Image and Video Compression

177Compressive Sensing

For the two-dimensional (2D) case, the denition of MSE is modied as

follows:

MSE

NN

IijIij

i

N

j

N

1

(, )

ˆ

(, )

11

2

∑∑

=

×

−

==

(8.17)

where xIij xI

ij

;(,)&

ˆ

;

ˆ

(,

)

represents the original and reconstructed sparse

signal in 1D and 2D, N; N × N is the signal dimensionality for 1D and 2D

cases, respectively. The results also indicate the total signal reconstruction

10009008007006005004003002001000

–1

–0.8

–0.6

–0.4

–0.2

0.2

0.4

0.6

0.8

1

0

(a)

10008006004002000

–2.5

–2

–1

–1.5

–0.5

0.5

1.5

2

0

1

(b)

FIGURE 8.10

Results of

1



minimization with equality constraints (basis pursuit): (a) original one-dimen-

sional signal: N = 1024, k = 50; (b) improperly reconstruc ...