Bayesian modeling with locally adaptive prior parameters in small animal imaging

Abstract

Medical images are hampered by noise and relatively low resolution, which create a bottleneck in obtaining accurate and precise measurements of living organisms. Noise suppression and resolution enhancement are two examples of inverse problems. The aim of this study is to develop novel and robust estimation approaches rooted in fundamental statistical concepts that could be utilized in solving several inverse problems in image processing and potentially in image reconstruction. In this study, we have implemented Bayesian methods that have been identified to be particularly useful when there is only limited data but a large number of unknowns. Specifically, we implemented a locally adaptive Markov chain Monte Carlo algorithm and analyzed its robustness by varying its parameters and exposing it to different experimental setups. As an application area, we selected radionuclide imaging using a prototype gamma camera. The results using simulated data compare estimates using the proposed method over the current non-locally adaptive approach in terms of edge recovery, uncertainty, and bias. The locally adaptive Markov chain Monte Carlo algorithm is more flexible, which allows better edge recovery while reducing estimation uncertainty and bias. This results in more robust and reliable outputs for medical imaging applications, leading to improved interpretation and quantification. We have shown that the use of locally adaptive smoothing improves estimation accuracy compared to the homogeneous Bayesian model.

Full text

EDITED BY
Mehdi Ostadhassan,
Northeast Petroleum University, China
REVIEWED BY
Nikos Efthimiou,
Massachusetts General Hospital and Harvard
Medical School, United States
Kyeong Min Kim,
Korea Institute of Radiological and Medical
Sciences, Republic of Korea
*CORRESPONDENCE
Robert G. Aykroyd
R.G.Aykroyd@leeds.ac.uk
RECEIVED 09 October 2024
ACCEPTED 03 February 2025
PUBLISHED 04 March 2025
CITATION
Zhang M, Aykroyd RG and Tsoumpas C (2025)
Bayesian modeling with locally adaptive prior
parameters in small animal imaging.
Front. Nucl. Med. 5:1508816.
doi: 10.3389/fnume.2025.1508816
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© 2025 Zhang, Aykroyd and Tsoumpas. This is
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terms of the Creative Commons Attribution
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these terms.
Bayesian modeling with locally
adaptive prior parameters in small
animal imaging
Muyang Zhang1, Robert G. Aykroyd1*and
Charalampos Tsoumpas1,2
1
Department of Statistics, School of Mathematics, University of Leeds, Leeds, United Kingdom,
2
Department of Nuclear Medicine and Molecular Imaging, University Medical Center Groningen,
University of Groningen, Groningen, Netherlands
Medical images are hampered by noise and relatively low resolution, which
create a bottleneck in obtaining accurate and precise measurements of living
organisms. Noise suppression and resolution enhancement are two examples
of inverse problems. The aim of this study is to develop novel and robust
estimation approaches rooted in fundamental statistical concepts that could
be utilized in solving several inverse problems in image processing and
potentially in image reconstruction. In this study, we have implemented
Bayesian methods that have been identified to be particularly useful when
there is only limited data but a large number of unknowns. Specifically, we
implemented a locally adaptive Markov chain Monte Carlo algorithm and
analyzed its robustness by varying its parameters and exposing it to different
experimental setups. As an application area, we selected radionuclide imaging
using a prototype gamma camera. The results using simulated data compare
estimates using the proposed method over the current non-locally adaptive
approach in terms of edge recovery, uncertainty, and bias. The locally adaptive
Markov chain Monte Carlo algorithm is more flexible, which allows better
edge recovery while reducing estimation uncertainty and bias. This results in
more robust and reliable outputs for medical imaging applications, leading to
improved interpretation and quantification. We have shown that the use of
locally adaptive smoothing improves estimation accuracy compared to the
homogeneous Bayesian model.
KEYWORDS
Bayesian modeling, inhomogeneous parameter, image processing, Markov random
field, Markov chain Monte Carlo
1 Introduction
As a non-invasive method, medical imaging is extensively used for diagnosing and
monitoring various medical conditions (1). However, the loss of information during the
scanning and image acquisition processes often creates an observed image that is
blurred and contains noise (2). The systematic relationship between the observed and
true image is often modeled linearly using a transformation matrix. However, this
transformation matrix is typically large and ill-posed, so directly solving a system of
linear equations to obtain the exact image is infeasible.
In medical image processing, Bayesian modeling transforms an ill-posed problem into
a well-posed problem by introducing a prior distribution as a form of penalization or
regularization. Moreover, this method holds potential for application in biomedical
image reconstruction (3,4). Most approaches, however, have the tendency to not only
TYPE Original Research
PUBLISHED 04 March 2025
DOI 10.3389/fnume.2025.1508816
Frontiers in Nuclear Medicine 01 frontiersin.org

