2010
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 10, OCTOBER 2014
Noise Reduction in Small-Animal PET Images
Using a Multiresolution Transform
Jose M. Mejia*, Humberto de Jesús Ochoa Domínguez, Member, IEEE,
Osslan Osiris Vergara Villegas, Senior Member, IEEE, Leticia Ortega Máynez, Member, IEEE,
and Boris Mederos
Abstract—In this paper, we address the problem of denoising
reconstructed small animal positron emission tomography (PET)
images, based on a multiresolution approach which can be implemented with any transform such as contourlet, shearlet, curvelet,
and wavelet. The PET images are analyzed and processed in
the transform domain by modeling each subband as a set of
different regions separated by boundaries. Homogeneous and
heterogeneous regions are considered. Each region is independently processed using different filters: a linear estimator for
homogeneous regions and a surface polynomial estimator for
the heterogeneous region. The boundaries between the different
regions are estimated using a modified edge focusing filter. The
proposed approach was validated by a series of experiments. Our
method achieved an overall reduction of up to 26% in the %STD
of the reconstructed image of a small animal NEMA phantom.
Additionally, a test on a simulated lesion showed that our method
yields better contrast preservation than other state-of-the art
techniques used for noise reduction. Thus, the proposed method
provides a significant reduction of noise while at the same time
preserving contrast and important structures such as lesions.
Index Terms—Image enhancement, multiresolution transform,
noise reduction, positron emission tomography (PET).
I. INTRODUCTION
P
OSITRON emission tomography (PET) is a technique of
nuclear medicine based on the tracer principle: A small
dose of a radioactive substance (radiotracer) is administered
into a patient. The radiotracer is a biologically active substance
for which a radioactive isotope replaces an atom or is added to
this substance with the help of a ligand. The most commonly
employed radiotracer is fluroro-deoxyglucose (FDG) for which
is added to a glucose molecule and subsequently, this glucose is metabolized and trapped in many tissues and is especially active in many cancer cells. The determination of the radiotracer distribution in the body is accomplished by detecting
Manuscript received March 21, 2014; revised May 29, 2014; accepted June
02, 2014. Date of publication June 09, 2014; date of current version September
29, 2014. Asterisk indicates corresponding author.
*J. M. Mejia is with the Department of Ingeniería Eléctrica y Computación, University of Ciudad Juarez, Chihuahua 32310, México (e-mail:
[email protected]).
H. J. Ochoa Domínguez, O. O. Vergara Villegas, and L. Ortega Máynez
are with the Department of Ingeniería Eléctrica y Computación, University of
Ciudad Juárez, Ciudad Juárez, Chihuahua 32310 México.
B. Mederos is with the Department of Física y Matemáticas, University of
Ciudad Juárez, Ciudad Juárez, Chihuahua 32310 México.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMI.2014.2329702
photons emitted by the substance disintegration process, where
positron decay will result in two photons emitted at 180 to each
other. In PET, two coincident photons must be detected to count
an event. Counts of events are processed by reconstruction algorithms to form the final image.
PET is widely used in clinical and preclinical research to
evaluate the metabolic evolution of molecular and biochemical
processes in patients, such as hibernating myocardium, cancer,
Alzheimer, epilepsy, and many others [1]. In addition, PET has
applications in the areas of biomedical and pharmacological
research. It provides quantitative and noninvasive imaging of
the biodistribution as well as pharmacokinetics [2] used in oncology, cardiology, molecular biology, drug discovery development, and genetics [3]. Preclinical studies are carried out on
small animals due to their genetic resemblance with humans,
and to avoid potential risk to human health. However, the presence of noise on the acquired data by PET scanners reduces the
quality of the reconstructed images.
In PET, the main source of noise is statistical noise (counting)
caused by a low activity injected dose and short scan times
which are chosen to minimize radiation doses to patients or animals. Although most methods for image reconstruction compensate for degradations such as random and scattered events
induced in the count process, the images could still be affected
by noise caused by other sources. This situation is exacerbated
even more in small-animal imaging, where a spatial resolution
less than 1.5 mm is needed in all directions when compared with
the 5-mm image resolution in typical human body studies.
