http://www.uib.no/People/nmaxt/papers/Lysaker_Lundervold_Tai_ieee_tip_dec_2003.pdf

http://www.uib.no/People/nmaxt/papers/Lysaker_Lundervold_Tai_ieee_tip_dec_2003.pdf
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
1579
Noise Removal Using Fourth-Order Partial
Differential Equation With Applications to Medical
Magnetic Resonance Images in Space and Time
Marius Lysaker, Arvid Lundervold, and Xue-Cheng Tai
Abstract—In this paper, we introduce a new method for image
smoothing based on a fourth-order PDE model. The method is
tested on a broad range of real medical magnetic resonance images, both in space and time, as well as on nonmedical synthesized test images. Our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail, even at poor signal-to-noise ratio. We have also compared our
method with related PDE models.
Index Terms—Fourth-order partial differential equations, MRI,
noise removal, nonlinear filtering and enhancement, restoration.
I. INTRODUCTION
D
URING the last two decades progress in medical magnetic
resonance imaging (MRI) technology has created a large
collection of imaging techniques available to both clinicians and
researchers. Each such pulse sequence exploits some specific
physical or chemical property of the hydrogen nuclei (protons)
of small, mobile molecules like water and lipids, and can depict
structural and functional information from living tissue at the
sub-millimeter scale.
Even if the scanner technology has undergone tremendous
improvements in spatial resolution, acquisition speed, and
signal-to-noise ratio, MR images are still hampered with
degradations like signal intensity inhomogeneities (bias fields),
noise, and other artifacts. One of the limiting factors regarding
performance and usefulness of quantitative MRI diagnostics,
such as voxel-based tissue classification, extraction of organ
shape or tissue boundaries, estimation of physiological parameters, e.g., tissue perfusion and contrast agent permeability from
dynamic imaging, is the amount of noise in the acquisitions.
A major source of this type of image degradation is random
thermal noise entering the MR data in the time domain [1]
(explained in more detail in Section II).
To overcome the deficiencies of so-called acquisition-based
noise reduction, such as increased acquisition time (i.e., time av-
Manuscript received August 8, 2002; revised June 26, 2003. This work was
supported by the Norwegian Research Council under Projects 135420/431 and
135302/320, Locus on Neuroscience (A.L.), and the NSF under Contracts NSF
ACI-0072112, NSF INT-0072863, and ONR-No0014–96-1–10277. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Giovanni Ramponi.
M. Lysaker and X-C. Tai are with the Department of Mathematics, University of Bergen, N-5009 Bergen, Norway (e-mail: [email protected],[email protected]).
A. Lundervold is with the Department of Physiology, University of Bergen,
N-5009 Bergen, Norway (e-mail: [email protected]).
Digital Object Identifier 10.1109/TIP.2003.819229
eraging over repeated measurements) or decreased spatial resolution (i.e., enlarging voxel volume), several post-processing
noise reduction methods have been proposed.
The ultimate goal of post-scanning noise removal methods in
MRI is to obtain piecewise constant, or slowly varying signals
in homogeneous tissue regions while preserving tissue boundaries. However, no single method has shown to be superior to all
others regarding noise removal, boundary preservation, robustness, user interaction, applicability to the different MR acquisitions techniques, and computation cost. Thus, improvements
are still needed.
In the literature, both statistical approaches and diffusion
filter methods have been used to remove noise from digital
images. An early study of “image improvement” in MRI using
statistical approach has been done in Godtliebsen [2] (see also
[3], [4]). Using Markov field models with maximum a posteriori (MAP) estimation of restored intensity values by iterated
conditional modes (ICM) and simulated annealing (SA), it
was found that application of the best of these methods to an
average of two measurements gave an image of about the same
quality as an average of four single measurements. We shall
also mention the work of Soltanian-Zadeh and coworkers [5],
where some nonlinear filter for multispectral (vector-valued)
MR image restoration using the zero-mean white Gaussian
model for statistical noise was introduced.
On the other hand, considerable research interest and numerous applications in MRI have been devoted to use nonlinear
diffusion filters; see [6]–[22]. One of the common features of
the nonlinear diffusion filters is to introduce a nonlinear diffuis small when
sion term where the diffusion coefficient
) is large. The images are filtered by exthe gradient (i.e.,
amining their evolutions under nonlinear PDE’s. In contrast to
linear diffusion filtering, i.e., Gaussian filtering, which is hampered with blurring and localization problems, the nonuniform
(anisotropic) process reduces the diffusivity at locations which
have a larger likelihood of being edges by using the “edge-stop. This idea was introduced in Peronaping” function
Malik [6] and it has been generalized to 3-D images by Gerig
et al. [7], and to vector-valued images by Sapiro and Ringach
in [23], making it suitable to both 3-D MRI and multispectral
MRI.
Black et al. [8] studied the relation between anisotropic diffusion and robust statistics. They implemented a robust estimation procedure that estimated a piecewise smooth image from a
noisy input image and demonstrated improved automatic stopping of the diffusion process with preservation of sharp bound-
1057-7149/03$17.00 © 2003 IEEE
1580
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
aries and better continuity of edges compared to the PeronaMalik method. Their rationale for applying robust statistics to
anisotropic diffusion was from the case of piecewise constant
images but was not extended to more general ones, like MRI,
where there can be regions with slowly varying signal intensities and unsharp tissue boundary zones due to partial volume
effects.