smooth out noise but also to smooth out the signal. This raises
the question of how to determine a prior distribution for
smoothness in order to avoid both under- and over-smoothing.
Homogeneous prior distributions have been found to be less
effective in scenarios with rapid changes, such as medical images
(5,6). Inhomogeneous Bayesian modeling aims to fully utilize
the distribution’s properties. Instead of employing different prior
distributions, one could consider using a prior distribution
with hyper-prior parameters (7). Therefore, we integrate
inhomogeneous factors into the modeling by updating our prior
distribution, introducing locally adaptive hyper-prior parameters,
with high dimensions, instead of a single global hyper-parameter.
2 Materials
Given the absence of real images in practical scenarios, certain
statistical measurements such as mean squared error (MSE)
evaluating the efficiency of statistical modeling by minimizing
the difference between real and estimated values are limited in
application. Hence, creating simulated data to mimic the real
image is required to bridge this gap.
2.1 Designed simulation with high contrast
The process for creating simulated data is as follows: we aim to
generate simulated data Xthat closely approaches the true image
represented as “real data.”We then apply random noise to create
the degraded observed data Y. In this case, instead of using the
projection data Pin the posterior, we can obtain estimations by
sampling from the posterior conditional distribution given Y.
This approach also allows for a comparison between estimations
and the corresponding real data X. The simulation is based on
the function minimum residual sum of squares (RSS), where we
can adjust the parameters to achieve simulations with different
levels of noise. The general expression of the function is
min
u
kYA
d
ðÞXr,c,z,TðÞ
e
k2,
where ris the collection of four objects’radius: r¼ðr1,r2,r3,r4ÞT;
c¼ðc1,c2,c3,c4ÞTand z¼ðz1,z2,z3,z4ÞTrepresent the
parameter vectors of the objects’central position of x-axis and
y-axis, respectively.
e
represents random errors introduced during
observation. Given the prior information, the density for each
cylinder is identical, denoted as T.
The elements within Arepresent the probabilities that pixels in
Xcan be transformed into corresponding information in Y. In our
application, aij follows a bivariate-normal distribution with zero
covariance:
aij N
m
,SðÞ,
where
m
¼ð0, 0Þand S¼
d
2I22(Iis an identity matrix). The
standard deviation
d
is highly dependent on the distance
between the scanner and the scanner object. The transformed
information from Xto Ydecreases with increasing distance.
Ideally, when there is no distance between the scanner and the
object, Ais an identity matrix, allowing for complete
information transformation. In medical imaging, despite the
scanner’spositionbeingfixed during an examination, the
distance may vary slightly depending on the size of the object.
Hence, when there is a considerable distance between the
scanner and scanned objects, information is missing due to
the reception of limited signals. Conversely, increasing the
distance introduces more blurring. Overall, it is crucial to
strike a balance between maintaining information and
reducing blurring simultaneously; in other words, deciding
how to set
d
becomes vital.
2.2 Designed simulation with high contrast
To generate simulations, we set values for parameters.
Afterward, random noise is applied to create the degraded
observation Y. In this case, we can obtain estimations by
sampling from the posterior conditional distribution given X(8).
This approach also allows for a comparison between estimations
and the corresponding real data X. The simulations are stored in
pixel matrices of the size 29 58.
Observed data Y, viewed as a degraded version of the actual
distributions Xwith blur and noise, is comparable to the
projection dataset in reality. As shown in Figure 1a,thetrue
distribution consists of four sharp regions with sharp
boundaries. The red lines indicate the regions of interest,
namely, the 20th row and the 36th column, which are applied
in the following pixel estimations. Figure 1b depicts observed
data with low contrast resolution; blur is evident around the
edge of each region.
The hot region and background pixels are constant, around
1,100 and 0, respectively. In the three simulations, the noise
presents in different ways. The first simulated dataset has the
lowest noise compared to the other two datasets since there is a
distinguished value gap between the background, where the
pixels are close to zero, and the hot region, where the pixels are
close to 1,100. In the second simulated dataset in Figure 2,it
creates an environment where the hot regions are soft tissues
with smooth edges; the average value of pixels in the background
is higher than in other experiences, while the peak values are
smaller than others. The red dots indicate the positions of pixels
applied in the following pixel estimation comparison. Finally, the
ones in Figure 3 describe the situation when there is a reduction
in scanning time. There is a higher noise level than the first
simulation. These simulations with artificial noise are regarded as
degraded images. The proposed methods are assessed using
simulated examples designed specifically to mimic real
experimental data collected as part of system
calibration experiments.
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2.3 Designed simulation with smoothing
edge
Supposing the objects within the circle have a soft edge instead
of a hard one. In reality, the edges are likely to be considerably
softer. Therefore, a smoothing pattern is obtained by applying a
Gaussian kernel filter to the datasets presented in Figure 2a.IfG
is a Gaussian kernel, we say that X1¼GX. The high blurring
around the high-contrast edge between the hot regions and the
background makes it difficult to detect the original edge.
2.4 Simulated experiment with lower
detected counts
Image quality improves with a reasonable extension of
scanning duration. Again, supposing the scanning time is
reduced, the observation image contains blurring. Another
degraded simulated image is created, as shown in Figure 3. The
high-contrast edge between the hot regions and the background
blurring and the observation image is expected to contain more
noise than the first observation image Y.
Ultimately, these simulated Y,Y1,andY2serve as our “observed
dataset”for the application sections in this paper, while the simulated
Xremains an unknown parameter that needs to be estimated.
However, as the values of simulated Xare obtained in advance, it
allows for a comparison between simulated Xand its estimations
from our defined posterior distribution in the following step.
3 Methods
In the case of medical imaging, the aim is to estimate a discrete
version of the unknown continuous emitter activity distribution X
FIGURE 1
Simulation datasets: true information Xand its correspondingly observation data Y.(a) Simulated image X.(b) Observation Y.(c) 3D simulated image
X.(d) 3D observation Y.
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from the single projection data Y. Suppose the unknown object X
is expressed as a set of mvolume pixels, X¼ðXj:j¼1, 2, mÞ,
where Xjrepresents the constant value of emitter activity in the
jth pixel. The data Y¼ðYi:i¼1, 2, nÞis related to the actual
activity through the deterministic equation EðYÞ¼fðXÞand
depending on the application being studied, fðXÞcan become a
linear function, or remain a non-linear function, especially when
scanning time and multilayer factors are considered (9–11).
3.1 Likelihood function
A Poisson form, identified as suitable for various image
processing with quantum noise, is particularly appropriate for
the -eyeTM camera projection data considered in our application
(12–14). The first -eyeTM scintillation camera developed by
BIOEMTECH (Athens, Greece) was used to generate two-
dimensional medical images (15).
The conditional distribution for observation Ygiven the
unknown true radionuclide distribution Xis as follows:
fYjXy1,y2,,ynjxðÞ¼
Y
n
i¼1
l
yi
i
exp 
l
i
ðÞ
yi!, (1)
where E[Yi]¼
l
i¼Pm
j¼1aijxj,j¼1, 2, , m. In other words, each
projection data value has an interaction with the whole vector X;
a known transformation matrix is denoted A¼[aij]nm. The
element aij is the probability that a gamma particle emitted from
pixel location iis recorded at pixel location j. The error
e
can be
expressed as an n1 vector with elements ð
e
i:i¼1, 2, , nÞ,
which may come from various types of unavoidable
measurement errors.
FIGURE 2
Simulation datasets with a reduction in scanning time: true image X1(left) and observed image Y1(right). (a) Simulated image X1.(b) Observation Y1.
FIGURE 3
Simulation datasets with scanning time reduction: observed image Y2and its corresponding actual image X.(a) Simulated image X.(b) Observation
image Y2.
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If the transformation matrix Ais square and non-singular,
image Xcan be easily realized by the least squares estimator
X arg minXkYAXk2, when the squared error function is
minimized:
^
X¼ATA

