, for a general non-

Decomposing g as the sum of g 0 and a gradient

integrable gradient field g , Cg = 0 .