It seems that Rudin-Osher-Fatemi [10] was the first one to
observe that if we minimize the total variation (TV) norm of
the image under some given conditions, we will get a nonlinear
diffusion filter. This idea gives a rigorous mathematical tool
to introduce nonlinear diffusion filters and has been used as a
regularization method for many applications where one needs
to identify discontinuous functions. Motivated by the TV-norm
filter, many similar filters have been proposed in the literature,
see (e.g., [11]–[20], [22], etc.). Our approach also belongs to
this class of filters. We shall introduce a fourth-order PDE noise
suppression method that handles edges and works on 1-D signals, as well as 2-D, 3-D, and 3-D+time images. The motivation
for proposing the new filter is to overcome the staircase effect
that occurs with the TV-norm filter (see Fig. 2) and better preserve the fine details in nonblocky images (see Fig. 5). In order
to test its practical potential we have applied our method to a
wide range of real images, including structural and functional
MRI data sets. The main strength of our method is the ability
to process signals with a smooth change in the intensity value.
This is often the case for MRI data (see Fig. 13). Compared with
some other fourth-order nonlinear filters, see [16], [18], [21],
[20], [22], [24], our approach is simpler and we only need to
know the approximate noise level. We shall also compare our
method with some related works, and it seems that our scheme
is rather robust in removing noise and handling edges.
The rest of this paper is organized as follows. Since MRI is
main target for our noise-removal method, Section II is devoted
to the principles of MR Fourier transform image formation and
how noise is introduced into the reconstructed images. Section III introduces a general formulation of the PDE model and
in Section IV the fourth-order PDE is adapted to this framework.
Section V explains the numerical technique we have employed.
We point out some advantages/disadvantages using higher order
PDE models for denoising in Section VI. Section VII is devoted
to numerical examples. First, on synthesized images and natural
scenes, then on a broad range of real MR images obtained from
MR phantoms, healthy volunteers, and patients. The test images
are acquired on clinical scanners from two different MR laboratories, and challenges our method by containing various amount
of noise and structural detail.
II. MR IMAGE FORMATION AND NOISE
The following figure illustrates the principles of 2-D Fourier
transform imaging on clinical MR scanners.
Today, variants of this Fourier transform imaging technique
is the most widespread MR method for obtaining structural and
functional information from the living human body.
Noise in MRI enters the data samples in -space. Here the
noise voltage competes with the NMR signal and is due to
random fluctuations in the receiving coil electronics and in the
Fig. 1. Principles of MR image formation. In MRI, we are sampling
complex-valued signals S k ; k
in the spatial frequency domain
(k -space), being time-domain signals from two independent electronic
sources (here denoted the “real” and “imaginary” channel). The signal
values in k -space (i.e., raw data) can be expressed as S k ; k
x; y
i xk
yk
dx dy , where x; y is the spatial
distribution of proton spin density that we want to measure. “Filling” of
k t ;k t
k -space is done by designing k -space trajectories, t
G d; G d , where the functions G t and G t are
programmable time courses of the magnetic field gradients (cf. the coils in
upper right insert). is the gyromagnetic ratio of the imaged nucleus (protons).
Thus, during an imaging examination k -space is traversed by proper activation
of the gradient system. Usually, k -space points are separated by k and k
in the two orthogonal directions, and the NMR signals can be sampled on a
regular 2-D grid. We see from the first equation above, that spatial distribution
of proton spin density x; y in complex-valued image space ( and
images), and the NMR signal S k ; k in complex-valued spatial frequency
space ( and data matrices) define a Fourier pair. Thus, the image of proton
spin densities can be formed by applying a discrete, inverse Fourier transform
to the data matrix of k -space samples. Generalization to 3-D imaging involves
application of the third gradient coil (G t ) in the same fashion. For visual
inspection or further processing the magnitude image (Mag) is used, and
sometimes also the phase image (Pha) (with possible phase unwrapping).
denotes the static magnetic field. M ; t is the magnetization vector at
position , rotating at (resonance) frequency ! . The total magnetic field
is under control of the pulse sequence program activating the gradient- and
RF (radio-frequency) coils. In receive mode, these coils are also used to pick
up the weak NMR signals.
( ) exp f0 (
(
()
(
)
+
)g
) =
(
( )
7 ( ( ) ( )) =
!
()
()
() )
1
R
I
( )
(
1
X
)
Y
()
B
B
r
(r )
= B
patient body (e.g., Brownian motion of spins). The variance
of this thermal noise can be described as the sum of noise
variances from independent stochastic processes, representing
the body, the coil and the electronics [1], i.e.,
The noise-removal obtained by averaging
measurements,
where noise in -space is additive i.i.d. zero-mean Gaussian
with standard deviation , yields an improvement in noise stan. This procedure of averaging,
dard deviation of
on expense of measurement time, is used in the experimental
part of this paper to obtain the “true” MR image for evaluation
purposes. Notice that in reconstructed magnitude images,
the images used in most medical applications, the noise will
generally not be Gaussian distributed. This is because computation of the magnitude image from the real and imaginary
image, reconstructed by the inverse Fourier transform, involves
a nonlinear operation which maps a Gaussian distribution to
a Rician distribution (e.g., [25]). Moreover, at high SNR the
LYSAKER et al.: NOISE REMOVAL USING FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATION
Fig. 2.
