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IEEE SIGNAL PROCESSING MAGAZINE [154] MARCH 2015

■ data handling since the size of vectorized data and the

associated dictionary B R

easily becomes prohibitively

large (see the section “Large-Scale Data and the Curse of

Dimensionality”), especially for tensors of high order.

Fortunately, tensor data are typically highly structured, a per-

fect match for compressive sampling, so that the CS framework

relaxes data acquisition requirements, enables compact storage,

and facilitates data completion (i.e., inpainting of missing samples

due to a faulty sensor or unreliable measurement).

KRONECKER-CS FOR FIXED DICTIONARIES

In many applications, the dictionary and the sensing matrix

admit a Kronecker structure (Kronecker-CS model), as illustrated

in Figure 7(a) [84]. In this way, the global composite dictionary

matrix becomes

,WW W W

() ( ) ()NN11

777g=

where each

term W B

() () ()nnn

U= has a reduced dimensionality since

B R

()n II

and .R

()nMI

Denote MMM M

N12

g= and

,III I

N12

g= then, since ,M I

# ,, , ,nN12f= this reduces

storage requirements by a factor of ()/().I MMI

nn n

R The compu-

tation of Wg is affordable since g is sparse; however, computing

W y

is expensive but can be efficiently implemented through a

sequence of products involving much smaller matrices W

()n

[85].

We refer to [84] for links between the coherence of factor matri-

ces

()n

and the coherence of the global composite dictionary

matrix .W

Figure 7 and Table 3 illustrate that the Kronecker-CS model

is effectively a vectorized TKD with a sparse core. The tensor

equivalent of the CS paradigm in (13) is therefore to find the

sparsest core tensor

G such that

,WW WYG

() () ( )

## #g, (14)

with ,KG

# for a given set of modewise dictionaries B

()n

and

sensing matrices

()

U ,, ,().nN12f= Working with several

small dictionary matrices, appearing in a Tucker representation,

instead of a large global dictionary matrix, is an example of the

use of tensor structure for efficient representation; see also the

section “Large-Scale Data and the Curse of Dimensionality.”

A higher-order extension of the OMP algorithm, referred to as

the Kronecker-OMP algorithm [85], requires

K iterations to find

the K nonzero entries of the core tensor .G Additional computa-

tional advantages can be gained if it can be assumed that the K

nonzero entries belong to a small subtensor of ,G as shown in

Figure 7(b); such a structure is inherent to, e.g., hyperspectral

imaging [85], [86] and 3-D astrophysical signals. More precisely, if

the

K L

= nonzero entries are located within a subtensor of size

(),LL L## #g where ,L I

% then, by exploiting the block-

tensor structure, the so-called N-way block OMP algorithm

(N-BOMP) requires at most NL iterations, which is linear in N

[FIG7] CS with a Kronecker-structured dictionary. OMP can

perform faster if the sparse entries belong to a small subtensor,

up to permutation of the columns of ,W

()1

()2

and .W

()3

[FIG8] The multidimensional CS of a 3-D hyperspectral image

using Tucker representation with a small sparse core in wavelet

bases. (a) The Kronecker-CS of a 32-channel hyperspectral image.

(b) The original hyperspectral image-RGB display. (c) The

reconstruction (SP = 33%, PSNR = 35.51 dB)-RGB display.

THE WORLD’S NEWSSTAND

THE WORLD’S NEWSSTAND

(3)

(2)

(1)

⊗

Sparse Vector Representation (Kronecker-CS)

Measurement Vector (CS)

Sparse

Vector

× I

)

× M

)(M

× l

)(l

× l

)(M

× l

)

× I

)

Measurement Tensor (CS)

Block Sparse

Core Tensor

Block Sparse Tucker Representation

⊗

(a) Vector Representation

(b)

Tensor Representation

(a)

(b)

(c)

(1,024 × 1,024 × 32) (256 × 256 × 32)

× M

)

(

× l

)

(3)

× I

)

(1)

(2)T

= 32

= 585

= 1,024

= 32

= 1,024

× M

)

(

× I

)