1ATY,
where ^
Xis the least square estimate of X. However, in image
processing, the transformation matrix Atypically has a complex
structure with high dimensions and is rectangular with
unavailable pseudo-inverse. This results in ill-posed and ill-
conditioned issues (16).
By imposing additional constraints in terms of prior
knowledge, a Bayesian approach transforms an ill-posed inverse
problem into one that is well-posed (17). During the modeling
process, a prior distribution is constructed to capture the
statistical properties of the image, and then estimation uses a
posterior distribution derived by the combination of prior and
likelihood. The uncertainty between Xand Yis captured by
likelihood function fYjXðyjxÞand the posterior density
fXjYðxjyÞis used for inference after incorporating prior
knowledge
p
XðxÞ.
3.2 Prior distribution
Discrete images comprise elements of finite product spaces,
and the probability distributions on such sets of images as prior
information are a critical part of image processing. The
efficiency of prior distributions depends on the available first-
hand information. Regarding informative priors, it is generally
expressed as the Gibbs measure, which was borrowed from
statistical physics (18). The primary goal of introducing this
type of probability is to describe the features relative to
“macroscopic”physical samples, such as an “infinite system”
(19,20).
The Gibbs probability distribution has gradually found
applications in various fields, including “Gibbs Sampling”in
Bayesian modeling. The Gibbs distribution is defined as
p
XxjBðÞ¼Z1exp B
k
xðÞðÞ,Z¼ðx
exp B
k
xðÞðÞdx,
X[Rm,B.0,
(2)
where Zis the normalization for the Gibbs distribution; the energy
function is
k
, representing the energy of the configuration of pixels;
and Bis a non-negative smoothing parameter. Furthermore, the
energy function can be rewritten as the sum of local energy
functions FðÞ:
k
xðÞ¼
X
m
j¼1
FjxðÞ, (3)
where FjðÞ represents the local energy function to corresponding
Xj¼xj.
3.2.1 Markov random field for pixel differences
Briefly, the primary assumption for Markov random field
(MRF) models is that a variable is only related to its adjacent
variables while being conditionally independent of the others
(21,22). Specifically, the clique-based structure makes MRF
models well-suited for capturing local pixel relations in
images. It proposes a lattice system, denoted as G¼ðV,EÞ,to
represent the connections between pixels (as illustrated in
Figure 4). For instance, assuming the yellow node represents
the object under analysis in the first-order system, its four
closest neighbors are located to its left, right, bottom, and top
sides, as indicated by the black solid lines. In the second-order
system, an additional four neighborhoods are considered,
located at the top-left, top-right, bottom-left, and bottom-right
corners around the yellow node, as indicated by the dashed
lines. In this system, pixels are represented as nodes (V), and
edges (E) connect all the nodes. While the shape of the grid is
not required to be rectangular, it is the most common
in applications.
FIGURE 4
Two-dimensional rectangular grid G. The blue and yellow nodes in the lattice represent pixels, while the solid lines and dashed lines describe first-
order and second-order neighborhoods, respectively.
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Once we define the MRF prior, the local function in Equations
2,3is rewrite a linear combination of differences between the pixel
and its neighborhoods:
FjxðÞ¼ X
t[@jðÞ
wjt
f
xjxt

, (4)
where wjt represents the weight for each paired comparison, as the
increased order in MRF, and wjt may change according to the
interaction within the neighborhood. For example, for the first-
order MRF, wjt ¼0:5 if there is only four neighbors to consider.
The set of nodes @ðjÞin a finite graph Xwith edges jt.
Finally, after employing the Markov random field for pixel
difference, the updated prior distribution is
p
XxjBðÞ¼Z1exp BX
m
j¼1X
t[N@jðÞ
wjt
f
xjxt

0
@1
A:(5)
Here, only local characteristics are considered when it comes to
estimating the individual pixel, which can be briefly divided into
two properties. First, in comparison to the global character which
includes all the pixels, we only study the smaller number of pixel’s
neighbors; for instance, the first-order neighborhood is adopted in
the prior distribution. Second, given the local property of the prior
distribution, we assume that the posterior is sensitive to the local
property of the prior distribution. If we define the potential
functions as absolute value function, then
f
ð
m
Þ¼j
m
jrespectively.
Thereby, the corresponding priors are quantified Markov random
fields with absolute function (LMRF) combined with Equations 4,5:
p
Xj
t
xj
t
ðÞ¼
Y
m
j¼1
1
2
t
j

exp Pt[@jðÞ
jxjxtj
t
j

,
xj0,
t
j.0,
(6)
where X¼{xj,j¼1, 2, , m}andxjis conditional based on the
neighbor’svaluesand
t
¼{
t
j,j¼1, 2, , m} is the local conditional
variance in the prior distribution, which accounts for the value
variances among individual pixels and its four neighbors. As a
comparison to Equation 6,wedefine a homogeneous prior variance
t
to capture the global variance between the pixel differences:
p
Xj
t
xj
t
ðÞ¼
Y
m
j¼1
1
2
t
ðÞ
exp Pt[@jðÞ jxjxtj
t