Fig. 3.
1581
One-dimensional signal evaluation. (a) Original curve, (b) noisy curve, (c) m = 1, (d) m = 2, and (e) m = 3.
A small portion of the results achieved with different PDEs. (a) 2th order (18). (b) 4th order (24). (c) 4th order (13). (d) 4th order (11).
Rician distribution approaches the Gaussian distribution, and
at low SNR it tends to be the Rayleigh distribution. Thus, there
are signal-dependent noise characteristics in reconstructed
magnitude MR images. However, our noise suppression PDE
method, like most other de-noising methods, does not incorporate these noise-statistics subtilities in the model (but see
[26] for a “Rician noise” model using wavelets), and we have
used addition of i.i.d Gaussian noise in the experiments with
nonmedical test images to evaluate our algorithm. For more
details on MR imaging principles and the presence of noise,
see [1], [27], [28].
III. PDE-DENOISING
Let
be a digital image and
be its obserfor
. Noise is
vation with random noise
superimposed on the pixel intensity value by the formula
. Assume the noise level is
approximately known, i.e.,
(1)
Since noise can be recognized as fast oscillating signals over
small areas, one essential idea for denoising digital images is to
1582
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
the total variation norm of
. We propose two different functionals to measure oscillations in a noisy data:
(5)
Here, we describe the models for 2-D problems, but a generalization to higher dimensions are given by
Fig. 4. Noise introduced in right part.
filter out high frequency signals while preserving the important
can be used to
features in the images. Different functionals
measure the oscillations in an image and a general formulation
of the noise removal problem (cf. [18], [10]) is to solve
(2)
subject to
We search for a new image by minimizing
constraints involving the statistics of the noise. Rudin-Osher. In
Fatemi [10] proposed the functional
[12], [15], [16], some modified TV-norm filters were applied.
Higher-order derivatives have recently been used to measure the
oscillations. You-Kaveh [20] proposed the functional
and
.
The main difference between the two functionals is that
is rotational invariant while
is not. However,
is simpler for higher dithe implementation with
mensional problems. Another alternative would be choosing
. This coincide with choosing
in (3), see [20]. Below we supply all the details when the
is used and point out the corresponding
functional
without going into the details. A technique
formulas for
subject to the noise level
for finding minimum values of
constraint (1) is based on a Lagrangian functional
(6)
we must have
,
At any critical point of
. Here, the differentiation means Gauteaux
derivatives. The first optimality condition gives
(7)
and this essentially recovers constraint (1). From the definition
, we can calculate
of
(3)
denotes the Laplacian operator and Chan, Marquina
where
and Mulet [18] used
(4)
is an elliptic operator and they restrict themselves
where
. As the functionals get more and more
to work with
complex, the computing time grows as well. This is usually not
an important issue for 1-D and 2-D data, but as the number of
unknowns increases, the choice of functional will have a significant impact on the computing time. In medical diagnostics,
2-D+time, 3-D and even 3-D+time MR images are used, and the
denoising process is quite expensive. We therefore search for an
image processing model with low computing time, which suppresses noise and handles edges in best possible way.
IV. FOURTH-ORDER PDE
In the original TV-norm filter [10], one tries to minimize the
total variation norm of the function . Our idea is to minimize
(8)
By using integration by parts twice for the two first terms in
(8), the second optimality condition implies that satisfies the
following nonlinear partial differential equation
(9)
and the boundary conditions are given as
(10)
is the outward normal direction on
.
where
The above boundary conditions are correct only if the domain
. To solve the nonlinear (9)
is a rectangular domain
LYSAKER et al.: NOISE REMOVAL USING FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATION
1583
Fig. 5. Evaluation of contour plot. (a) Original contour, (b) noisy contour, (c) 2th order (18), (d) 4th order (24), (e) 4th order (13), and (f) 4th order (11).
Fig. 6. Two-dimensional image evaluation with different PDEs. (a) Noisy image, (b) 2th order (18), (c) 4th order (13), and (d) 4th order (11).
Fig. 7.
Two-dimensional image evaluation with rotated objects. (a) Input SNR
we use a parabolic equation with time as an evolution parameter,
i.e., we solve for
(11)
The value of needs to be determined in such a way that the above
equation has a steady state, and condition (1) must be fulfilled at
the steady state. Using the same idea as in [10], we multiply (11)
and use integration by parts twice over to get
by
(12)
4; (b) 4th order (11); (c) Input SNR 4; and (d) 4th order (11).
If we apply the Lagrangian method for
and go through a
similar calculation as described above, we will end up with the
nonlinear PDE:
(13)
Here,
we
have
used
the
notation
.
The
boundary
condition for the above equation is rather complicated.
Moreover, it is not easy to describe the discretized realization
1584
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
Fig. 8. Evaluation of our smoothing algorithm, implemented for isotropic 3-D images. Only one of the 32 contiguous slices, transectioning the lateral ventricles,
is shown. (a) AC = 4, (b) Input AC = 1, and (c) Output 4th order PDE.
Fig. 9. Detail from Fig. 8. (a) AC = 4, (b) Input AC = 1, and (c) Output 4th order PDE.
