Compressive Acquisition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 19

N denote an image. Any particular pixel of X is written as X n , m .The
discrete directional derivatives on X are defined pixel-wise as

N ×

Let X ∈ C

( X x ) = X n , m − X n − 1 , m

n , m

( X y ) n , m = X n , m − X n , m − 1 .

N × N × 2

Based on these, the discrete gradient operator ∇ where ∇ X ∈ C is defined as

( ∇ X ) n , m =(( X x ) n , m , ( X y ) n , m ) .

From these operators, one can define the discrete total-variational operator TV or
| ∇ | on X as

( TV [ X ]) n , m =( | ∇ | ( X )) n , m

= | ( ) |

2

+ | ( ) |

2

,

X x X y (3.1) n , m n , m

from which one can also define the total-variation seminorm of X as

X TV = TV ( X ) 1 .

It is said that X is K -sparse in gradient (or in the total-variational sense) if

| ∇ | ( K .

The objective is to recover an image X that is K -sparse in gradients from a set of

X ) 0 =

N 2

M Fourier measurements. To that end, define a set Ω of M two-dimensional

frequencies ω k =( ω x , k , ω y , k ) ,1 ≤ k ≤ M chosen according to a particular sampling

2 .Let

pattern from { 0 , 1 ,···, N − 1 } F denote the two-dimensional DFT of X

m = 0 X ( n , m ) ex p − 2π i n ω x

N − N −

1
n = 0

1

m ω y

N

F ( ω x , ω y )=

N ,

F − 1

and its inverse

ω y = 0 F ( ω x , ω y ) ex p 2π i n ω N ,

1
ω x = 0

N − N − 1

1
N 2

m ω

y

F − 1 {F ( ω x , ω y ) } = X ( n , m )= .

N

N × N M

Next define the operator F Ω : C → C as

( F Ω X ) k =( F X ) ω k (3.2)

F Ω i.e. Fourier transform operator restricted to will represent its conjugate

adjoint. Equipped with the above notation, the main problem considered in [108]
can be formally stated as follows:

Ω .

N 2 frequencies and Fourier observations of a K -sparse in 3.1. Given a set Ω of M

gradient image X given by F Ω X , how can one estimate X accurately and efficiently?