,
xj0,
t
.0:
The inhomogeneous prior relies on the contrast levels in the image’s
segmentation, making it locally adaptive to various image densities.
In order to cover all scenarios among pixel neighborhoods, the
modeling with inhomogeneous hyper-parameter
t
can tolerate a
large value fluctuation and detect a small variation within the
smoothing areas.
3.2.2 High-dimensional Markov random field
The image dataset presents a two-dimensional image in our
application. Hence, we only consider the first-order system in
Markov random field priors, where the pixel and its four closest
neighbors are on the same planet. However, if the application
dataset from the projection data to the tomography image, where
pixels within two-dimensional space transfer into voxels within
three-dimensional space, we can introduce another two
neighbors of pixels based on the first-order system. As seen in
Figure 5, the left side, as a comparison to the right side, displays
the Markov random field within two-dimensional space. There
are two additional red nodes applied based on the first-
order system.
The Markov random field prior distribution can still be written
as a general form of Gibbs distribution:
p
XxjBðÞ¼kBmexp BX
m
j¼1X
t[N@jðÞ
wjt
f
xjxt

0
@1
A,
xj0,
t
.0,
where the number of neighbors in N@ðjÞincreases from four to six
neighbors in three-dimensional space.
3.2.3 Hyper-prior distribution
In the inhomogeneous model,
t
is a vector of unknown local
parameters which can either be defined by allocating a series of
artificial values or by introducing another level of modeling that
correctly incorporates the additional uncertainty. The first
solution requires advanced information for the assignment which
is not usually available. Alternatively, the second action employs
a hyper-prior distribution,
pt
ð
t
Þ, which is diffuse unless there is
FIGURE 5
Markov random field in three-dimensional space. Left image: Markov
random field in two-dimensional space. Right image: Markov
random field in three-dimensional space. The nodes with different
colors in the lattice represent pixels, while the dashed lines
describe the connection between the objective pixel (orange
node) and its neighbors.
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more informative information available. Here, an exponential
distribution is used for each
t
jindependently but with a common
rate parameter,
g
, that is,
t
jj
g
Expð
g
Þ. The value of
g
is chosen
to produce a long-tailed distribution to cover varied scenarios. As
well as promoting smaller values of
t
jcompared to a non-
informative uniform distribution, it also avoids the need to impose
an arbitrary upper range limit. The distribution for
pt
j
g
ð
t
j
g
Þ:
pt
j
gt
j
g
ðÞ¼
Y
m
j¼1
g
exp 
gt
j

,
g
.0:(7)
The elements in
t
¼{
t
j;j¼1, 2, , m} are the collection of local
prior variances and
g
refers to the rate parameter in the hyper-
prior distribution, a smaller value for the rate parameter indicates
the more flat hyper-prior distribution.
3.2.4 Rate parameter in the hyper-prior
distribution
The fundamental strategy for establishing a hierarchical
Bayesian multilevel model is specifying prior distributions for
each unknown parameter, which enables estimation for each
parameter based on the other prior distributions from different
levels. Therefore, it can incorporate more prior knowledge and
hence improve estimation accuracy. However, once additional
prior levels are involved in the model, it can result in a
prolonged computation time and a high demand for supportive
information. Figure 6 explains the multilevel structure in our
hierarchical Bayesian model.
Now that an additional hyper-parameter,
g
, has been
introduced, further modeling must be considered. Due to a lack
of supportive prior knowledge, it is common to utilize a uniform
distribution or a Jacobian transformation of a uniform
distribution. Another option is to introduce a conjugate prior
distribution which here would be the Gamma distribution with
shape and scale parameters.
For the first method, the uniform prior allocates the same
probability to each value within its defined range; it is subjective
but hard to estimate values when the value is extremely small (23,
24). However, overly complex models have a high risk of poor
performance (25). The use of a Gamma distribution unavoidably
expands the number of unknown parameters. We expect, however,
that the density peak of the Gamma distribution to be a small value
and therefore, an exponential distribution with a small value for the
rate parameter,
u
, should be adequate: ρexp(Θ).
The result of these multilevel models, combining likelihood
function in Equation 1, prior distribution in Equation 6, and
hyper-prior distribution in Equation 7, produces the following
posterior distribution:
fX,
t
,
g
jYx,
t
,
g
jyðÞ
¼fYjXyjxðÞ
p
Xj
t
xj
t
ðÞ
pt
j
gt
j
g
ðÞ
pgg
ðÞ
/Y
n
i¼1
l
pi
i
exp 
l
i
ðÞ
pi!Y
m
j¼1
1
2
t
j