For image problems, we will take
,
and
is the size of image support. In Table I a survey of the notations
that occur throughout this section is given.
is introduced to avoid numerical
The parameter
divisions by zero numbers. In accordance with Table I we approximate a solution for (11) by the scheme:
(14)
Fig. 10.
Difference between input and output image, i.e., the entire slice.
of the boundary condition. We shall omit the details about this
in the present work.
V. DIFFERENCE EQUATION
A digital image is often defined on a rectangular domain
with a regular mesh
To incorporate the boundary conditions (10), the approximations at the boundaries need to be treated properly.
To explain this in detail, let us introduce two functions
,
. In the implementations,
for a given function are evaluated
the values of and
according to the following formulas:
if
if
if
if
if
or
(15)
(16)
Boundary condition (10.a) is incorporated in (15), and
boundary condition (10.b) is incorporated in (16). Similar
treatment should be done for the -spatial variable.
At each iteration we have to update in accordance with
we use the
(12). To discretize
LYSAKER et al.: NOISE REMOVAL USING FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATION
1585
2
Fig. 11. The plastic straws MR phantom used to test our de-noiser’s preservation of edge information. Each pixel is approximately 0.15 0.30 mm . The k -space
average of eight measurements (AC = 8) was taken to represent the “ideal” image. Only a portion of one of the five transaxial slice images is shown. (a) AC = 8,
(b) Input AC = 1, and (c) Output 4th order PDE.
Correspondingly, given a noisy signal
sion, the algorithm of [10] takes the form:
in one dimen(19)
From (11) and (13) we see that our fourth-order schemes in the
one dimensional case reduce to:
(20)
To see what kind of fundamental differences these two updating
formula may have, we look at two related linear parabolic equations, i.e.,
(21)
Fig. 12. Axial slice from a dynamic susceptibility contrast MR perfusion
imaging study. The curve shows signal intensity versus time in a selected pixel
located in perfused brain parenchyma.
approximation as indicated in Table I and shown in (17) at the
bottom of the page.
VI. THE FOURTH-ORDER PDE COMPARED WITH
SECOND-ORDER PDE
A
We shall compare our proposed algorithm with the
well-known algorithm of [10]. To give an indication that our
scheme might have better properties than the scheme of [10],
we shall consider some one dimensional problems. The scheme
.
of [10] is obtained from the functional
Similar to (11), we need to solve the following parabolic
equation to steady state for the approach of [10].
(18)
and fix the nonThese equations are obtained if we let
and
to be 1 in (19)
linear diffusion coefficient
and (20) respectively. By using separation of variables the solutions for the two linear equations in (21) are
and
Suppose the initial values
posed of signals of different frequencies, i.e.,
and
the solutions of (21) are given by
(22)
are com. Then
(23)
, the signal is damped out by a
For a given frequency
at time by the second-order parabolic equation
factor
at time by the fourthand it is damped out by a factor
order parabolic equation. Higher-order PDE’s damp out signal
with high frequency faster, and we expect the nonlinear case
(17)
1586
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
Fig. 13. 4th order PDE smoothing of 2D+time data from a MR perfusion imaging study. (a) Original pixel time course. (b) Smoothed time course. (c) Difference
between (a) and (b).
TABLE I
SURVEY
OF THE NOTATIONS THAT
THROUGHOUT SECTION V
OCCUR
also has this important property. By a similar argument as given
’th order PDE is able to damp
above it is easy to show that a
by a factor
. However, there
out a frequency
might be some drawbacks by using higher-order PDE’s. Let us
’th order parabolic equation in 1-D:
consider the following
Here
denotes the ’th order derivative of and
is an
acts as a penalty term to
integer. The term
the right hand side equals zero if
oscillations, and for
is constant. If
the penalty term equals zero also if
is linear, and for
the penalty term is zero when is
quadratic. Assume we want to identify a homogeneous region
which is exposed with noise. There is no reason to expect that
, 3 is able to do a better job than
. On the other
, 3 would give a better result
hand, we shall expect that
for nonblocky images. In the next section we shall give a quantitative evaluation for different values with a one dimensional
problem; see Fig. 2.
VII. EXPERIMENTAL RESULTS
A. Synthesized Images and Natural Scenes
To compare the effect for schemes with different ’s, we
),
have implemented and tested the second-order scheme (
) and the sixth-order scheme
the fourth-order scheme (
) in 1-D. A function of constant, linear and curved re(
gions was exposed with noise and used as observation data. For
each scheme we used
in accordance with the CFL-condition
described later. Otherwise, all input parameters were identical
and each scheme was solved to steady state.
. HomoAs expected the staircase effect is visible for
. The scheme
geneous regions is not restored properly for
seems to have a good balance between these things;
with
do
see Fig. 2. In [29] they stated that linear schemes for
not obey the minimum-maximum principle and we assume the
same also is true for the nonlinear case. Even so, we have never
was
experienced any artifacts or stability problems as long as
chosen small enough (CFL-condition is discussed below). The
rapid oscillation around some jumps in Fig. 2(e) may indicate
.
that artifacts is produced with
Below we report some 2-D experimental results where we
compare our fourth-order methods with two related works [10],
[20]. The second-order method in [10] is already given by (18).
