Improved Image Reconstruction Based on Block Compressed Sensing

Qiaoling Wu1,Lin Ni1,Delong He1 1Department of Electronic Engineering and

Information Science University of Science and Technology of China

Hefei, China

qiaolwu@mail.ustc.edu.cn

Abstract—Constrained by traditional sampling theory,it’s

difficult to obtain high resolution image and the sampling

data is great. The theory compressed sensing combines

the sampling and compressing together under the

assumption that the signal is compressible or sparse in a

certain sparse transform domain.Compressed sensing

needs fewer measurements and it can successfully recover

original signal using an optimization process,which will

greatly reduce the complexity of sampling and

calculation.Since traditional algorithm to sample the

whole image is time-consuming and it requires huge

storage space,we study block compressed

sensing.According to the properties of coefficients, only

the high-pass coefficients are measured,then the original

image is reconstructed using the orthogonal matching

pursuit method.Compared with the original algorithm,

simulation result demonstrates that high resolution image

can be obtained with the proposed algorithm,which

reduces the sampling and storage data.The quality of the

reconstruction image is greatly improved.

Keywords-Compressed Sensing; Block; Image

Reconstruction

I. INTRODUCTION

In order to recover the original signal, in the conventional sampling theory ,the sampling rate must

be at least twice the maximum signal frequency (the

Nyquist rate) to avoid signal distortion. The Nyquist theory has become one of the main bottlenecks for

further development in information field. The process

of sampling, compressing and transmitting needs large data storage space,which leads to the waste of resources. In many practical applications,the sampling

efficiency is low and the cost of Nyquist sampling

hardware is high. For above reasons, studying how to

break through the Nyquist sampling theory has become

the key promote development in information field.

Recent years have seen great interest in the paradigm of compressed sensing[1] (CS),which is a new

sampling theory proposed by D. Donoho, E. Candes and

T. Tao. The CS theory states that structured signals can be reconstructed exactly with a sampling rate far

smaller than the Nyquist sampling rate. It proposes to

sample the signal in a space with dense information and to reconstruct it via an optimization process. The CS

theory has been widely used in signal processing,

medical imaging[2-4]etc..However,as the sampling process requires to access the entire target at once and

the process of sparse normally needs multi-layers

wavelet transform,it’s not suitable for real-time sensing and the reconstruction algorithms are generally very

expensive.

In this paper,we propose an improved image reconstruction algorithm based on block compressed

sensing. The basic idea is that the original image is

divided into small blocks and each block is sampled independently using the same measurement matrix and

we only measure the high frequency coefficients.This

algorithm can speed up the reconstruction process and the measurement matrix can be easily stored,so it’s

more advantageous for real-time applications.

The article is organized as follows. In Section II we briefly introduces CS theory .Section III proposes

the improved image reconstruction algorithm based on

block compressed sensing. We present experimental results in Section IV and conclusion in Section V.

II. COMPRESSED SENSING THEORY

A. Sparse Representation If a signal has only a few non-zero elements,then

the signal is sparse.Generally,the natural signal is not This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61172157.

978-1-4577-1964-6/12/$31.00 ©2012 IEEE

sparse,so we first need the sparse representation of the signal.Most coefficients of the transform(usually

DCT,Wavelet etc.) of a signal are small,and we

suppose the transform basis is sparse. Consider a one dimensional ,length-N,real valued

signal 1Nx R ×∈ .We can express the signal x as

1

N

k kkx y yψ

== = Ψ∑ (1)

Where y is a 1N × column vector of

coefficients , Hk k k

y x xψ ψ=< >= ,the N N× basis

matrix1 2

{ , ,... }N

ψ ψ ψΨ = ,in which

, 1, 2...k

k Nψ = is a 1N × column vector.We

assume that the basis Ψ is orthogonal.If y has

only k N elements are non-zero,we call the

signal x k − sparse. B. Measurement Matrix

One of the cores of CS is how to construct

random measurement matrix,making it can maintain the necessary information of the original signal during

random projection.The measurement method should be

non-adaptive and the measurement times should be as little as possible.Donoho point out that the

measurement matrix Φ should satisfy properties

CS1-CS3.Candes prove that Φ must satisfy the

restricted isometric property(RIP)[5].To any k − sparse signal v , Φ should satisfy