exp Pt[@jðÞjxjxtj
t
j


g
exp 
gt
j

"#
exp 
ug
ðÞ
,
(8)
where X¼{Xj,j¼1, 2, , m} represents the unknown radionuclide
distribution,
t
¼{
t
j,j¼1, 2, , m} are the locally adaptive prior
variance parameters,
g
is the hyper-parameter modeling
t
, and
Y:{Yi,i¼1, 2, , n} is the observed data.
There is a concern about the potentially unreliable posterior
estimations from the hierarchical model, especially for estimating
small values. Hence, instead of estimating the hyper-prior
parameter
g
, we consider a calibration experiment where a series
of values for
g
, from 104to 103, are used to investigate the
trend in the mean squared error under different
g
. In other
words, instead of bringing another level of hyper-prior
FIGURE 6
Factor graph for hierarchical Bayesian model.
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distribution,
g
is fixed, but based on the calibration experiments
rather than requiring prior knowledge.
As shown in Figure 7, a similar increased pattern of MSE
occurs among the three data applications with different levels of
noise and blurring. The applications are the three designed
examples. The estimation using small
g
, smaller than 0:1 say, is
robust and based on mean squared error performance.
Combining the posterior estimation for
g
with the hierarchical
Bayesian model, we can conclude that a rate parameter
g
around
102is a robust hyper-prior parameter value for our current
application and hence in what follows
g
¼102.
4 Results
4.1 Estimation strategy
Once the posterior distribution involving likelihood, prior, and
hyper prior distribution is defined, as seen in Equation 8.A
metropolis Hasting algorithm is used for the estimation. This is
an example of the general Markov chain Monte Carlo (MCMC)
approach that is able to handle complex distributions where
other estimation methods fail. Details of the estimation process
of our application can be found in Estimation algorithm of
Markov chain Monte Carlo, Appendix Table A1. Apart from
interest in the posterior estimation of the unknown radionuclide
distribution, X, the locally-adaptive hyper-prior parameters τ
must be estimated simultaneously. In general, the single global
prior variance τshould capture the global variance between
pixels. However, in the locally-adaptive model the hyper-prior
variances τmeasure each pixel’s variation within the
corresponding neighbourhood, and hence we expect the elements
{τ=τ
j
,j=1,2,...,m} to be non-identical.
4.2 Posterior estimation
Figure 8 shows the posterior estimates for the three scenarios,
with X,X1, and X2. For comparison, posterior estimates using the
homogeneous model (left) are shown along with the estimates
using the inhomogeneous, locally adaptive prior model (right).
After incorporating prior flexibility, estimation improvements are
apparent for deblurring and denoising, particularly in the second
and third applications of simulation datasets with high-level blurring.
In the first application, where more accurate and sufficient
first-hand information Yis available, the results from both
models (the homogeneous and inhomogeneous models,
respectively) are similar. However, in the third application,
despite the foundational truth being the same as in the first
example, the first-hand information Y2contains high levels of
noise and blurring. In this case, the hierarchical Bayesian
modeling successfully produces an image based on Y2that is
closer to the truth compared to the outcomes from the
homogeneous modeling.
In addition to posterior estimates of X, standard deviation and
bias values are shown in Figures 9,10.
For standard deviation, the results from the homogeneous
models (left) show higher values than the inhomogeneous
models (right), especially for the hot regions and smoothing
background. The variation can still be found in high-contrast
FIGURE 7
MSE of posterior estimation with fixed hyper prior parameter
g
. Applications refer to the employment of simulation datasets (Y,Y1, and
Y2), respectively.
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edges while there is a reduction within the hot regions and smooth
backgrounds where the pixel differences are assumed to be small.
The bias in Figure 10 provides clear additional evidence that
the estimation accuracy of the locally adaptive model is improved
compared to that of the homogeneous model. For instance, in
Figure 10f, the bias around the high-contrast edge and
smoothing hot regions is hardly detected in the right image as
compared to the image on the left side. Generally, the image
patterns of standard deviation and bias on the right side have
less information observed compared to the left side. In other
words, the adjusted model captures the variation and bias within
the posterior estimations.
FIGURE 8
Comparisons of image processing for three simulations, X,X1and X2, respectively. From the right side, (a,c,e) present the image processing with a
global hyper-prior variance from the homogeneous Bayesian modeling. While (b,d,f) show the image processing under hierarchical Bayesian
modeling with local hyper-prior variances.
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4.3 Comparisons of posterior pixel
estimation
Figures 11a,bshow box plots of three pixels from the
background and three from the hot regions, respectively. The
distribution in the blue box plots (left of each pair) is close to
the red dashed line, representing the truth. In contrast, the ones
in the orange box plots (right of each pair) have significant bias.
Overall, the Bayesian model with locally adaptive hyper-prior
variance introduces estimation flexibility to realize a more
accurate outcome in each application.
Figures 12–14 present comparisons of estimation pixels
between the posterior containing homogeneous and
inhomogeneous hyper-prior parameters in each simulation
FIGURE 9
Standard deviation of posterior estimations for true X,X1and X2in image pattern. (a,c,e) present the image patterns of standard deviation that come
from the homogeneous Bayesian model with a global hyper-prior variance. (b,d,f ) show the image patterns of standard deviation originate from
hierarchical Bayesian modeling with local hyper-prior variances.
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application (X,X1, and X2, respectively). In general, the
estimations with a global hyper-parameter (left side) tend to have
higher variations and broader credible intervals than those with
locally adaptive hyper-parameters (right side), especially for the
hot regions. Although the wider credible interval is more capable
of covering variation in the values in comparison to the narrow
credible interval, we noticed that there are some severe
estimation fluctuations, especially within the smoothing pixel
region, for example, in Figures 12,13.
Furthermore, the value difference between the estimations
from the locally adaptive model and the truth is smaller than
the homogeneous ones. For instance, without sufficient first-
hand information in the third simulation experience, as shown
in Figure 14, the modeling with local hyper-prior variance
produces a more accurate estimation as opposed to the one
with global hyper-prior variance. Table 1 shows the
corresponding estimation measurements of eight selected
pixels in the 20th row in the third simulation application.
FIGURE 10
Bias of posterior estimations for truth X,X1, and X2in image pattern. (a,c,e) present the image patterns of bias that come from the homogeneous
Bayesian model with a global hyper-prior variance. (b,d,f) show the image patterns of standard deviation originating from hierarchical Bayesian
modeling with local hyper-prior variances.
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Overall, the Bayesian model with locally adaptive hyper-prior
variance introduces estimation flexibility to realize a more
accurate outcome in each application.
5 Modeling application in small animal
imaging
In the previous simulation examples, locally adaptive
Bayesian modeling proves the advantages of estimation accuracy.
We now aim to apply this technique to images obtained
from mouse scanning by using -eyeTM to confirm the
conclusion obtained from the former sections. Figure 15a
shows the image of a mouse injected with a technetium-99m
labeled radiotracer acquired with -eyeTM, and Figure 15b
presents the correspondingly designed dataset for
estimation application.
The results are presented in Figures 16b,c. Here, we assign
the rate parameter
g
¼102in the hyper-prior distribution.
As in the previous examples, the estimation with the locally
adaptive hyper-prior variance performs better compared to the
homogeneous model in terms of deblurring and denoising; for
instance, the smoothing edge (red circle) between the
background and hot region is clearer in Figure 16c than in
Figure 16b.InFigure 16d, the estimation of hyper-prior
t
is
displayed, showing the clear non-identical value distribution of
t
.
The high-dimension
t
introduces flexibility when estimating
pixel variance.
The locally adaptive Bayesian model with inhomogeneous
hyper-prior parameters can specifically describe the
FIGURE 11
Box plots of posterior distributions. The box plots in blue show the results of the model with locally adaptive hyper-prior parameters, and the box plots
in orange represent estimations from the homogeneous model. The red dashed line is the true pixel value. (a) Box plots of three background posterior
distributions from the second example. (b) Box plots of three hot-region posterior distributions from the second example.
Zhang et al. 10.3389/fnume.2025.1508816
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probabilistic distribution of each unknown pixel xj.Beyond
pixel-wise posterior estimation, these inhomogeneous hyper-
prior variances enabled a more detailed exploration of the
outcomes. For instance, plotting the estimated hyper-
prior variances directly reveals spatial information about
the pixels. In conclusion, the locally adaptive Bayesian
modeling constructs a hierarchical network that encompasses
multiple levels of parameters. This network effectively
integrates information from estimated parameters across
different levels.
6 Tikhonov regularization and real
image application
The application of Bayesian modeling in cylinder simulation
datasets demonstrates advantages in terms of deblurring and
denoising. Therefore, to confirm the applicability of Bayesian
modeling, another real-world data application is required.
Tikhonov regularization has been identified as useful as it
introduces the homogeneous regularization term into ill-
conditioned problems, specifically in the context of inverse
problems (26–28). Therefore, a comparison between the
estimations from Bayesian modeling and Tikhonov regularization
is necessary.
6.1 Tikhonov regularization comparison
Tikhonov regularization for medical image processing, which
holds the linear relationship between observation image and real
image, can be written as
min kYAXk2
2þ
l
kLAk2
2