In [20] they proposed the functional
and the Euler equation was solved through the following gradient descent procedure:
(24)
and k is an image dependent pawhere
rameter introduced to avoid numerical instabilities in planar re, it is
gions. In order to produce such a nonlinear diffusion
should be
worthy to note that the corresponding
. The gradient descent method
(24) produces speckles in the processed image and a despeckle
algorithm needs to be introduced. We shall not give the details of
the despeckle algorithm. However, we have used it in the comparisons in Fig. 3.
To evaluate the performance of the different PDE
models we look at the signal-to-noise-ratio (SNR) before and after processing. SNR is given by the formula
First we did a
similar test as described in [20] in which i.i.d Gaussian noise
was added to the Lena image. To achieve the result in Fig. 3(b)
we have used all parameters and the despeckle algorithm as
. To really visualize different
reported in [20] for
performance of the schemes only a small portion of the images
is depicted here.
Steady state was achieved for all schemes in less than 350
iterations, except fourth-order scheme (24) where we fixed the
number of iterations to 1000 in accordance with [20]. The major
differences between the results in Fig. 3 are the staircase effect
LYSAKER et al.: NOISE REMOVAL USING FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATION
1t
1587
TABLE II
, SNR, AND L -NORM FOR EACH OF THE SCHEMES
visible in Fig. 3(a) (see Lena’s cheek) and some speckles seen in
Fig. 3(b). For all the schemes, we have observed that there exists
such that the schemes are unstable when
.
a
, the iterative solution explodes. On the other
For
hand, the solutions of the schemes go to a steady state if we
. To find
, we do many simulations
choose
using the same initial data. Starting with a small , we increase
bigger than
it by a small constant for each simulation. For
a constant, the scheme becomes unstable and this is the
we report in Table II. In this test, the Lena image was used as
initial data. In Table II we also report the SNR and the -norm
and
for each of the schemes. Before processing
.
The fourth-order schemes (13) and (11) have the best performances with respect to both SNR and -norm. Regarding
the computational efficiency there are minor changes between
second-order (18) and fourth-order scheme (11) regarding the
cost per iteration. Evaluation of the mixed derivatives in fourthorder scheme (13) and the use of despeckle algorithm in fourthorder scheme (24) makes these two schemes slower.
To further distinguish the different schemes we use a special
test where noise is only introduced in one half of the image, depicted in Fig. 4. The restoration algorithm should now preserve
the left part as it is, and recover a denoised version of the right
part. A contour plot is used to visualize the ability to detect edges
and remove noise, see Fig. 5. We did several simulations with
to perform best.
fourth-order scheme (24) and found
. A careful inFor the despeckle algorithm we have used
spection of the contours makes it clear that Fig. 5(e)(f) matches
the original better than Fig. 5(c)(d), in both part of the image.
The second-order scheme (18) is known to work for blocky
images (i.e., the image is piecewise constant). The purpose for
the next experiment is to evaluate how our two fourth-order
schemes process the image given in Fig. 6. It is a nontrivial case
to process an image like this since some of the blocks are as
small 2 2 pixels and other as thin as 1 10 pixels.
The solution set of the second-order PDE (18) allows discontinuities, and this is in accordance with what we observe in
Fig. 6(b) with sharp contrasts between bright and dark objects.
However, our fourth-order PDE’s (11) and (13) do not allow
discontinuities in the solution set. This can be seen as ghosts
around the smallest objects in the restored images.
In the final example in this section we do a special test concerning the fourth-order scheme (11). We know that (11) is not
rotation invariant. To unveil the consistency of this, we process
an image both before and after rotation (20 degrees). Artifacts
were not observed in either of the processed images; see Fig. 7
In conclusion, these experiments demonstrate that our PDE
method is robust, and the only information that goes into the
algorithm is the approximate noise level of the input image. If
the noise level is not known, we choose this parameter by trial
and error. In fact this is the case when we work with MR images in the next section. If we use a bigger value for the noise
level the restored image will be smoother. If we use a smaller
value for the noise level the algorithm is adding less diffusion
to the image. Accordingly, the restored image will have better
edge preservation, but the amount of noise removed from the
image is also less in such a case. Even if fourth-order scheme
(11) is not rotational invariant, it seems to perform almost as
good as the rotation invariant fourth-order scheme (13). Due to
extra computation cost and complexity of this scheme we suggest to use fourth-order (11) for processing data of dimension 3
or higher. All results given in the rest of this paper are achieved
with fourth-order scheme (11).
B. Medical MRI Images
Most digital 2-D images have nearly the same spatial resolution along the and directions. However, in medical MRI
applications, volumetric (3-D) images, and even time series
of volume images (4D), are acquired. The spatial resolution
cells of these volumes are very often nonisotropic. Moreover,
in dynamic imaging the temporal sampling rate of the volume
acquisitions will depend on the performance of the gradient
system, pulse sequence characteristics, and the selected number
of slices making up each volume in time. Fortunately, the
difference scheme used to solve (11) can easily be modified to
handle higher dimensional data, even with different spatial and
temporal resolution units.
We shall first evaluate two different three-dimensional MRI
datasets. Both datasets were acquired on a Siemens Vision 1.5
Tesla MR scanner.