Several approaches exist to reduce noise in the reconstructed
images. In [4], the PET image was transformed using wavelets
followed by a cartoon (U) plus texture (V) decomposition (UV),
and it was expected that the cartoon part contains mostly true
signal while the noise remains in the texture part. In [5], the
authors proposed application of a bilateral filter to the PET image.
They compared its performance with traditional moving average
filters and found better denoising with edge-preservation results.
However, the shortcoming was that the bilateral filter needed
a careful selection of parameters. In [6], the PET image was
processed using the Anscombe transform to normalize data.
Next, it was transformed into the wavelet domain. Finally, Bayes
shrinkage, with threshold dependent on the noise variance, was
applied to the wavelet coefficients to denoise the data. The method
had a good preservation of the structures in the images, but it did
not remove most of the noise close to the edge zones.
While the reviewed approaches have achieved promising results, most of them are based on the application of adapted
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MEJIA et al.: NOISE REDUCTION IN SMALL-ANIMAL PET IMAGES USING A MULTIRESOLUTION TRANSFORM
existing methods originally developed for moderate Gaussian
noise, and these techniques generally do not consider the particular characteristics of the PET images such as:
• very low signal-to-noise ratios;
• low resolution and lack of fine details (texture);
• shapes defined by the internal anatomy of small animals.
Taking into account the above-mentioned remarks, in this study,
a technique to reduce the noise of post-reconstructed PET images is proposed. The technique is based on three steps. First, a
detection of regions with similar features, corresponding to different organs or tissues defined by the internal anatomy is carried out. Second, because of the lack of fine details, a classification of regions is performed by considering only smooth and
homogeneous regions. Finally, adaptive denoising is achieved
on each region.
These steps are accomplished by working in a multiresolution transform domain. A multiresolution analysis offers sparse
representation of images, contributing to the emergence of homogeneous zones within the subbands of the transform. In addition, the orientability of its basis functions allows gathering
adjacent edge points into contour segments [7] useful to detect
edges, thus, giving separation between regions and its contours.
The proposed algorithm determines regions in each subband
of the transformed image by locating boundaries with an edge
detector. Afterwards, regions are classified, according to its
regularity, into homogeneous or heterogeneous. Homogeneous
areas are denoised by finding the best linear estimator, while
in a heterogeneous region, a point is denoised by fitting a
polynomial surface over its neighborhood. The estimation of
the point value corresponds to the magnitude of the surface at
that location.
The contributions of this study are as follows.
• A technique for PET image denoising with the ability to
separate regions from edges, aiming to denoise each region
adaptively.
• A measure to approximate the regularity of a region in an
image by using the coefficients of a multiresolution transform.
• Preservation of the geometrical characteristics of the radiotracer distribution on smooth regions by using polynomial
surface fitting.
The rest of the paper is organized as follows. Section II introduces to the multiresolution transform. In Section III, a model
of regions and boundaries is described and the proposed estimators to denoise each region are presented. In Section IV, the
performance of the algorithm is evaluated. Finally, conclusions
and future work are discussed in Section V.
II. THEORY
A. Multiresolution Transforms
Multiresolution representations provide a framework to analyze data at different resolution levels. In signal processing, the
most common approach is the wavelet transform and its discrete
implementation, called the discrete wavelet transform (DWT).
The typical DWT implementation for 1-D signal consists of a
filter bank, where scale coefficients are obtained by decimation
2011
Fig. 1. Filter bank to implement a three-level DWT decomposition. Input
, LP and HP are the low-pass and high-pass filters, respectively.
signal is
indicates decimation by two.
Fig. 2. Filter bank to implement one level of 2-D DWT. The input image is
, LP and HP are the low-pass and the high-pass filters, respectively.
indicates decimation by two. Subbands
,
,
, and
are obtained
by 1-D convolutions of the rows and columns of with the 1-D HP and LP
filters.
along with a low-pass filter, while detail coefficients are determined from decimation along with a high-pass filter. The filters
are recurrently used to obtain transform coefficients at diadic
resolution levels. Fig. 1 illustrates this process, which is straightforwardly generalized to higher dimensions.