,
where Ais the transformation matrix, and Yand Xare the
observation dataset and real unknown dataset, respectively. The
regularization parameter
l
controls the trade-off between the
model fitness and the regularization term. The regularization
matrix Lcontains the prior information about the solutions.
Here, we employ the identity matrix as the regularization matrix
Lbecause of the lack of supportive prior information.
In the context of the regularization parameter
l
, the criterion
of cross-validation has been applied in various regularization
algorithms, including Tikhonov regularization. Cross-validation
FIGURE 12
Pixel posterior distributions under homogeneous (left) and locally adaptive (right) models with hyper-prior variance parameters, using the first
simulation dataset. The posterior distributions for the 20th row are shown at the top, while those for the 36th column are shown at the bottom.
(a,b) Pixel estimates under the homogeneous and inhomogeneous modelling for the 20th row; (c,d) Pixel estimates under the homogeneous and
inhomogeneous modelling for the 36th column.
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selects the optimum regularization parameter
l
by identifying the
minimum estimation residuals.
1
The estimation outcome from
Tikhonov regularization, using observation Yfrom within the
first simulation dataset, is shown in Figure 17. The left side
displays the observation of the first simulation dataset, while the
right side presents the estimation from Tikhonov regularization.
Compared to the Bayesian application shown in Figure 8, the
blurring in Figure 17b can still be detected around the high-
contrast area between the background and hot region. Hence, the
estimation for real image Xis not accurate, since regularization
applies to the whole information not only noise but also pixel
values. In other words, the smoothing effect from regularization
applies globally to pixels within both background and hot
regions simultaneously, regardless of varied pixel densities. The
estimated pixels with a large value of regularization (represented
in green) are smoother than those with small regularization
(represented in blue). Furthermore, some non-negative pixels
from the background are unavoidably transformed into negatives
after applying regularization.
Similarly, the estimations of specific columns and rows within
the pixel matrix from Bayesian modeling and Tikhonov
regularization are presented in Figure 18. Compared to the
Bayesian modeling, Tikhonov regularization is only based on the
pixel point estimation without consideration of the pixels’
environment. Furthermore, unlike estimations from Markov
chain Monte Carlo within the Bayesian framework, the
distribution of estimated pixels and the quantified information,
including the confidence intervals of estimations, are not available.
6.2 Real medical image application
Here, the employed medical image for the mouse kidney was
obtained by using a dimercaptosuccinic acid scan (DMSA).
Compared to the -eyeTM camera, the DMSA scan with
technetium-99 m labeled radiotracer is well-known for its valuable
capability in identifying patients’kidney shape and location.
The information of the region of interest, where the kidneys are
located, is clearer in the reprocessed image in Figure 19b compared
to the observed image in Figure 19a.
FIGURE 13
Pixel posterior distributions under homogeneous (left) and locally adaptive (right) models with hyper-prior variance parameters, employing the second
simulation dataset. The posterior distributions for the 20th row are shown at the top, while those for the 36th column are shown at the bottom. (a,b)
Pixel estimates under the homogeneous and inhomogeneous modelling for the 20th row; (c,d) Pixel estimates under the homogeneous and
inhomogeneous modelling for the 36th column.
1
The estimation algorithm for regularization parameter “cv.glmnet”and the
application of Tikhonov regularization “glmnet”can be find in R package
“glmnet.”
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FIGURE 14
Pixel posterior distributions under homogeneous (left) and locally adaptive (right) models with hyper-prior variance parameters, employing the third
simulation dataset. The posterior distributions for the 20th row are shown at the top, while those for the 36th column are shown at the bottom. (a),(b)
Pixel estimates under the homogeneous and inhomogeneous modelling for the 20th row; (c),(d) Pixel estimates under the homogeneous and
inhomogeneous modelling for the 36th column.
TABLE 1 The list includes estimation measurements.
Position ucl.H ucl.In lci.H lci.In mean.H mean.In
1 45.36 20.60 0.90 18.47 15.80 19.58
4 32.58 20.09 0.48 16.81 11.46 18.67
8 35.69 23.06 0.59 13.86 12.18 18.36
14 1,219.05 1,136.55 1,045.48 1,086.63 1,132.85 1,111.66
18 79.04 30.05 2.13 15.27 29.53 21.53
22 62.27 24.30 1.66 16.50 23.93 20.46
24 48.84 25.52 0.55 16.01 16.94 20.55
26 123.85 43.67 3.40 9.73 44.13 24.86
“H”indicates the posterior estimation from Bayesian modeling with the global LMRF, while “In”indicates the posterior estimation from Bayesian modeling with locally adaptive LMRF. The
outcomes are stored to two decimal places.
FIGURE 15
Scan of a mouse using -eyeTM:(a) real scan and (b) a simulated dataset.
Zhang et al. 10.3389/fnume.2025.1508816
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FIGURE 16
Image processing under the Bayesian modeling. (a) shows the simulation image with noise and blurring. (b) displays image processing under the
Bayesian modeling with homogeneous hyper-prior parameter
t
.(c) displays image processing under the Bayesian modeling with inhomogeneous
hyper-prior parameter
t
.(d) shows the posterior estimation of locally adaptive hyper-prior variances in the image pattern.
FIGURE 17
Observation image and the corresponding estimation from Tikhonov regularization with the optimum regulation. (a) Observation Y.(b)
Tikhonov regularization.
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FIGURE 18
Estimation comparison between Bayesian modeling and Tikhonov regularization. The estimation from Bayesian modeling highlighted in the dark color
has a credible interval in gray. The estimations from Tikhonov regularization are presented with two regularization options: the green line represents
the optimum regularization defined by the cross-validation method, while the blue line represents manual regularization with
l
=0.1 applied. Here, the
pixel estimations for the 20th row are shown at the top, while those for the 36th column are shown at the bottom. (a) Estimation comparison between
Bayesian modeling and Tikhonov regularization I. (b) Estimation comparison between Bayesian modeling and Tikhonov regularization II.
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7 Conclusions
We extended the hierarchical Bayesian model for image
processing by introducing the locally adaptive hyper-prior
variance
t
, replacing a single homogeneous hyper-prior variance
t
. The locally adaptive model adjusted the hyper-prior variances
based on the different local spatial conditions, effectively allowing
the hyper-prior variances to vary for each location estimation.
This adaptation provided the model with greater flexibility in
estimation, subsequently improving accuracy. In our exploration
of hyper-prior parameter estimation, we found that weakly
informative prior distributions, such as a relatively flat
exponential distribution, performed more efficiently compared to
non-informative priors. This was evidenced by higher
convergence rates and lower estimation correlations.
The locally adaptive Bayesian model with inhomogeneous hyper-
prior parameters can specifically describe the probabilistic distribution
of each unknown pixel. Beyond pixel-wise posterior estimation, these
inhomogeneous hyper-prior variances enabled a more detailed
exploration of the outcomes. For instance, plotting the estimated
hyper-prior variances directly revealed spatial information about the
pixels. In conclusion, the locally adaptive Bayesian approach
constructs a hierarchical model that encompasses multiple levels of
parameters. This approach effectively integrates information from
estimated parameters across different levels, leading to improved
image estimation. Consequently, there is the potential for
enhancements in quantification, diagnosis, and treatment
monitoring in medical imaging applications.
Data availability statement
The original contributions presented in the study are included
in the article/Supplementary Material, further inquiries can be
directed to the corresponding authors.
Ethics statement
The manuscript presents research on animals that do not
require ethical approval for their study.
Author contributions
MZ: Conceptualization, Data curation, Formal Analysis,
Investigation, Methodology, Software, Validation, Visualization,
Writing –original draft, Writing –review & editing. RGA:
Conceptualization, Data curation, Investigation, Methodology, Project
administration, Resources, Software, Supervision, Validation,
Visualization, Writing –review & editing. CT: Conceptualization,
Data curation, Investigation, Methodology, Project administration,
Resources, Software, Supervision, Validation, Visualization, Writing –
review & editing.
Funding
The authors declare that no financial support was received for
the research, authorship, and/or publication of this article.
FIGURE 19
A real application in a medical image using DMSA (left) and the corresponding posterior estimations from Bayesian modeling with LMRF prior
distribution (right). (a) Observed image from DMSA. (b) Estimation from Bayesian modeling.
Zhang et al. 10.3389/fnume.2025.1508816
Frontiers in Nuclear Medicine 18 frontiersin.org