In the previous section we could compare the processed
image with the known original. In the case of medical images,
the evaluation of noise suppression is not this simple due to
lack of an original image to compare our result with. Therefore
an acquisition-based noise reduction method is used to obtain
the ”ideal”, or “true” image, as was also done in [2]. The
) repeated
”ideal” image is obtained by averaging 4 (
measurements. For our algorithm, we just pick one of the 4
measurements and process it. From [25] we know that Rician
distribution approaches a Gaussian distribution at high SNR.
By averaging over repeated measurements the SNR will
increase and the Rician distribution should tend to a Gaussian
1588
distribution. This averaging process should therefore provide
us with the desirable reference image.
The first MR data set consisted of isotropic 1 1 1
FLASH 3-D head acquisitions from a healthy volunteer (
,
,
,
, 32 slices).
To our disposal we had the magnitude images from -space averaged data using 1, 2, 3, and 4 measurements from the same
slice positions. Input data to the algorithm is the data set with
,
). Our result
one measurement (
is evaluated together with the observed average of 4 measure,
). The “ideal” image, the
ments (
input image, and the output image produced by our algorithm
are shown in Fig. 8(a)–(c), respectively. Due to the isotropic
is used
voxels in this data set, the partition
in this calculation. One arbitrary slice from the 3-D data set is
given in Fig. 8. To better evaluate the performance of our algorithm, only a portion of the slice is depicted. The transacted
test-tubes located in the anterior part of the head coil were filled
with reticulated foam and polystyrene spheres and used for texture analysis, reported elsewhere.
On the anterior, right side of the head, three of the foamfilled test tubes are visible, and their content are recognized as
rather homogeneous regions in Fig. 8(c). This is in accordance
with the “ideal” image in Fig. 8(a). When zooming further, it is
possible to see that even the image based on four measurements
is affected by noise. This is not the case for the output image
from our restoration algorithm; see Fig. 9.
In addition to visual comparison of the output with the “ideal”
image, we have also computed the difference between the input
and the output. This will better reveal where details have been
lost. The difference image is shown in Fig. 10.
Here, the major difference between input and output is noise,
as it should be. However, some weak contours of the skull and
the test tubes can be seen, showing that our de-noiser has a slight
smoothing effect also at tissue boundaries.
The second MRI data set is a MR phantom of a tube filled with
plastic straws embedded in Gd-doped agarose gel with tissue
equivalent T1 and T2 relaxation times. These straws phantom
were imaged with a spin-echo (SE) pulse sequence using the
,
,
same MR scanner as above (
,
,
,5
slices). The SE pulse sequence was designed to generate 8 single
measurements together with 7 -space averages from measure, respectively, before
ments
magnitude images were calculated (cf. [30]). We applied our
restoration algorithm to one of the seven single-measurement
), which all had poor SNR compared to the
datasets (
eight averaged data sets; see Fig. 11
This peculiar, high spatial resolution data set is chosen to test
if our restoration algorithm is capable to work as locally as it
should. Notice how thin the straws walls in Fig. 11 are. These
tiny, circular contours could thus be sensitive to all kind of algorithmic influence and smoothing effects of our de-noiser. From
the results depicted in Fig. 11(c), it is clear that our restoration
algorithm handles the thin walls well, while the interior of the
straws are getting smoothed. The de-noised image is remarkably similar to the “ideal” image, which has almost three times
( ) better SNR than the input image.
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
A desirable generalization of our noise removal algorithm
is to include the time domain as well. Time series of images
are obtained in various situations, such as fMRI brain activation studies using the so-called Blood Oxygenation Level Dependent (BOLD) contrast technique, and for tracking a bolus of
intravenously injected contrast agent while it passes through a
vascular tree or a capillary bed in breast, prostate, kidney, heart
or brain.
We have tested a spatio-temporal implementation of
our de-noiser on a routine brain perfusion study. We
used 3-D+time data from a gradient-echo echo-planar
imaging (GE EPI) sequence on a Siemens Vision 1.5 T
scanner (
,
,
,
,
,
,
).
Fifteen slices were repeatedly measured every 2 s, for a period of 1 min and 20 s, so that each slice was scanned 50 times
in succession. After about three measurements a rapid injection
of an MR contrast agent was administered intravenously. As the
bolus passed the microcirculation of the brain, a magnetic susceptibility-induced signal drop could be observed in those voxels
covering perfused brain tissue. Due to expensive computation of
4D data, we selected one slice, transectioning the brain at the level
of the lateral ventricles, and applied our 2-D+time de-noising algorithm to this image time series. In the upper part of Fig. 12 this
. In the lower part, the signal intenslice is shown at time
sity versus time from a selected pixel is shown. Tissue covered
by this pixel is clearly perfused as we see a marked decrease in
signal intensity from time frame 10, with signal minimum about
20 s (frame 13) after bolus injection.
Another (annoying) source for the variation in the pixel time
course is noise. Notice how the signal oscillates even before the
contrast agent has reached the brain (frame 1 to frame 8), and it
is also hard to decide where in time the signal recovers toward
baseline after bolus passage. Since estimation of local cerebral
blood flow (rCBF), local cerebral blood volume (rCBV), and
mean transit time (MTT) of plasma in the capillary bed is essentially dependent on the decision of start and stop points in
time for the signal drop, and also on the area under the signal intensity curve during bolus passage, time-course restoration and
proper de-noising will be very helpful. In this medical context,
we therefore tested our 3-D (i.e., 2-D and time) noise removal
algorithm with different step size in time and space.