2

2

1 1v

vω ω

Φ− ≤ ≤ +

(2)

in which 0ω > . We have known Gaussian random measurement matrix and Bernoulli measurement

matrix[6] as follows:

1: ( , ) , (0,1)M Nij ij

H R H i j h h NM

×∈ = ∼ (3)

1 11: ( , ) , 1 1

2 2

M Nij ij

G R G i j g gM

×−⎛ ⎞

⎜ ⎟∈ =⎜ ⎟⎝ ⎠

∼

(4)

For above two matrix,we can reconstruct a signal

accurately with a certain number of measurements.The Gaussian random measurement matrix is not correlated

with any sparse signal ,so we use it commonly.

C. Reconstruction Algorithm In CS theory,the measurement can be denoted by

, M Ns x R ×= Φ Φ ∈ (5)

s y y= ΦΨ = Θ (6)

Where1 2

{ , ,... }N

ϕ ϕ ϕΦ = is the measurement matrix

and M N . Φ multiply the signal x to pick up some information of x ,so each line of Φ can be seen as a

sensor.Since M N ,the reconstruction of x from the equation(5) is generally ill-posed.However,the signal y is sparse,we can get the sparse coefficients y

by solving the inverse problem of (6),then reconstruct x by the equation (1).

We first define the p -norm of the vector

1 2{ , ,... }

Ny y y y= as ( )1

1

ppN

p iiy y

== ∑ .

According to CS theory,Candes prove that the

reconstruction process is solving the minimum of

0l -norm.

0min ly . .s t s y= ΦΨ (7)

Unfortunately,this problem is difficult to solve because

0l -norm is combinatorial.Solving (7) can sometimes be

attacked using heuristic greedy pursuit algorithm,such as orthogonal matching pursuit,matching pursuit,or

sovling the following convex program instead

1min ly . .s t s y= ΦΨ (8)

namely basis pursuit(BP)[5],can be cast as a linear programming problem and solved using modern

interior-point methods,simplex methods,or other

thchniques.The OMP[7] algorithm is a greedy iterative method of image reconstruction.The steps of existing

OMP algorithms are as follows:

1)Initialize the residual0

r s= ,the index set

0Λ = ∅ ,and the iteration count 1t = .

2)Find the index1,... 1

arg max ,t j d t j

rλ ϕ= −

= < > .

3)Augment the index set and the matrix of chosen

atoms: 1

{ }t t t

λ−

Λ = Λ ∪ ,1

[ , ]tt t λ

ϕ−

Φ = Φ .

4)Solve a least squares problem to obtain a new

signal estimate: 2

arg mint x t

v s v= − Φ .

5)Calculate the new approximation of the data and

the new residual: t t t

a v= Φ ,t t

r s a= − .

6)Increment t and return to step 2 if t m< .

7)The estimate x̂ for the ideal signal has nonzero

indices at the components listed inm

Λ .

III. IMPROVED IMAGE RECONSTRUCTION BASED

ON BLOCK COMPRESSED SENSING

In CS,we first need the sparse representation of the image.The wavelet basis is usually used for sparse

basis.In original algorithm,how many layers of wavelet

decomposition has great impact on reconstructed image.This is because the image is decomposed into

low frequency and high frequency sub-bands and the

high frequency sub-bands can be seen as spares.The low frequency sub-band cannot be seen as sparse

because it’s the approximation signal of the original

image at different scales.It will destruct the correlation of the coefficients if we measure both the high and low

frequency coefficients.On the other hand,we hope the

real-time computing of the whole image.But if the image is large scale,the data of measurement matrix is

large and it’s not conductive to real-time

processing.This paper proposes an improved image reconstruction algorithm based on block compressed

sensing in response to the above shortcomings.This

algorithm is beneficial to real-time processing and the improved algorithm is as follows:

1)Consider a N N× image,the image is first divided into /N n small blocks with size n n× .Let

( 1, 2,... / )i

I i N n= represent each block.

2)Choose appropriate M and construct the same measurement matrix(Gaussian matrix) for each

blocki

I .For the whole image,the equivalent

measurement matrix is thus a block diagonal matrix

,which will reduce the computing data and speed up the reconstruction process.