Acknowledgments
We thank Steve Archibald and John Wright from the
University of Hull for providing us with the data which John
performed using 99mTc.
Conflict of interest
CT declares collaboration with BIOEMTECH where he undertook
a secondment of about 6 months in 2018 sponsored by the EU Horizon
2020 project: Vivoimag (https://vivoimag.eu/). The remaining authors
declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a
potential conflict of interest. The authors declared that they were an
editorial board member of Frontiers, at the time of submission. This
had no impact on the peer review process and the final decision.
Generative AI statement
The authors declare that no Generative AI was used in the
creation of this manuscript.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated organizations,
or those of the publisher, the editors and the reviewers. Any product
that may be evaluated in this article, or claim that may be made by its
manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found
online at: https://www.frontiersin.org/articles/10.3389/fnume.
2025.1508816/full#supplementary-material
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Appendix
Estimation algorithm of Markov chain
Monte Carlo
The hyper-parameter
t
is a global variable, and we assume
the distribution for a global hyper-parameter
t
is a uniform
distribution. However,
t
¼{
t
j,j¼1, 2, , m} becomes a vector
of unknown parameters. Therefore, there are
corresponding changes in terms of distribution
pt
j
g
ð
t
j
g
Þand
estimation process: MCMC. We initially define the
Gamma hyper-prior distribution with artificial values
j
¼1
and
u
¼2:
g
Gammað
j
¼1,
u
¼2Þ.Ifwewantto
estimate
t
for the kthiterationintheMarkovchain
Monte Carlo and update unknown parameters, follow the
steps below:
Once the chain has burned in, a process to drop a bunch of
initial iterations whose value probabilities are low, the estimation
for each parameter is supposed to be the sampling mean, we
seek, i.e., Eð
g
Þð
Pk
g
ðkÞÞ=k.
Estimation measurement
The estimation Xjcan be assessed by averaging the application
outcomes for the last Kstationary MCMC iterations. The most
popular and common indexes for accurate measurement are the
MSE, bias, and standard deviation (SD). Furthermore, residual
sum squares (RSS), the modeling fitness measurement, is widely
applied among model comparisons. For the subset of the whole
parameters, the forms of measurement are seen in Table A2.
The entire image is of size m, while any image sub-region is of
size R, i.e., Rm. The average MCMC estimation is EðxjÞand ^
xK
j
refers to the estimation of corresponding pixels in the Kth iteration.
The denominator in each of the expressions varies based on the
number of pixels considered.
TABLE A1 MCMC for modeling with locally adaptive hyper-parameter
g
.
Algorithm MCMC for modelling with locally adaptive
hyper-prior parameters (iteration k)
Input: A list of initial values {X1¼xð0Þ
1,X2¼xð0Þ
2,,Xm¼xð0Þ
m};
A list of initial values {
t
1¼
t
ð0Þ
1,
t
2¼
t
ð0Þ
2,,
t
m¼
t
ð0Þ
m};
An updated positive constant
s
ðkÞ,
s
ðkÞ
t
,
s
ðkÞ
g
.
For j¼{1, 2, , m}
1. Propose a new value xðkÞ
jNðxðk1Þ
j,ð
s
ðkÞÞ2Þ;if and only if xk
j0
2. Generate
m
unifð0, 1Þ
3. Accept xk
jwith probability
a
¼min 1, fXjY,
t
ðxðk1Þ
1,xðk1Þ
2,,xðkÞ
j,,xðk1Þ
mjY,
t
ðk1ÞÞ
fXjY,
t
ðxðk1Þ
1,xðk1Þ
2,,xðk1Þ
j,,xðk1Þ
mjY,
t
ðk1ÞÞ