In contrast to previous cases, we do not have an “ideal”
time course to use in an evaluation. Even worse, the injected
contrast agent forces each pixel to change its signal in time,
and recirculation and leaking effects might occur, so averaging
over repeated measurement would not be feasible. However,
we would expect in the “ideal” situation that the pixel-intensities should be almost constant both before and after the
signal drop at bolus passage. If the restored image time-series
approaches this “ideal” situation we would judge the method
to be potentially useful. Another way to assess the behavior of
our de-noiser is to compute and display the difference between
the observed time course and the restored time course.
The measured perfusion time series and the PDE smoothed
time series are given in Fig. 13(a) and (b), respectively. It should
be easy to estimate the start and the end of the V-shaped signal
LYSAKER et al.: NOISE REMOVAL USING FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATION
drop in the output. We will also remark how well the minimum
point of the signal drop is preserved. The fast oscillating, low
amplitude signals in Fig. 13(c) indicate that main difference between input and output is noise.
We do not believe the second-order scheme (18) manage to
recover the V-shaped signal in a similar way due to the staircase
effect as observed in Figs. 2(c) and 3(a). The authors in [20] reports that their fourth-order scheme tend to leave the processed
image with isolated black and white speckles, see Fig. 3(b). We
think the same effect will occur if fourth-order scheme (24) was
used to process signals like Fig. 13(a). Our proposed schemes
avoid these undesired effects.
We would like to add that our algorithm seems to work for
a large set of perfusion pixel time-courses being tested. A next
step will therefore be to conduct a more rigorous evaluation on a
larger set of perfusion examinations. To be relevant for clinical
use, estimation of the physiological parameters described above
(e.g., [31]), should also be incorporated.
ACKNOWLEDGMENT
The authors would like to thank M. Bock and L. Schad at
the German Cancer Research Center (DKFZ), Heidelberg, Germany, for providing the MR images depicted in Figs. 8 and 11
with support from the European COSTB11 action. Thanks to J.
Berntsen for valuable discussions about fourth-order PDE’s.
REFERENCES
[1] E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatesan,
Magnetic Resonance Imaging: Physical Principles and Sequence Design. New York: Wiley, 1999.
[2] F. Godtliebsen, “Study of Image Improvement Techniques Applied to
NMR Images,” Ph.D., The Norwegian Institute of Technology (NTNU),
Division of Mathematical Sciences, 1989.
[3]
, “Noise reduction using Markov random fields,” Journal of Magnetic Resonance, vol. 92, pp. 102–114, 1991.
[4] A. Lundervold and F. Godtliebsen, “Noise reduction and brain tissue
classification in MR images,” Proceedings of the 1992 IEEE Nuclear
Science Symposium and Medical Imaging Conference, pp. 1265–1267,
October 1992.
[5] H. Soltanian-Zadeh, J. P. Windham, and A. E. Yagle, “A multidimensional nonlinear edge-preseriving filter for magnetic resonance image
restoration,” IEEE Transactions on Image Processing, vol. 14, no. 2, pp.
147–161, 1995.
[6] A. Perona and J. Malik, “Scale-space and edge detection using
anisotropic diffusion,” IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990.
[7] G. Gerig, O. Kubler, R. Kikinis, and F. Jolesz, “Nonlinear anisotropic
filtering of MRI data,” IEEE Transactions on Medical Imaging, vol. 11,
no. 2, pp. 221–232, 1992.
[8] M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust
anisotropic diffusion,” IEEE Transactions on Image Processing, vol.
7, no. 3, pp. 421–432, 1998.
[9] A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image denoising,” in Energy Minimization Methods
in Computer Vision and Pattern Recognition, Third International Workshop, LNCS 2134, M. A. T. Figueiredo, J. Zerubia, and A. K. Jain, Eds.,
Sophia Antipolis, France, September 3–5, 2001, pp. 19–34.
[10] L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based
noise removal algorithms,” Physica D, vol. 60, pp. 259–268, 1992.
[11] M. E. Alexander, R. Baumgartner, A. R. Summers, C. Windischberger,
M. Klarhoefer, E. Moser, and R. L. Somorjai, “A wavelet-based method
for improving signal-to-Noise ratio and contrast in MR images,” Magnetic Resonance Imaging, vol. 18, no. 2, pp. 169–180, 2000.
1589
[12] L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing
and edge detection by nonlinear diffusion:II,” SIAM Journal of Numerical Analysis, vol. 29, no. 3, pp. 845–866, 1992.
[13] P. Blomgren, T. F. Chan, P. Mulet, L. Vese, and W. L. Wan, “Variational
PDE models and methods for image processing,” in Numerical Analysis
1999 (Dundee). Boca Raton, FL: Chapman & Hall/CRC, 2000, pp.
43–67.
[14] F. Catt, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective
smoothing and edge detection by nonlinear diffusion,” SIAM Journal of
Numerical Analysis, vol. 29, pp. 182–193, 1992.
[15] A. Chambolle, “Image segmentation by variational methods: Mumford
and shah functional and the discrete approximations,” SIAM J. Appl.
Math., vol. 55, no. 3, pp. 827–863, 1995.