For image transformation, the 2-D DWT can be implemented
using a filter bank composed of low- and high-pass filters followed by decimation [8]. One level of DWT decomposition
is obtained by applying the 1-D DWT first along the rows of
the image and then along the resulting columns, yielding four
subbands: one scaling subband that is an approximation of the
input image and three directional subbands representing details
aligned in vertical, horizontal, and diagonal directions, respectively. More levels of decomposition can be obtained by iterating the scaling subband through the same structure of the filter
bank, as shown in Fig. 2. The number of directional subband per
scale is insufficient to represent the important and unique features of multidimensional signals such as images [9].
More general multiresolution transforms exist that allow decomposing an image in an arbitrary number of directional subbands per level such as contourlet, shearlet, and the curvelet
[9]–[11], which yield an appealing multiscale space-frequency
localization as well as a better directional selectivity of features
within the image than the wavelet transform. In this study, nondecimated versions of these transforms are used because of the
invariance to translation property, which makes them suitable
for denoising algorithms as pointed out in [12].
Throughout this paper, the number of decomposition levels
or scales is denoted by , while subbands are denoted by
with
and
, where
is the number
of subbands at level . Moreover, the coarser subband, which
contains the low-frequency information, is expressed as
.
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 10, OCTOBER 2014
Finally, the inverse transform is performed on the resulting
subbands to obtain a noise-reduced PET image.
In the following subsections, a detailed description of each step
is given.
A. Domain Transformation
Fig. 3. Example of two levels of DWT decomposition with three directions per
is the scaling subband.
level;
Fig. 3 shows an example of a multiresolution decomposition
(wavelet) applied to a PET image of a rat. Two levels of DWT
decomposition are used and three directional subbands per level
are obtained.
The multiresolution domain is represented by a set of subbands
which corresponds to different
scales and orientation. The domain of each subband
is
partitioned into a set of regions
and border segments
. Each subband
is represented by a function
defined as follows:
(1)
III. METHODS
Small animal PET images consist of defined structures of organs with specific irregularity separated by boundaries. Therefore, it is appealing to model the structures as regions with different geometry, morphology, and regularity. These regions are
well differentiated in a multiscale representation due to the good
space-frequency localization information available at each subband [13]. Moreover, this domain offers sparse representation
of images that contributes to the emergence of homogeneous
regions. In addition, the directional subbands allow gathering
nearby edge points into contour segments [7] that are useful to
detect edges that separate regions.
Taking advantage of the above-mentioned properties of
the multiresolution transform and the underlying structure of
the images consisting of anatomical regions, in this study, a
denoising algorithm that processes the subbands of the transformed image separating it into regions and boundaries is
proposed. The boundaries are detected by means of a multiscale edge detector and the remaining areas of the subband
correspond to different regions. Each region is classified into
homogeneous or heterogeneous according to its regularity and
smoothed based on its classification. The proposed algorithm
consists of the following steps.
1) Domain Transformation. The image is transformed to a
multiresolution domain obtaining a low-pass
and the
detail subbands
. The
subband
is not processed.
2) Boundary and Region Detection. A boundary set is estimated using the large magnitude coefficients in the subbands. The different regions of a given subband are determined as the complement of the boundary set.
3) Classification. The global regularity of each region is estimated by the expression (2) that measures the average
decay of the coefficients across the scales. According to
this estimate, the regions are classified as homogeneous or
heterogeneous.
4) Smoothing. A filter is applied depending on the estimated
global regularity of the region. If the region is homogeneous, then a representative value is determined using a
linear estimator. If the region is heterogeneous, then a regularized least square polynomial fitting is performed in a
neighborhood of each coefficient to estimate its true value.
Number of regions in the subband
.
Number of border segments in the subband
.
Position vector in
.
Value of
on the region
Value of
on segment
at .
at .
Noise model.
and
Indicator functions for
, and
.
respectively, i.e.,
if
if
if
if
.
Note that
and for each subband, a different partition is obtained.
B. Boundary and Region Detection
When a multiresolution transform is employed to represent
the image, the boundaries of the relatively large regions can
be observed with high accuracy at the coarser scales, while the
small regions tend to have a diffused representation and barely
appear at these scales. However, they start to rise at finer scales
and get better definition as the scale decreases.
The multiresolution transforms encompass information for
different directions
and allow accurate gathering
of coefficients that comprise information of edge segments oriented in a wide range of directions. Therefore, to estimate a
boundary set in a particular region at a particular scale , the
MEJIA et al.: NOISE REDUCTION IN SMALL-ANIMAL PET IMAGES USING A MULTIRESOLUTION TRANSFORM
edge information that corresponds to transform coefficients with
larger absolute values is used.
Based on the previous discussion, a multiscale edge-detection
method that explores similar ideas used in the edge-focusing
algorithm [14], aiming to efficiently detect the boundary set
on each subband , is proposed.
The proposed edge-detection algorithm relies on the idea
that it is easier to localize edges at coarser scales, and then use
this information to find edges in the next fine scale, that is, a
coarse-to-fine edge-detection approach is considered. Once the
edge coefficients
comprising the boundary of a large region
are determined at a coarse scale, the method focuses on finding
new edge set
at the next fine scale, only searching initially
for edge coefficient on the region limited by the coefficients in
computed at the previous scale.
The edge-detection method is implemented as follows: Beginning at the coarser level, the edge set is estimated by determining the coefficients with higher absolute value in all the orientations, that is, the coefficients that are greater than a threshold
producing the set . To promote an edge set
that is a
connected curve, the coefficients adjacent to coefficients in ,
which were not included, are also added by employing the following schema.
1) The set of connected coefficients
to
(i.e., coefficients in an eight-neighborhood of each coefficient is )
is determined.
2) From
, only coefficients greater than a threshold
are added to
(i.e.,
).
3) The threshold
is decreased by a unit, and steps (1) and
(2) are repeated until
.
4) The final set of edges
is set to the resulting .
The parameter
was taken as 97% of the maximum absolute
value of the coefficients in the subband. The lower limit on
was obtained after exhaustive tests to obtain optimal results.
Subsequent steps were performed analogously for every level.
The previous schema was carried out on each scale
obtaining the set of edges
In the proposed scheme, the influence of the noise is minimized as a consequence of first starting to find edges at the
coarsest scales, which present low noise levels. Then this more
reliable border information is used to lead the detection method
to the next scale where noise increases. Fig. 4 compares the
proposed method with other edge-detection algorithms at scales
and
in a nonsubsampled contourlet decomposition.
C. Classification
is classified into homogeneous or heteroEach region
geneous according to its global regularity, which depends on
the behavior of the function
. In the homogeneous region,
has slow oscillations, while in the heterogeneous regions,
there are high oscillations.
The following expression is proposed as a measure of global
regularity:
(2)
2013
Fig. 4. Edges detected at scales
and
in a nonsubsampled con(b) a Sobel
tourlet decomposition, using: (a) a Sobel edge detector in
, (c) a Prewitt edge detector in
, (d) a Prewitt edge
edge detector in
, (e) proposed edge detector in
, and (f) proposed edge
detector in
.
detector in
Fig. 5. Assessment of expression (2) for different regions (enclosed by black
rectangles). Note that the measure is positive for homogeneous regions and as
the surface becomes more irregular (noise and texture), the measure decreases.
which gives an insight of the average decay of the coefficients
over the region
, where
denotes absolute value and
is the number of elements in
. For regions
with high regularity, coefficients certainly decay fast; thus, (2)
tends to be positive as shown in Fig. 5. However, in low regularity regions, (2) tends to be negative. Therefore, a region is
classified as homogeneous in case (2) is positive and as heterogeneous otherwise. Fig. 5 shows an assessment of the proposed
measure with several images.
In addition, (2) is calculated only between the two coarsest
scales because these subbands are less affected by noise. Therefore, at fine scales, regions inherit the classification of regions
at coarser scales localized at the same position.
The common approaches to estimate the pointwise and local
regularity rely on the determination of the local Holder’s exponents of the images at each point. This is carried out by analyzing the decay of the coefficients across the scales [8], [15].
However, the calculation of Holder’s exponents is computationally expensive.