4. Compare the
m
with the calculated
a
,
5. if:
m

a
then
6. Accept the proposal value xj¼xðkÞ
j
7. else xj¼xðk1Þ
j
end update x
8. Propose a new candidate value
t
k
jNð
t
k1
j,ð
s
ðkÞ
t
Þ2Þ;if and only if
t
k
j0
9. Generate
m
1unifð0, 1Þ.
10. Accept
t
ðkÞwith probability
a
1¼min 1, f
t
jXð
t
ðk1Þ
1,
t
ðk1Þ
2,,
t
ðkÞ
j,,
t
ðk1Þ
mjxk,
g
ðk1ÞÞ
f
t
jXð
t
ðk1Þ
1,
t
ðk1Þ
2,,
t
ðk1Þ
j,,
t
ðk1Þ
mjxk,
g
ðk1ÞÞ

,
11. if:
m
1
a
1then
12. Accept the proposal value
t
j¼
t
ðkÞ
j,
13. else
t
j¼
t
ðk1Þ
j
end update
t
14. Propose a new candidate value
g
kNð
g
k1,ð
s
ðkÞ
t
Þ2Þ;if and only if
g
k0
15. Generate
m
2unifð0, 1Þ.
16. Accept
g
ðkÞwith probability
a
2¼min 1, f
g
jX,
t
ð
g
kjxk,
t
ðkÞÞ
f
g
jX,
t
ð
g
k1jxk,
t
ðkÞÞ

,
17. if:
m
2
a
2then
18. Accept the proposal value
g
¼
g
ðkÞ,
19. else
g
¼
g
ðk1Þ
end update
g
20. end if
Repeat the above steps until it receives enough sampling size.
TABLE A2 List of statistical measurements.
Measurement Equation
Absolute bias ðPm
j¼1jxjEðxjÞjÞ=m
Regional absolute bias Pj[RjxjEx
j

j

=R
Standard deviation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PK
k¼1Pm
j¼1jxkðÞ
jEx
j

j2
q=KmðÞ
Regional standard deviation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PK
k¼1Pj[RjxkðÞ
jEx
j

j2
q=KRðÞ
Mean square error ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PK
k¼1Pm
j¼1jxkðÞ
jEx
j

j2
q=KmðÞ
Regional mean square error ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PK
k¼1Pj[RjxðkÞ
jEðxjÞj
2
q=ðKRÞ
Residual sum of squares PK
k¼1Pm
j¼1yjykðÞ
j

2=KmðÞ
Regional residual sum of squares PK
k¼1Pj[RyjykðÞ
j

2=KRðÞ
Zhang et al. 10.3389/fnume.2025.1508816
Frontiers in Nuclear Medicine 20 frontiersin.org