[16] A. Chambolle and P.-L. Lions, “Image recovery via total variation
minimization and related problems,” Numer. Math., vol. 76, no. 2, pp.
167–188, 1997.
[17] T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method
for total variation-based image restoration,” SIAM J. Sci. Comput., vol.
20, no. 6, pp. 1964–1977, 1999.
[18] T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based
image restoration,” SIAM Journal on Scientific Computing, vol. 22, no.
2, pp. 503–516, 2000.
[19] K. Majava, “Optimization-Based Techniques for Image Restoration,”
Ph.D., University of Jyvaskyla, Finland, 2001.
[20] Y.-L You and M. Kaveh, “Fourth-order partial differential equation for
noise removal,” IEEE Transactions on Image Processing, vol. 9, no. 10,
pp. 1723–1730, 2000.
[21] T. Tasdizen, R. Whitaker, P. Burchard, and S. Osher. (2002) Geometric
Surface Processing via Normal Maps. UCLA. [Online]. Available:
http://www.math.ucla.edu/applied/cam/index.html
[22] G. W. Wei, “Generalized Perona-Malik equation for image restoration,”
IEEE Signal Processing Letters, vol. 6, no. 7, pp. 165–167, 1999.
[23] G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. on Image Processing, vol. 5, pp. 1582–1586, 1996.
Solutions of a Class of Fourth Order Non[24] J. Greer and A. Bertozzi,
linear Equations for Image Processing, 2002, to be published.
[25] J. Sijbers, A. J. den Dekker, J. Van Audekerke, M. Verhoye, and D. Van
Dyck, “Estimation of the noise in magnitude MR images,” Magnetic
Resonance Imaging, vol. 16, no. 1, pp. 87–90, 1998.
[26] R. D. Nowak, “Wavelet-based Rician noise removal for magnetic resonance imaging,” IEEE Transactions on Image Processing, vol. 8, no. 10,
pp. 1408–1419, 1999.
[27] W. S. Hinshaw and A. H. Lent, “An introduction to NMR imaging: From
the Bloch equation to the imaging equation,” Proceedings of the IEEE,
vol. 71, pp. 338–350, 1983.
[28] D. L. Parker and G. T. Gullberg, “Signal-to-noise efficiency in magnetic
resonance imaging,” Medical Physics, vol. 17, no. 2, pp. 250–257, 1990.
[29] G. Gilboa, Y. Y. Zeevi, and N. Solchen. (2003) Image Sharpening
by Flow Based on Tripel Well Potentials. [Online]. Available:
http://tiger.technion.ac.il/gilboa
[30] A. M. Fenstad, A. Lundervold, M. Bock, and L. R. Schad, “How does
signal-to-noise ratio influence texture measure?,” in Proceedings of the
Eighth Annual Scientific Meeting of the International Society for Magnetic Resonance in Medicine, Denver, CO, April 1–7, 2000, p. 61.
[31] A. G. Sorensen and R. Reimer, Cerebral MR Perfusion Imaging – Principles and Current Applications. Stuttgart: Thieme, 2000.
H
Marius Lysaker received his master degree in
applied mathematics in 2001 from the University
of Bergen, Norway. The topic of the thesis was
to develop algorithms to denoise MRI images via
partial differential equations. He is currently a PhD.
student in applied mathematics at the University of
Bergen, Norway.
His research interests includes constructions,
denoising and segmentation of medical images
(MRI/PET). During the last few years he has mainly
focused on level set methods. From 2004 he will
hold a Post.doc position at Simula Research Laboratory AS in Oslo at the
department of Scientific Computing.
1590
Arvid Lundervold The focus of Lundervold’s
research is on image processing and pattern recognition in structural and functional medical magnetic
resonance images (MRI) and its application in clinical neuroscience and neurobiological research. He
got his BSc in mathematics and philosophy (1976)
and a Medical degree (1982), both at the University
of Oslo. In 1995 he obtained a PhD in medical image
analysis at the University of Bergen. Presently he
his Associate professor in the Neuroinformatics and
Image Analysis Group, Department of Physiology,
University of Bergen. Dr. Lundervold has published more than thirty papers
in journals and conference proceedings on experimental neurophysiology,
neuroinformatics, and image analysis in medical ultrasound and MRI. He is
affiliated with the Bergen fMRI group and the Image processing group at the
Department of Mathematics, UoB. He his reviewer in several medical imaging
journals and an editorial board member of Computerized Medical Imaging and
Graphics.
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 12, DECEMBER 2003
Xue-Cheng Tai received Licenciate degree in 1989
and PhD in 1991 in applied mathematics from
Jyvaskalya University in Finland. The subject for
his Phd Thesis was on inverse problems and parallel
computing. After holding several research positions
in Europe, he was employed as an associated
professor in 1994 at the University of Bergen,
Norway and as a professor since 1997. He has also
worked as a part time Senior Scientist at a private
company “Rogaland Research”. He is now a member
of “Center for Mathematics for Applications” in
Oslo and a member of “Center of integrated Petroleum Research” in Bergen.
His research interests include Numerical PDE for image processing, multigrid
and domain decomposition methods, iterative methods for linear and nonlinear
PDE problems and parallel computing. He has educated numerous master and
Phd students, published more than 60 scientific papers. He has been reviewer
and editor for several international journals.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement