###### Abstract

Image fusion helps in merging two or more images to construct a more
informative single fused image. Recently, unsupervised learning based
convolutional neural networks (CNN) have been utilized for different types of
image fusion tasks such as medical image fusion, infrared-visible image fusion
for autonomous driving as well as multi-focus and multi-exposure image fusion
for satellite imagery. However, it is challenging to analyze the reliability
of these CNNs for the image fusion tasks since no groundtruth is available.
This led to the use of a wide variety of model architectures and optimization
functions yielding quite different fusion results. Additionally, due to the
highly opaque nature of such neural networks, it is difficult to explain the
internal mechanics behind its fusion results. To overcome these challenges, we
present a novel real-time visualization tool, named $\it{FuseVis}$, with which
the end-user can compute per-pixel saliency maps that examine the influence of
the input image pixels on each pixel of the fused image. We trained several
image fusion based CNNs on medical image pairs and then using our FuseVis
tool, we performed case studies on a specific clinical application by
interpreting the saliency maps from each of the fusion methods. We
specifically visualized the relative influence of each input image on the
predictions of the fused image and showed that some of the evaluated image
fusion methods are better suited for the specific clinical application. To the
best of our knowledge, currently, there is no approach for visual analysis of
neural networks for image fusion. Therefore, this work opens up a new research
direction to improve the interpretability of deep fusion networks. The FuseVis
tool can also be adapted in other deep neural network based image processing
applications to make them interpretable.

###### keywords:

Image Fusion; Convolutional Neural Networks (CNN); Saliency visualization;
Jacobians; Graphical User Interface (GUI).

xx 1 5 Received: date; Accepted: date; Published: date FuseVis: Interpreting
neural networks for image fusion using per-pixel saliency visualization
Nishant Kumar 1∗, Stefan Gumhold 1 Nishant Kumar, Stefan Gumhold
Correspondence<EMAIL_ADDRESS>Tel.: +49-351-463-38354

##  1 Introduction

The recent development of state-of-the-art imaging modalities has
revolutionized the way we perform our everyday activities. For example, in
self-driving cars, infrared images from the respective camera sensors
positioned at the vehicle help to detect obstacles such as pedestrians at
night. Remote sensing satellites, on the other hand, acquires multi-spectral
and multi-resolution images that are needed for object detection and
recognition from high altitudes. In medical diagnosis and treatment, Magnetic
Resonance Images (MRI), provides a detailed view of the internal structures of
the human brain such as white and gray matter whereas Positron Emission
Tomography (PET) and single photon emission computed tomography (SPECT) images
provide functional information like glucose metabolism and extent of cerebral
blood flow (CBF) or perfusion activity in the specific regions of the brain.
However, it is challenging to analyze such complementary information provided
by these image modalities individually. Multi-modal image fusion based image
processing technique solves this problem by combining two or more pre-
registered images from single or multiple imaging modalities into a fused
feature space. Ideally, a fusion algorithm should have low computational
overload and it should also preserve input features for usage in the
aforementioned real-world applications.

In the early years of image fusion research, most of the methods had been
proposed in a three-step approach. Firstly, the input images were converted
into several feature maps using an appropriate image transformation method
such as pyramid decomposition, wavelet transformation, and sparse
representation. Then, the coefficients of the multimodal feature maps were
combined using a pre-defined fusion strategy to get fused feature maps.
Finally, the fused image was reconstructed using the inverse transformation
applied to the fused feature maps. However, for an ideal fusion model design,
these methods mainly focused on enhancing the transformation and fusion
strategies to pursue good perceptual results by defining some intricate design
rules. Recently, several unsupervised machine learning based CNNs have been
proposed to achieve real time image fusion results. However, these image
fusion based neural networks lacks the trust of the end-users since there are
no tools to analyze the quality of the fused images as these networks act as
blackboxes.

To overcome this challenge, the neural network-based image fusion methods
evaluate the quality of the results using performance metrics such as
Structural Similarity Index (SSIM) ref9  that compares the perceptual
similarity between the input images and the fused image. However, the
shortcoming of such evaluation is that these metrics do not visualize the
influence of input image features on the features of the fused image as it
does not consider the underlying heuristics of the hidden network layers
thereby not analyzing the internal mechanics of these networks. Therefore,
image fusion using neural networks that have been evaluated on these
performance metrics have low interpretability.

Recently, several visualization techniques have been proposed that broadly try
to interpret the neural network decisions. For example, the gradient based
visualization methods backpropagate the gradients through the hidden layers of
the neural network to interpret the per-pixel influence of the input image on
the output predictions. However, such methods are specifically suited for
problems such as image classification where the aim is to visually explain the
class decision made by the neural network using jacobian based saliency map
for the class score with respect to the input image. On the contrary, the
primary goal for interpreting per-pixel decisions in a fusion problem is to
compute the jacobian based saliency map for each pixel of the fused image with
respect to each pixel of the input image. However, such analysis leads to very
high computational overload since the number of backpropagation iterations is
equivalent to the number of fused pixels. To circumvent this challenge, the
deep learning-based software frameworks such as PyTorch and TensorFlow
implicitly aggregate the per-pixel gradients from the input tensor elements to
compute a single gradient value for each of the output tensor elements. In
image fusion, this is not helpful since the gradient information related to
how each pixel in the input image influences the machine decision is lost.
Hence, the per-pixel saliency visualization by computing gradients of a neural
network based fusion model remains untapped in popular literature.

Therefore, in this paper, we present a novel per-pixel saliency based
visualization approach with real time capabilities that will guide in the
interpretability of image fusion based neural networks. The major novelties of
this work are summarised below:

  * •

We trained several state-of-the-art fusion based unsupervised CNNs under the
same learning objective. Considering the importance of understanding the
fusion blackboxes in life sensitive domain such as medical imaging, we focused
on interpreting the trained neural networks specifically for MRI-PET medical
image fusion.

  * •

We performed fast computation of per-pixel jacobian based saliency maps for
the fused image with respect to input image pairs. To the best of our
knowledge, it is a first-of-its kind technique to visualize fusion networks by
considering the backpropagation heuristics that helps it to be more
transparent in a real-time setup.

  * •

We constructed guidance images for each input modality by using gradients of
the fused pixel with respect to the input pixel at the corresponding location
in the input image. We also interpreted the gradient values in each of the
guidance images with the grayscale intensities of the fused image by combining
the images in the color channels of an RGB image.

  * •

We computed scatter plots between the gradients of the guidance images which
provides a visual overview of the correlation between the influence of each of
the input modalities. For example, a positive correlation will show that the
input modalities influence the fused image equally.

  * •

We developed an interactive Graphical User Interface (GUI) named
$\it{FuseVis}$, that combines all the visual interpretation tools in an
efficient way. The FuseVis tool allows to compute saliency maps in real-time
on the mouse over at the pixel pointed to by the mouse pointer. Our code is
available at https://github.com/nish03/FuseVis.

  * •

Finally, we performed clinical case studies on MRI-PET image pairs using our
FuseVis tool and visually interpreted the fusion results obtained from several
different neural networks. We showed the usefulness of FuseVis in identifying
the capability of the evaluated neural networks to solve clinically relevant
problems.

Section 2 provides a detailed literature review of the existing works relevant
to the problem of image fusion and visualization of neural networks. In
Section 3, we explain the visual analysis goals, the proposed visualization
concepts, and our FuseVis tool layout in detail. In Section 4, we provide the
experimental details of the training setup, the hardware used, the
architecture of the neural networks for image fusion, loss function used, and
the hyperparameter studies conducted. In Section 5, we present a clinical
application of MRI-PET medical image fusion and discuss key visualization
requirements for case studies in this field. In Section 6, we interpret the
visualization results obtained from each of the fusion based neural networks
using our FuseVis tool. In the same section, we also show the frame rates
achieved during the mouseover operation that supports the real-time
capabilities of our tool. Finally, in Section 7, we summarise the major
contributions of this work and discuss its applicability in the field of real-
time visualization of neural networks in general.

##  2 Related Work

In this section, we provide an overview of the literature relevant to this
paper. For this, we divide this section into five categories. The first sub-
section summarises the classical image fusion approaches that involve non-
machine learning based techniques proposed in the recent past along with some
relevant review papers related to image fusion problem. The second sub-section
gives an overview of the methods that applied the classical image fusion
approach along with pre-trained deep neural networks to attain the fusion
results. The next sub-section discusses recent methods in the field of
unsupervised end-to-end deep learning based image fusion specifically in the
area of multi-focus and multimodal image fusion. The fourth sub-section gives
a detailed overview of the recent state-of-the-art visualization techniques
that have been proposed for interpreting and explaining the black-box neural
networks in general. Finally, we end the related work discussion with current
literature on the fast computation of gradients in a neural network setup.

###  2.1 Classical Image fusion approaches

Many review papers in the past such as ref1 ; ref2 ; ref10  provided an
overview of different methods for multimodal and multi-focus image fusion. It
is quite pertinent from these papers that the classical method for the fusion
of multimodal images involves activity level measurement and pre-defined
fusion rules. Activity level measurement aims to transform the input images to
obtain salient features by either performing coefficient based activity
measurement ref4 , window-based activity measurement ref36 , or region based
activity measurement ref15 . The transformation strategies were mainly
categorized into domains namely multi-scale decomposition, sparse
representation, and hybrid transformation among others. This was followed by
pre-defined fusion rules to output a fused image. Subsequently, the quality of
the fused image was evaluated by using performance metrics that involve human
visual perception and some objective statistical assessments.

Multi-scale decomposition (MSD) ref4 ; ref11 ; ref12 ; ref13 ; ref18 ; ref14 ;
ref15 ; ref16 ; ref17 ; ref19 ; ref20 ; ref21 ; ref22 ; ref23 ; ref24 ; ref25
separates the source image into low and high frequency sub-bands in order to
separately analyze the base and detail level features. Some of the popular
transformation strategies under multi-scale image decomposition has been
pyramids ref4 ; ref11 , discrete wavelets ref12 ; ref13 ; ref14 , dual-tree
complex wavelets ref15 , contourlets ref16  and shearlets ref17 ; ref18 .
Multi-scale geometric analysis based MSD methods such as non-subsampled
contourlet (NSCT) ref19 ; ref20  and non-subsampled shearlet transform (NSST)
ref21 ; ref22 ; ref23  have also been proposed due to their effectiveness with
image representation. Edge preserving filters such as bilateral filters ref24
and guided filters ref25  have been popular in multi-scale decomposition based
literature since they retain the salient features from the input images.
Sparse representation (SR) based transformation strategies ref5 ; ref27 ;
ref28 ; ref29 ; ref30 ; ref42a  do not decompose the original image into low
and high frequency bands but instead assume that both frequency bands have
similar sparse coefficients. Hybrid transformation strategies aim to define
more than one transformation procedure such as curvelet-wavelet ref31 ,
combination of multi-scale decomposition and sparse representation based
methods ref32 ; ref33 ; ref34 , Intensity-Hue-Saturation (IHS) and Principal
Component Analysis (PCA) ref35  among others.