D. Smoothing
Given a region
, its coefficients
are considered as random variables contaminated by an unknown independent and identically distributed noise
as stated in
(1). According to the classification step, the region is labeled
as either homogeneous or heterogeneous, and consequently, a
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 10, OCTOBER 2014
different smoothing algorithm is used in each case to estimate
the true value of
.
• Linear estimation (homogeneous region). Each subband
coefficient of
with
can be expressed
as
where
(10)
More specifically, the functional (10) can be rewritten as
(3)
Owing to the homogeneity of the region, the magnitude of
are considered constant with an
the coefficients
unknown value
(4)
and
Note that the second term of (10) prevents the overfitting
controlled by the parameter. This is known as ridge regression [16]. The minimum is found by taking the partial derivative of with respect to the components
of
the vector and equating to zero
(5)
where
denotes the expected value.
The main goal is to estimate the value of in the region
. As the probability density function of (4) is unknown, the proposed estimator is constrained to be linear
(11)
such that
for all
the following linear system:
, which leads to solving
(6)
and unbiased, and
unbiased estimator
. Therefore, the best linear
is calculated as follows:
(7)
is a column vector of the elewhere
ments of
, is one’s column vector, is the identity
matrix, and is the unknown variance of
of the noise.
After some algebraic manipulation on (7), the following
expression for the estimator is obtained:
(8)
which corresponds to the sample mean of the coefficients
in the region.
• Regularized polynomial fitting (heterogeneous region): The heterogeneous regions present a moderate
variation on the values of the function
, and the
pair
with
is considered as
smooth surface. Therefore, to perform the denoising of
a coefficient on this region, a regularized polynomial
approximation
to the values of
on a local
neighborhood
of each point
is carried
out. The value of
is used to estimate
.
To obtain this approximation, a polynomial of order 3 is
employed
(9)
of is dewhere the coefficient’s vector
termined by solving the following optimization problem:
(12)
where
A set of 20 realizations of the reconstructed rat phantom
was obtained with the SimSet software as described in
Section III-E. For this set, different runs of the proposed
method were assessed with different degrees of the fitting polynomial. Fig. 6 shows a plot of the PSNR (peak
signal-to-noise ratio) of the resulting images after applying
the proposed method for each run.
The PSNR increases along with the order of the polynomial. However, polynomials of degree greater than 3 do
not contribute to a significant gain in PSNR.
E. Simulation Method
PET scans of phantoms were simulated using SimSET [17],
[18] in 2-D data acquisition mode. Noise was implicitly added
by the SimSET software when the physical process of scanning
was modeled.
Two digital phantoms were simulated. The first is a rat
phantom that consists of regions with activities of 3:1, 4:1, and
MEJIA et al.: NOISE REDUCTION IN SMALL-ANIMAL PET IMAGES USING A MULTIRESOLUTION TRANSFORM
Fig. 6. Test of the proposed algorithm using different degrees of the fitting
polynomial (top) and a zoom of the plot (bottom).
2015
Fig. 7. Denoising of a set of 20 images using NSCT, wavelet, curvelet, and
shearlet.
5:1 with respect to the background. The second, a mouse thorax
phantom was designed using the anatomical atlas described
in [19] and a small circular object of 1 1 mm was placed in
the liver to simulate a lesion having a contrast to background
ratio of 4:1. A set of 250 image realizations was obtained using
the rat phantom. The average of the images was taken as the
ground truth and a subset of 20 images from the set of image
realizations was used to tune the algorithm parameters, and
for PSNR evaluation. For the mouse thorax phantom, seven
realizations were simulated and each one was reconstructed
using the expectation maximization (EM) algorithm using
25 iterations or when the average of the differences between
iterations was less than 0.0002.
F. Tuning of Parameters
The free parameters of the algorithm are transform domain,
basis functions, and levels of decomposition. Parameter tuning
is conducted by assessing the performance, changing one parameter at a time, and leaving the other two constants. The
metric used is the PSNR defined as
(13)
is the maximum possible intensity value for a pixel
where
in the image and
is the mean squared error between a
reference image (original) and the processed image. The metric
is a ratio between the maximum power of the signal and the
power of the noise corrupting it. Because of its low complexity,
it is widely used for evaluating image-denoising algorithms.
In [20], it has been shown that for compression algorithm optimization, the PSNR is a valid performance metric and an indicator of the variation of quality only when it is evaluated within
a specified algorithm and fixed content. This is also applicable
to denoising algorithms, as in compression, where the PSNR is
used to evaluate distortions introduced in the original signal.
In this study, a set of 20 simulated realizations of the rat
phantom was used for parameter evaluation purposes, the PSNR
metric is used for parameter optimization because it provides
a pixel-to-pixel comparison giving an overall measure of the
quadratic error, and there is no need to define particular structures in every image.
1) Types of Multiresolution Analysis: To analyze the algorithm under different multiresolution analyses, the following
Fig. 8. Test of the algorithm under the NSCT domain and different number of
levels (top) and a zoom of the plot (bottom).
transforms were selected: the nonsubsampled contourlet transform (NSCT) [21], the nondecimated shearlet [22], the curvelet
[11], and the nondecimated discrete wavelet. Fig. 7 shows a
graph of the PSNR attained in each domain; four levels of decomposition and Daubechies 7 basis functions were used in all
transforms, and four directions per level were used in the NSCT
and the shearlet.
The remaining parameters in the curvelet transform were left
with the library defaults, except for the number of levels. As
the NSCT gave a better performance, the rest of the tests were
carried out using this transform.
2) Decomposition Levels: Fig. 8 shows the effectiveness of
the algorithm using different number of decomposition levels of
the NSCT, which are limited by the matrix size of the image.
3) Basis Filters: Finally, the performance was evaluated
using Daubechies 9-7, maxflat, pyramidal (pyr), and pyramidal
exchanged (pyrex) filters along with the NSCT. Fig. 9 shows
that the selection of a particular filter has a minimum effect on
the performance of the algorithm.
IV. RESULTS
The performance of the proposed method is studied. Image
quality was evaluated using the metrics defined in the NEMA
NU 4-2008A standard [23]. Details about image quality are
given in Section IV-A.
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 10, OCTOBER 2014
TABLE I
MEASURES IN THE UNIFORM REGION
Fig. 9. Test of the algorithm using the NSCT domain and different basis filters
(top). Zoom of the graph (bottom).
Simulated and acquired data sets of small-animal PET images
are presented to show how the proposed post-filtering algorithm
deals with noise and structures such as lesions and borders.
For comparison purposes, the images were also processed with
Gaussian kernels and two state-of-the-art post-filtering algorithms: the UV method [4] and the bilateral filter [5]. The input
parameters for the UV method were
,
,
and
. For the bilateral filter, standard deviations of 1.0 and 0.51 were used for the intensity and the spatial domain, respectively. For the proposed method, the NSCT
was used with five levels of decomposition, four directional subbands per level, and the pyramidal basis filters. The results obtained from simulated and acquired data are presented and analyzed in Sections IV-B and IV-C, respectively.
Fig. 10. Images of the slices of the uniform region summed together, (a) original, (b) processed with our method, and (c) axial profile through slices.
A. Image Quality
Image quality assessment was performed using the NEMA
small animal phantom, as described in [23]. This phantom consists of two chambers: the first is a large cavity of 30 mm diameter and filled with isotope (hot region) from where uniformity
is measured. The cavity also houses two smaller cavities filled
with water and air, respectively (cold regions), which are used
to quantify the spillover ratio. The second part of the cylinder
contains five cavities with diameters of 1, 2, 3, 4, and 5 mm, respectively, filled with isotope (hot regions) that are used to measure the noise and recovery coefficients. The phantom was filled
with 7.4 MBq of
. Data were acquired for 20 min using a stationary quad-HIDAC small animal PET scanner. The acquired
data were reconstructed using three iterations and 10 subsets
of the EM algorithm without using a smoothing filter. Image
matrix has a dimension of 128 128 256 and the voxel size
is 0.5 0.5 0.5 mm. The reconstructed image was compared
with its post-processed version obtained with the proposed algorithm.
To measure uniformity, a 22.5-mm-diameter by 10-mm-long
cylindrical volume of interest (VOI) was taken at the center
of the uniform region. The average activity concentration, the
maximum and minimum values, and the percentage standard
deviation are shown in Table I. Fig. 10, shows the images obtained after summing together the slices of the uniform region.
Fig. 11. (a) Transverse view of the projections of maximum intensity of the
rods, and axial profiles along the (b) 5-mm rod, (c) 4-mm rod, (d) 3-mm rod, (e)
2-mm rod, and (f) 1-mm rod.
Circular ROIs were drawn around each rod in an image of the
average of slices covering the central 10-mm length of the rods.
Transverse image pixel coordinates of the locations with the
maximum ROI values were used to create line profiles along the
rods in the axial direction. Fig. 11 shows the profiles along the
rods in the axial direction. As the rod radii decrease, the number
of counts in that region decreases. Results of the contrast recovery (CR) and the percentage standard deviation (%STD) of
each rod are shown in Table II. Reductions of up to 26.4% in
%STD were observed.
The spillover ratio (SOR) is calculated as the ratio of the
average value of each cold region to the average of the hot
uniform region. The VOI taken in the cold regions has a diameter of 4-mm (half the physical diameter of the cylinders)
and encompass the central 7.5-mm in length of the waterand air-filled cylindrical inserts. Table III shows the results
of SOR and %STD for the cavities filled with water and air.
Fig. 12(a) and (b) show the sum of slices containing the cold
MEJIA et al.: NOISE REDUCTION IN SMALL-ANIMAL PET IMAGES USING A MULTIRESOLUTION TRANSFORM
TABLE II
RECOVERY COEFFICIENTS AND %STD
2017
TABLE IV
LESION CONTRAST OBTAINED AFTER SEVERAL SIMULATION RUNS
Fig. 13. Image of the mouse thorax phantom with a simulated lesion in the
liver; (a) original reconstructed image with arrow indicating the lesion; processed images with (b) Gaussian kernel, 2 mm, (c) Gaussian kernel, 1 mm, (d)
bilateral filter, (e) UV method, and (f) proposed method.
Fig. 12. Slices in cold regions summed together; (a) original, (b) processed
with the proposed method, and (c) activity profiles of the cross-sections.
TABLE III
SOR AND %STD MEASURED IN THE COLD REGIONS
regions in the original, and the postprocessed image, respectively. Fig. 12(c) shows the profiles of the dotted lines drawn
in Fig. 12(a) and (b). A reduction of the counts in the cold
chambers is observed.
B. Simulated Data Comparisons
1) Structure Preservation Measure: The following scheme
was used to quantitatively measure structure preservation in the
images processed. The seven realizations of the mouse thorax
phantom were processed with the denoising methods and its performance was evaluated by calculating the lesion to background
contrast (
) as a figure of merit [24], which is defined by
(14)
denotes the mean value within the lesion and
inwhere
dicates the mean value in a 1.6 1.6 mm rectangular region in
the liver outside the lesion. The resulting contrast, after applying
each method in seven realizations, is reported in Table IV. In
Figs. 13 and 14, a 2-D image of the phantom and a 3-D mesh
of the lesion of each method are shown, respectively. Note how
the proposed method preserves the lesion structure while maintaining the contrast between the lesion and the background.
Fig. 14. Mesh plots of the simulated lesion of Fig. 13 showing: (a) the original
reconstructed image and the processed image with the (b) Gaussian kernel, 2
mm, (c) Gaussian kernel, 1 mm, (d) bilateral filter, (e) UV method, and (f) proposed method.
2) PSNR Comparison: In addition, for completeness, the
software phantom described in Section III-F was used to compare the PSNRs attained by each algorithm. A plot of the results
from a set of 20 images is shown in Fig. 15. Fig. 16 shows the
ground truth image; and profiles attained by each method are
shown in Fig. 17.
3) Processing Times: The average processing times for different matrix sizes were computed using a 3.6-GHz Quad Core
Opteron processor with 3.9 GB of memory. In Table V, the bilateral filter and UV method take less than 10% of the total time
2018
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 10, OCTOBER 2014
Fig. 15. Comparison of PSNR resulting from the algorithms with the software
phantom.
Fig. 18. Image of a rat acquired with the microPET Focus 120. (a) Original, A
and B are the structures in Table VI, (b) processed with Gaussian kernel, 2 mm,
(c) processed with Gaussian kernel, 1 mm, (d) processed with the UV method,
(e) processed with bilateral filter, and (f) processed with the proposed method.
TABLE VI
MEASURED IN TWO DIFFERENT STRUCTURES
Fig. 16. Ground truth image for the rat phantom.
Fig. 17. Comparison of profiles resulting from the algorithms with the software
phantom. (a) Noisy, (b) Gaussian kernel, 1 mm, (c) Gaussian kernel, 2 mm, (d)
UV method, (e) bilateral, and (f) proposed method.
TABLE V
PROCESSING TIMES
required to finalize the processing when compared with the proposed method. The main factors contributing to the processing
time in the proposed method are: the matrix size, the computation of the multiresolution transform and its inverse, and the
process of finding and classifiying the different regions in the
image.
C. Acquired Data Comparisons
Fig. 18 shows the results using Gaussian kernels of 2 and
1 mm, respectively, the UV filter, the bilateral filter, and the
Fig. 19. Image of a rat acquired with the quad-HIDAC PET scanner. (a) Original, (b) processed with Gaussian kernel, 2 mm, (c) processed with Gaussian
kernel, 1 mm, (d) processed with the UV method, (e) processed with bilateral
filter, and (f) processed with the proposed method.
proposed method. The original image is shown in Fig. 18(a)
and was acquired using a microPET Focus 120 located in the
PET Center in Universidad Nacional Autonoma de Mexico.
Fig. 18(b) and (c) shows that the Gaussian kernel results have
the highest levels of blur. The UV, bilateral, and proposed filters
are shown in Fig. 18(d), (e), and (f), respectively. Table VI
shows the resulting
on two different structures on the rat
shown in Fig. 18.
Fig. 19(a) shows a reconstructed image of a rat. The data
were acquired from a 500 g
rat (17 MBq), located at the
center of the FOV of the scanner and acquired with the quadHIDAC in rotating mode with 30-min acquisition time and reconstructed using the EM, 0.5-mm sided voxels, one iteration,
MEJIA et al.: NOISE REDUCTION IN SMALL-ANIMAL PET IMAGES USING A MULTIRESOLUTION TRANSFORM
Fig. 20. Zoom of the rat acquired with the quad-HIDAC PET scanner. First
column shows the original noisy data and its zoom view. Second column shows
the data after being processed with the proposed algorithm.
and 50 subsets, and including only geometric normalization.
No corrections for attenuation, random, and scatter events were
considered. As shown in Fig. 19, the Gaussian kernel results
of Fig. 19(b) and (c) show the highest levels of blur. The UV
and bilateral filters, presented in Fig. 19(d) and (e), respectively,
exhibited better structure preservation than Gaussian kernels.
However, the noise was still present. The proposed filter, shown
in Fig. 19(f), preserved the structures and maintained low noise
levels. Fig. 20 shows a zoom of a section of Fig. 19(a) and its
denoised version obtained with the proposed algorithm.
V. CONCLUSION
In this study, a denoising method for reconstructed smallanimal PET images was presented. Furthermore, the use of a
multiresolution representation was proposed to separate the homogeneous and heterogeneous regions and be able to perform
adaptive smoothing.
The proposed method is able to reduce the variance introduced by the noise while maintaining the average levels of radioactivity concentrations (counts). Moreover, as a result of the
region’s adaptive denoising, border and structure preservations
are possible without incurring excessive over-smoothing; this
contributes to a better delineation between organs and improves
the visual quality of reconstructed images, which facilitates further processing. Thus, one potential use of the proposed method
is in the context of pattern recognition where it could become
a powerful tool as a preprocessing step for further supervised
and unsupervised analyses such as image registration, segmentation, and detection. However, one drawback of the algorithm
is the processing time, which mainly depends on the employed
multiresolution transform.
In future work, the plan is to improve the performance of this
method by taking advantage of the geometry and information of
the whole available volumetric data in the 3-D PET image. In
addition, testing the algorithm using PET images of the human
body is also part of the plan.
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