3)Make sparse representation for each block

( 1, 2,... / )i

I i N n= and get the low-frequency

coefficients ( 1,2,... / )i

L i N n= ,high-frequency

coefficients , , ( 1,2,... / )i i i

H V D i N n= .Then only

measure the high-frequency coefficients. 4)Reconstruct the high-frequency coefficients

using OMP algorithm and we get the high-frequency

coefficients , , ( 1,2,... / )i i i

H V D i N n= .Then make

inverse wavelet transform with the low-frequency

coefficients ( 1,2,... / )i

L i N n= and we get the

reconstructed block images.

5)Mosaic the block images and get the reconstructed image.

IV. EXPERIMENTAL RESULTS

In this paper,a 256 256× lena image is used to conduct experiment.Figure (b1) and (c1) are

reconstructed images by original CS algorithm.We can

see that image(b1) is vague seriously ,which is reconstructed by only one-layer wavelet transform.

Figure(c1) is much better with four-layer wavelet

transform,which show that the wavelet decomposition level plays an important role on the reconstructed

image.Figure (d1), (e1) and (f1) are the images

reconstructed by the our algorithm block in 8*8,16*16 and 32*32.Figure(1) is the reconstructed images with

sampling rate 0.5.Figure(2) is the enlarged images of

figure(1).We can see that (c1) has significant block

effect.We use PSNR to evaluate the reconstruction image.The PSNR results are shown in TableI.Vertical

comparison can be seen that the PSNR is getting higher

as the sampling rate increasing.Horizontal comparison can be seen that the reconstructed image by our

algorithm is better.TableII is the running time of

different algorithm. Observing the vertical form can be seen that the higher the sampling rate,the longer the

time consumed and horizontal comparison can be seen

that our algorithm needs less time which is conductive to real-time processing.

(a1)original image(b1) (c1)original CS algorithm(1 and 4 layer

wavelet transform)

(d1) (e1) (f1)proposed algorithm(block in 8*8,16*16,32*32)

Fig.1 Reconstructed images of the sampling rate /M N =0.5

Fig.2 Enlarged image of(a1),(c1)-(f1)

TABLE I. DATA PSNR

TABLE II. RUNNING TIME

V. CONCLUSIONS

The compressed sensing theory enriches the signal acquisition greatly, and provides a new technology for the research of other related fields.According to CS theory,we study the reconstruction of high resolution image,and proposes an improved image reconstruction algorithm based on block compressed sensing.The experimental results demonstrate that our method is feasible,and the detail of image can be maintained better,and the time consuming is less.

REFERENCES

[1] Donoho D L. Compressed sensing[J]. Information Theory, IEEE

Transactions on. 2006, 52(4): 1289-1306. [2] Guerquin-Kern M, Haberlin M, Pruessmann K P, et al. A Fast Wavelet-Based Reconstruction Method for Magnetic Resonance Imaging[J]. Medical Imaging, IEEE Transactions on. 2011, 30(9): 1649-1660. [3] Potter L C, Ertin E, Parker J T, et al. Sparsity and Compressed Sensing in Radar Imaging[J]. Proceedings of the IEEE. 2010, 98(6): 1006-1020. [4] Paredes J L, Arce G R. Compressive Sensing Signal Reconstruction by Weighted Median Regression Estimates[J]. Signal Processing, IEEE Transactions on. 2011, 59(6): 2585-2601. [5] Candes E J, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[J]. Information Theory, IEEE Transactions on. 2006, 52(2): 489-509. [6] Candes E J, Tao T. Decoding by linear programming[J]. Information Theory, IEEE Transactions on. 2005, 51(12): 4203-4215. [7] Tropp J A, Gilbert A C. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit[J]. Information Theory, IEEE Transactions on. 2007, 53(12): 4655-4666.

sampling rate

/M N PSNR(dB)

/M N =0.25 /M N =0.5

64M = 128M =

original CS (single layer)

original CS (four layer)

block in8*8

block in16*16

block in32*32

5.4081 8.4238

14.6769 26.4967

27.8359 29.9789

28.3536 31.3884

28.6170 33.0879

sampling rate

Time(s) /M N

/M N =0.25 /M N =0.5

64M = 128M =

original CS (single layer)

original CS (four layer)

block in8*8

block in16*16

block in32*32

9.847218 44.167879

8.362016 30.171616

5.222684 5.693933

2.723498 3.169885

2.495036 3.574955