Finally, the image fusion rules after the image transformation have
traditionally been defined to combine the features obtained from multi-scale
transform approaches into a single fused image and been a crucial factor that
helps to provide state-of-the-art fusion performances ref10 . The three
different components which together comprised a robust fusion rule in the past
are coefficient grouping, coefficient combination as well as consistency
verification. The coefficient grouping and combination strategies include
choose-maximum rules ref37 , weighted-averaging ref4  and guided-filtering
based weighted averaging ref25 . Subsequently, the consistency verification
aimed to ensure that the neighborhood coefficients are fused using the same
fusion rule by refining the calculated weight map based on some priors ref38 .
At last, the fused image obtained from the classical methods were evaluated
with several performance metrics ref39 ; ref40 ; ref41 ; ref42 ; ref9  since
there is no gold standard for an ideal fused image.

###  2.2 Mixture of classical and deep learning based fusion approaches

In recent years, there have been several works that use deep learning based
fusion networks ref3  to further enhance the fusion image quality. However,
these works focused on using pre-trained neural networks to extract high and
low-level image features for activity level measurement while the fusion rules
were still manually defined. For example, in ref43 , a CNN for medical image
fusion was used to perform activity level measurement and based on these
measurements, coefficient grouping and coefficient combination were performed
manually to obtain the fused image. On the other hand, in ref44 , the infrared
and visible input images were first separated into low and high-frequency
features using a pyramid based multi-scale decomposition approach, and then
the high-frequency features were fed into a pre-trained deep learning based
framework to extract multi-channel high-frequency features. Finally, the low-
frequency features were fused using weighted averaging while the high-
frequency multi-channel features were fused using a pre-defined max-selection
rule. In ref7 , low-resolution grayscale images were divided into high and
low-frequency features using wavelet transformation, and the high-frequency
features were fed into a trained neural network which provided the high-
resolution version of the input. Then, the absolute-max strategy was applied
to the high-resolution outputs while the simple averaging was done to fuse the
low-frequency features. Finally, the inverse wavelet transformation was done
to retrieve a high-resolution output image. ref8  on the other hand, used
multi-scale convolutional neural networks to obtain multi-scale feature maps
that are fused and then post-processed to obtain the decision maps which are
segmentation results of the fused image using watershed transformation. This
is done to evaluate the quality of the fused image from different image fusion
algorithms.

###  2.3 Unsupervised end-to-end deep learning based fusion approaches

All the fusion methods discussed up to now use the traditional approach of
defining the transformation and fusion strategies while some use deep neural
networks to extract feature maps from its hidden layers to perform the
activity level measurement. Based on the feature maps generated, a final
fusion rule is defined. The second observation with these methods is that the
network is trained on image distributions which possess properties quite
distinct from the inferred images. Though such usage of convolutional neural
networks has provided good results in these approaches, it undermines the true
ability of a standalone end-to-end learning algorithm capable of providing a
final fused result by eliminating pre or post-processing steps such as
traditionally used transformation and fusion strategies. Recently, there are
end-to-end deep learning based image fusion methods that introduce an
optimization strategy using a neural network that models all the
transformation, fusion, and reconstruction strategies in its hidden network
layers and learns a fused image in an unsupervised manner using loss functions
such as SSIM ref9 . ref6 ; ref45 ; ref50  were some of the early works in the
field of multi-focus image fusion which used end-to-end convolutional neural
networks that jointly generated activity level measurements and fusion rules.
However, these methods trained the network in a supervised environment by
leveraging the training data with available groundtruth from image
classification databases. ref61  on the other hand, proposed a simple encoder-
decoder based architecture for directly mapping the source images with a
single camera focus to get a multi-focus fused image while ref54 ; ref59
proposed a Conditional Generative adversarial network (CGAN) based approach
for multi-focus image fusion. In the field of multimodal medical image fusion,
ref46  proposed a novel end-to-end fusion network to fuse MRI and PET image
pairs by extracting high and low-frequency features within the network layers
with SSIM as the loss function. ref47  used a similar approach for fusing
multi-exposure RGB images. However, they first converted the RGB images into
YCbCr color space and used an end-to-end neural network to fuse only the luma
component $Y$ of the input images while the Cb and Cr components were fused
using weighted averaging strategy. Finally, the fused YCbCr color space was
converted into the RGB fused color space. There have been several works in the
field of visible and infrared image fusion as well that utilize unsupervised
deep learning based fusion framework. ref48  trained an end-to-end
convolutional neural network using visible and infrared pedestrian images from
both day and night to attain robust fusion results. ref49  used a different
approach to fuse infrared and visible images where they used only a single
modality at once to optimize the common weights. Secondly, the feature
extraction and feature reconstruction layers involved the trainable weights
while the fusion layer did not contain any trainable parameters. ref51  went a
step further and defined a trainable fusion layer. ref53 ; ref57 ; ref60
applied the GAN approach for fusing infrared and visible images using end-to-
end neural networks. ref55  was another such GAN based approach but they used
a single modality and fused multi-resolution input images. Finally, there were
some works ref56  which used a novel densely connected network using
unsupervised learning strategy to fuse source images for multiple fusion tasks
whereas ref58  proposed a new GAN based method that combined the identity of
one input image and the shape of another input image.

###  2.4 Visualization techniques

Recently, there have been several methods that interpreted the decisions made
by neural networks by visualizing the model predictions. ref62  proposed a
method for visualization of classification models by computing the gradients
of the class predictions with respect to the input images. ref63  on the other
hand, computed the gradients for activation neurons in the hidden layers of
the network with respect to the input image by visualizing the specific
regions of the image that activates a given neuron the most. ref64  extended
this work and generalized the gradient-based visual explanations proposed in
ref63  for different types of neural network architectures. ref65  proposed an
alternative visualization method where they used a relevance score approach
that visualized the contributions of each pixel of the input image to the
classifier predictions. ref66  presented a method inherently similar to ref65
but they also visualized the output prediction by calculating the
contributions of all neurons in the network to every feature of the input
image.

Perturbation based visualization algorithms, however, do not need
backpropagation heuristics and is a model agnostic approach. ref67  was one of
the first perturbation based approaches that tried to implement occlusion in
the input image and examined the output of the classification based neural
networks. The results showed that the model was not localizing the objects in
the image as the probability of the correct class reduced significantly when
the object was occluded. ref68  was another model agnostic method where the
perturbations were performed in the input image around its neighborhood and
the behavior of the model’s predictions was later analyzed. Finally, the
method weights the perturbed data with respect to its proximity to the
original image and learns an interpretable model based on the old and the new
predictions. ref70  interpreted the model predictions by changing the pixel
intensities of the input images with some noise like blurring or occlusion and
modeled the change in the prediction probability of the output image. ref69
showed that there are two important characteristics of a visualization method
to properly explain the regions of the input that are important for the
model’s decisions. These characteristics are gradients and implementational
invariance. To satisfy both the properties, the method integrated the
gradients from multiple inputs due to which the method was able to satisfy the
sensitivity and implementational invariance of a visualization method.
Therefore, the method was able to visualize the prediction of a neural network
with respect to its inputs and required just a few instances of gradient
computations. ref71  proposed a visualization approach specifically for
medical image fusion where they computed a heat map showcasing the mutual
information between the source and the fused image. However, this method
cannot interpret and explain the learning based fusion networks itself since
there are no backpropagation heuristics involved. Although these perturbation
based visualization approaches are generally applicable to a wide range of the
problem, they still need several iterations to get the visualization results
due to which they are not yet suitable for real-time deployment.

Crucially, none of the visualization methods can be generalized for fusion
based deep neural networks since, in a fusion problem, the number of output
predictions is equal to the number of output pixels. This has huge memory and
time requirements due to the very high number of backpropagation heuristics
involved. ref72  efficiently computed the gradients of the loss function by
conducting the backpropagation heuristics for per-training example. However,
the idea only works on a neural network for image classification that has a
single class prediction, and therefore, it cannot be applied to an image
fusion problem where there are a large number of predictions in the form of
fused pixels. Therefore, an efficient method is needed to decrease the
computational cost for the visualization of high dimensional output
predictions provided by deep neural networks.

##  3 Method

In this section, we will present the key visual analysis goals of a neural
network for image fusion. Next, we will describe the proposed visualization
concepts along with their mathematical formulations that help in achieving
these visual analysis goals. Finally, we will present an overview of our user
interface, FuseVis, and describe its user interaction capabilities.

###  3.1 Visual analysis goals

A fused image provides important information related to the input image
modalities in combined feature space by preserving features from both the
input images. However, by visualizing only the fused images, it could be
challenging to interpret the sensitivity of the fusion methods for the input
images. It is especially difficult to interpret which of the fusion methods
reproduce features from which input image in a better way and how sensitive it
is to changes in pixel intensities of the input images. Hence, this analysis
is important to analyze the stability of a fusion network to the pixel
intensity changes in the specific regions of the input images.

Figure 1: The figure shows a layout of FuseVis’s user interface.

###  3.2 Visualization concepts

We present several visualization strategies that help to understand the fusion
results. The most important of those strategies is to compute gradient based
saliency heat maps that indicate the relevance of each input pixel to a single
pixel in the fused image. These heatmaps are particularly valuable since they
allow for an easy and intuitive investigation of what the respective fusion
model perceived to be important in the input image features. Therefore,
gradient based saliency heatmaps are useful in interpreting fusion blackboxes
and understanding the characteristics of image fusion based neural networks.
Using these saliency based heat maps, we can also estimate whether a neural
network for image fusion is robust to new examples. Therefore, to improve the
interpretability of such image fusion based neural networks, we present
several visualization techniques that help to understand the fusion results.

####  3.2.1 Jacobian images

We define per-pixel saliency heat maps, named as Jacobian images which
highlight the regions in the input images that most prominently influenced the
per-pixel fusion results. Let us interpret a fusion operator as a mapping
$f:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n}$ that maps two input images
$\boldsymbol{x_{1}}$ and $\boldsymbol{x_{2}}$ to a fused image
$\boldsymbol{y}=f(\boldsymbol{x_{1}},\boldsymbol{x_{2}})$, where all images
have the same dimensions and contain $n$ pixels. Then, the saliency analysis
is focused on a single pixel $i$ $\in$ $\\{1,...,n\\}$ of the fused image that
can be selected by the user interactively and is called principle pixel,
$y^{(i)}$, where we use a superscript enclosed in parenthesis to denote the
pixel of an image. For each input image $\boldsymbol{x}$ $\in$
$\\{\boldsymbol{x_{1}},\boldsymbol{x_{2}}\\}$, we define the jacobian image of
$y^{(i)}$ as the image of the partial derivatives with respect to the input
image $\boldsymbol{x}$:

$\frac{\partial y^{(i)}}{\partial\boldsymbol{x}}=\Bigg{(}\frac{\partial
y^{(i)}}{\partial x^{(1)}},\frac{\partial y^{(i)}}{\partial
x^{(2)}},....,\frac{\partial y^{(i)}}{\partial x^{(i)}},....,\frac{\partial
y^{(i)}}{\partial x^{(n)}}\Bigg{)}$ (1)

The jacobian computation involves backpropagating the gradients through the
hidden layers of the neural network using chain rule-based automatic
differentiation. A jacobian image visualizes the extent up to which each pixel
element of the input image $\boldsymbol{x}$ influences the principle pixel
$y^{(i)}$. Therefore, the jacobian images reveal the sensitivity of the
principle pixel $y^{(i)}$ in the fused image to the changes in the pixel
intensities of the input images. An example of a jacobian image is shown in
Figure 1 e). It can be expected that the fused principle pixel $y^{(i)}$ is
highly sensitive to changes in the grayscale intensities at $x^{(i)}$.
Additionally, the local neighborhood of the pixel element $x^{(i)}$ might also
have a direct influence on the fused principle pixel $y^{(i)}$. The jacobian
image based visualization concept also helps to compare the sensitivity of the
fused principle pixel with respect to changes in the pixel intensities of each
of the two input images as well as estimate which of the two input image has a
greater neighborhood influence on the fused pixel.

####  3.2.2 Guidance images

Based on the observation that the pixel $x^{(i)}$ has by far the largest
saliency value, we developed a visualization concept named Guidance images
which only considers the gradients of the principle pixel $y^{(i)}$ with
respect to the input pixel element $x^{(i)}$ located at the corresponding
coordinate in the input image:

$\frac{\partial\boldsymbol{y}}{\partial\boldsymbol{x}}=\Bigg{(}\frac{\partial
y^{(1)}}{\partial x^{(1)}},\frac{\partial y^{(2)}}{\partial
x^{(2)}},....,\frac{\partial y^{(i)}}{\partial x^{(i)}},....,\frac{\partial
y^{(n)}}{\partial x^{(n)}}\Bigg{)}$ (2)

While jacobian images allow per-pixel interpretation of fusion outputs,
guidance image aims to provide for each input image, a static overview of its
influence on the fused image. An example of a guidance image is shown in
Figure 1 c). The two static guidance images provide an overview of the
jacobian images and allow us to compare the major gradient values on an
absolute scale. This fosters comparison of the same pixel location in the two
input modalities as well as different pixel locations in the same input image.
The changes in the pixel intensities of the input image region with high
gradients will significantly change the pixel intensities in the corresponding
region of the fused image. On the contrary, it will require significant
changes in the pixel intensities of the input image region with low gradients
to have a similar effect on the fused features in the same region. Due to
this, the guidance images reveal which of the two input modalities have higher
influence on the fusion result in specific regions of interest as well as
where each input image has a large influence on the fusion result.

####  3.2.3 Guidance RGB images

During the interactive visual analysis with jacobian images and guidance
images, a typical analysis task is to compare the values in the guidance
images of the two modalities in order to find out which modality influences
the fused image more. Furthermore, one typically wants to compare a guidance
image with the fused image. To simplify these local comparisons, we provide a
single Guidance RGB image that encodes the two normalized guidance images as
well as the fused image in the three color channels red, green, and blue. We
perform a max-min normalization scaling of the guidance images to re-scale the
features with a distribution value between 0 and 1. For every guidance image,
the minimum value of that image gets transformed into 0, and the maximum value
gets transformed into 1. An example of a Guidance RGB image is shown in Figure
1 i). In red are the regions where the MRI gradient is high, the PET gradient
is low and the fused image has relatively low pixel intensities. By increasing
fused pixel intensities, the color changes from red to magenta. Similarly,
green and cyan correspond to high PET gradients with low MRI gradient values.
In blue regions, both gradient values are low and the fused pixels are bright.
The yellow depict regions where both gradients are high. In this way, the
Guidance RGB image allows us to find different gradient constellations fast
and the selection of the principle pixel is also possible on the Guidance RGB
image.

####  3.2.4 Scatterplot

As the Guidance RGB image gives a local overview of how the gradients of each
modality behave in comparison to the fused image, a scatterplot can depict the
correlation and more advanced statistical relationships between the gradient
values of the guidance images. For example, a positive correlation between
these gradients would mean that the input modalities equally influence the
fused pixels while a negative correlation will show that an increase in the
influence of one modality will lead to a decrease in the influence of the
other modality. For this statistical analysis of the gradient values in the
guidance images, we show a scatterplot with MRI gradients along the x-axis and
PET gradients along the y-axis with an example provided in Figure 1 k). The
overall scatterplot in black shows a rather complicated relationship between
the gradients. Therefore, we illustrated the points corresponding to pixels
around the principle pixel in green. In this way, the local statistical
relationships can be explored interactively.

Figure 2: The first and second row shows the effect of the gamma correction on
the Jacobian PET and Guidance PET images of MaskNet network with
$\gamma_{corr1}$ and $\gamma_{corr2}$ varying between 0.1 and 2.0.

####  3.2.5 Gamma correction

The jacobian images inherently have low luminance since the gradients of the
principle pixel is extremely bright compared to the gradient of its
neighborhood pixels. Therefore, to better visualize the small gradient values
in the jacobian images, we used the gamma correction, $\gamma_{corr1}$, on
jacobian images as $\Big{(}\frac{\partial
y^{(i)}}{\partial\boldsymbol{x}}\Big{)}^{\gamma_{corr1}}$. To also adjust the
luminance of the guidance images, we defined another gamma
correction,$\gamma_{corr2}$, on guidance images as
$\Big{(}\frac{\partial\boldsymbol{y}}{\partial\boldsymbol{x}}\Big{)}^{\gamma_{corr2}}$
in a 8-bit range of RGB images. We chose two separate gamma corrections for
jacobian and guidance images to independently fine-tune the luminance of these
images. Gamma correction is an efficient method to improve the luminance since
it only has a single parameter to fix for better visualization of the
underlying gradient intensities. Currently, the range of the gamma correction
values varies between $0.1$ and $2$ in our FuseVis tool. An example of the
effect of $\gamma_{corr1}$ and $\gamma_{corr2}$ on jacobian and guidance
images respectively is shown in Figure 2. It can be seen from the images that
as the $\gamma_{corr1}$ and $\gamma_{corr2}$ decreased below $1$, the
luminance of the images increased due to which the small gradients became
brighter while ${\gamma_{corr}}$ above 1 decreased the luminance of the small
gradients.

###  3.3 Overview of FuseVis tool

An overview of the FuseVis tool is shown in Figure 1 where the images related
to the MRI are placed at the top row, the images related to PET at the bottom
row whereas all the modality combining images are in the middle row. We placed
input and fused images on the left, guidance images in the middle, and the
jacobian images on the right side of the tool. The tool also consists of a
dropdown menu to select the fusion method and two separate slider widgets,
namely $Gamma1$ that represents $\gamma_{corr1}$ to fix the gamma correction
of the jacobian images and $Gamma2$ that represents $\gamma_{corr2}$ to fix
the gamma correction of the guidance images. The principle pixel can be chosen
in any of the input images, guidance images, or the fused image using the
mouseover operation. The user performs the mouseover interaction by pressing
and holding the left button of the mouse and hovering around the image. While
performing this operation, the current mouse cursor position defines the
principle pixel coordinate and the local environment around the principle
pixel is displayed in the zoomed-in versions of the images which is always
placed to the right of the respective images. The local environment of the
principle pixel is defined based on the red squared bounding box and the
current principle pixel coordinate, $i$, is positioned around the center of
the bounding box which is also shown as the red square in the zoomed images.

Figure 3: Loss curves of the evaluated fusion based neural networks trained on
MRI-PET image pairs.

##  4 Experimental Setup

As there are multiple real-world applications for image fusion, it is not
possible to interpret fusion based neural networks for each of these
experimental settings. Therefore, we focus on the evaluation of fusion based
neural networks on a specific clinical application that deals with MRI-PET
fusion for tumor resection planning. For this, we first comparably train the
different neural networks. Then, we present medical case studies where we
mention key visualization requirements of a robust fusion method. In the same
section, we select the features in the input images that are most important
for the clinical diagnosis. Finally, we use our tool to find out which of the
fusion methods is best suited for this specific clinical application.

###  4.1 Fusion networks

We visualized multiple end-to-end unsupervised learning based fusion
approaches available in popular image fusion literature using our FuseVis
tool. The image fusion based neural networks that we evaluated are FunFuseAn
ref46 , MaskNet ref52 , DeepFuse ref47  and DeepPedestrian ref48 . All of
these fusion based neural networks have distinct network architectures (as
shown in Figure 9), built specifically for different types of image fusion
applications. In addition to these four end-to-end unsupervised fusion based
neural networks, we also evaluated a non-machine learning based fusion
approach by implementing a simple weighted averaging technique and compared
its visualization results with those from these fusion based neural networks.
The weighted averaging based image fusion is a pixel-level method that assigns
higher weights to the sharpest pixels in each input image. Assuming pixel
$x_{1}^{(i)}$ from one input image to be extremely bright compared to pixel
$x_{2}^{(i)}$ from the other input image, then the weighted averaging method
will give higher weightage $w_{1}^{(i)}$ to $x_{1}^{(i)}$ and lower weightage
$w_{2}^{(i)}$ to $x_{2}^{(i)}$ , thereby resulting in a bright fused pixel
element $y^{(i)}$:

$w_{1}^{(i)}=\frac{x_{1}^{(i)}}{x_{1}^{(i)}+x_{2}^{(i)}},\;\;\;\;w_{2}^{(i)}=\frac{x_{2}^{(i)}}{x_{1}^{(i)}+x_{2}^{(i)}},\;\;\;\;y^{(i)}=w_{1}^{(i)}*x_{1}^{(i)}+w_{2}^{(i)}*x_{2}^{(i)}$
(3)

###  4.2 Hardware-software setup

The work was performed with Python (version 2.7) and deep learning library
PyTorch was used to implement fusion based neural networks. The Python wrapper
of the GUI library Tkinter was used to create the user interface of our
FuseVis tool. All the visualization results were derived from FuseVis tool
using a 64 bit Windows operating system equipped with a single GeForce GTX
1080 Ti GPU and 12 Intel Core i7-8700K CPU @ 3.70 GHz with 64-GB RAM. The same
hardware-software setup was used for the real-time computation of the jacobian
images. However, to fasten the training of the neural networks and to speed up
the static computations of guidance images, we used a more powerful hardware
setup containing six NVIDIA Tesla V100 GPUs.

###  4.3 Training dataset

For training each of the fusion networks in a uniform setup, we obtained
several MRI-T2 and PET image pairs of 50 different patients publicly available
at the Alzheimer’s Disease Neuroimaging Initiative (ADNI) ref73  with the age
of patients varying between 55-90 years among both genders. The patients were
chosen in such a way that the training dataset covers mild, moderate, and
severe stages of Alzheimer’s disease. All images were extracted as axial
slices with a voxel size of 1.0 x 1.0 x 1.0 mm3. The MRI-T2 images were N3m
MPRAGE sequences while PET images were co-registered, averaged, and
standardized with a uniform resolution for each of the subjects. We aligned
the MRI-PET image pairs using the Affine transformation tool of the 3D Slicer
registration library.

###  4.4 Network architectures

The architectures of the evaluated fusion based neural networks are shown in
Figure 9. A Conv 3 x 3 x 9 x 16 block, for example, means a convolution
operation performed on an input with nine channels that results in an output
with 16 channels. For this, 16 kernel filters each with size 3 x 3 and nine
channels are utilized. Each channel of the kernel filter is convoluted with
the corresponding input channel and subsequently summed up to generate a
single feature map. This operation is repeated for all the kernel filters to
extract output feature maps equal to the number of filters. The weights of the
kernel filter have been initialized based on a uniform distribution while the
striding and the padding operation have been performed in such a way that
there is no downsampling in any hidden layer. A BatchNorm means a batch
normalization operation and LReLU means a leaky ReLU activation function with
a slope 0.2. It is noticeable that each of the networks has different
architectures due to varying image fusion tasks that they solve. The
architecture of FunFuseAn ref46  is inspired by the classical non-machine
learning based image fusion technique of feature extraction, fusion, and
feature reconstruction with specific application to the MRI and PET image
pairs. MaskNet ref52  concatenates the input images and performs a dense
feature fusion in the initial layers of the network before constructing a
fused image. DeepFuse ref47  defines an architecture where the feature
extraction layers have tied weights for each of the input image modalities.
This helps in increasing the brightness of an underexposed input image by
using the luminance of an overexposed input image. DeepPedestrian ref48  also
concatenated the input images and utilized a deep architecture with multiple
residual blocks specifically for infrared and visible image fusion. Since we
aim to interpret the behavior of these image fusion based neural networks by
performing a per-pixel gradient based saliency visualization using our FuseVis
tool, we modified the original training setup of these fusion networks and
defined a common strategy to train these networks.

###  4.5 Loss Function

An image fusion approach aims to preserve as much of the input image features
as possible in the fused image. Therefore, the training objective should be
defined in such a way it could measure the similarity between the input and
the fused image and optimize the network parameters in order to achieve
perceptually similar results. Structural Similarity Index (SSIM) ref9  is an
efficient similarity metric that is also differentiable due to which it can be
adopted as a primary loss function for training each of the fusion based
neural networks. However, empirically setting only SSIM as the loss function
leads to the change of brightness in the final fused image. The $\ell_{2}$
loss helps to overcome this shortcoming by preserving the luminance component
better. Therefore, we define a combination of SSIM and $\ell_{2}$ losses to
train each of the neural networks as shown in Equation 4. The
$\boldsymbol{x_{1}}$ and $\boldsymbol{x_{2}}$ are MRI and PET training images
respectively, the regularization hyperparameter $\lambda$ controls the
weightage of the total $L_{SSIM}$ and total $L_{\ell_{2}}$ losses while the
hyperparameters $\gamma_{ssim}$ and $\gamma_{\ell_{2}}$ controls the weightage
between the individual SSIM and $\ell_{2}$ losses of each input image modality
respectively.

$\begin{gathered}L_{SSIM}^{MRI}=1-SSIM(\boldsymbol{x_{1}},\boldsymbol{y}),\;\;\;\;L_{SSIM}^{PET}=1-SSIM(\boldsymbol{x_{2}},\boldsymbol{y})\\\
L_{SSIM}=\gamma_{ssim}*L_{SSIM}^{MRI}+(1-\gamma_{ssim})*L_{SSIM}^{PET}\\\
L_{\ell_{2}}^{MRI}=||\boldsymbol{y}-\boldsymbol{x_{1}}||_{2},\;\;\;\;L_{\ell_{2}}^{PET}=||\boldsymbol{y}-\boldsymbol{x_{2}}||_{2}\\\
L_{\ell_{2}}=\gamma_{\ell_{2}}*L_{\ell_{2}}^{MRI}+(1-\gamma_{\ell_{2}})*L_{\ell_{2}}^{PET}\\\
L_{total}=\lambda*L_{SSIM}+(1-\lambda)*L_{\ell_{2}}\end{gathered}$ (4)

The loss convergence curves of all the evaluated unsupervised learning based
fusion networks are shown in Figure 3. Each of the fusion networks was trained
for 200 epochs, the batch size was fixed as 2 and the Adam optimizer was used
as the optimization function during the backpropagation step with a learning
rate of $2x10^{-3}$.

###  4.6 Hyperparameter tuning

Ideally, a fused image should preserve features from both the input images
equally and there should be minimal addition of the brightness artifacts. For
achieving this, we followed a hyperparameter tuning strategy where we aimed to
find a plausible balance between the $L_{SSIM}^{MRI}$ and $L_{SSIM}^{PET}$ as
well as $L_{\ell_{2}}^{MRI}$ and $L_{\ell_{2}}^{PET}$ loss curves by training
a single fusion network with multiple combinations of $\lambda$,
$\gamma_{ssim}$ and $\gamma_{\ell_{2}}$ values. Once a good balance was found
between these curves, we evaluated the best $\lambda$ configuration that leads
to the least brightness artifacts by analyzing the resultant fused images. We
initially trained the FunFuseAn network with $\lambda$ = 0.01, 0.2, 0.5, 0.8
and 0.99 with $\gamma_{ssim}$ and $\gamma_{\ell_{2}}$ each equal to 0.1, 0.3,
0.5, 0.7 and 0.9. Therefore, a total of 125 instances of FunFuseAn network
were learned in separate training environments. For each $\lambda$
configuration, we analyzed the combination of $\gamma_{ssim}$ and
$\gamma_{\ell_{2}}$ that nearly balances the SSIM and $\ell_{2}$ loss curves
for both MRI and PET images. Then, we fine tuned $\gamma_{ssim}$ and
$\gamma_{\ell_{2}}$ parameters for each $\lambda$ by training several more
instances of the FunFuseAn network in the vicinity of the previous best
$\gamma_{ssim}$ and $\gamma_{\ell_{2}}$ in order to find the next best
configuration that balances the loss curves even better.

Table 1: The table shows the partial loss values of the trained fusion networks after 200 epochs and the fine tuned hyperparameter configurations. Network |  $\lambda$ |  $\gamma_{ssim}$ |  $\gamma_{\ell_{2}}$ |  $L_{SSIM}^{MRI}$ |  $L_{SSIM}^{PET}$ |  $L_{\ell_{2}}^{MRI}$ |  $L_{\ell_{2}}^{PET}$
---|---|---|---|---|---|---|---
FunFuseAn |  0.99 |  0.47 |  0.5 |  0.2524 |  0.2147 |  0.0148 |  0.0094
MaskNet |  0.99 |  0.5 |  0.494 |  0.2208 |  0.2236 |  0.0109 |  0.0115
DeepFuse |  0.99 |  0.497 |  0.5 |  0.2824 |  0.2830 |  0.0385 |  0.0338
DeepPedestrian |  0.99 |  0.52 |  0.5 |  0.2164 |  0.2219 |  0.0148 |  0.0175

After fixing the best $\gamma_{ssim}$ and $\gamma_{\ell_{2}}$ configuration
for each $\lambda$, we determined which of the $\lambda$ values leads to
negligible brightness artifacts in the fused image. Therefore, we trained new
network instances with additional values of $\lambda$ = 0.9, 0.999, 0.9999,
0.99999, 0.999999 and 1.0 to properly evaluate the corner cases of our loss
function. We observed that the brightness artifact in the fused images
worsened as the values of $\lambda$ were increased above 0.999 with $\lambda$
= 1.0 having the most intense brightness in the fused image. $\lambda$ = 0.99
was one configuration where we observed negligible change in brightness of the
fused image which meant a weightage of 0.01 to $\ell_{2}$ was sufficient to
remove the brightness artifact problem from the SSIM loss function. $\lambda$
= 0.999 also had negligible brightness addition but the fused image from this
configuration lost some PET features due to which we fixed $\lambda$ = 0.99 as
the best hyperparameter for the $FunFuseAn$ network. Since $\lambda$ = 0.99
was sufficient to overcome the brightness artifact problem, we used the same
$\lambda$ = 0.99 value for all the other fusion networks and then evaluated
several instances of these networks to obtain the best $\gamma_{ssim}$ and
$\gamma_{\ell_{2}}$ that balances the loss curves. The loss curves in Figure 3
were generated with the best hyperparameter configuration for each of the
fusion based neural networks. The final hyperparameter values for the fusion
networks can be seen in Table 1. It is observable that all the networks were
able to balance the loss curves while DeepFuse network converged at higher
loss values compared to the other fusion networks.

##  5 Medical Case studies

In this section, we discuss the specific clinical application of the MRI-PET
image fusion in detail. Then, we perform case studies on pre-registered MRI
and PET image pairs by providing the pathological information about the
patients from whom these images were acquired. Finally, we will present key
visualization requirements that will make a fusion approach suitable for usage
in this specific clinical application.

###  5.1 Glioma and its pathological features

Glioma is a type of brain tumor that originates in the glial cells that
surround and support the neurons in the brain. MRI enables the clinicians to
estimate the size and the location of glioma, thereby acting as an initial
marker for malignancy. For example, in an MRI-T2 grayscale image, brighter
pixel intensities imply hyperintense signal abnormalities which are observable
in the brain regions with a tumorous lesion such as glioma. The grayscale
intensities in these tumorous lesions could be similar to the grayscale
intensities in the lateral ventricle region of the brain that mostly contains
cerebrospinal fluid (CSF) with high water content. However, there could be
several sub-regions within the boundary of high-grade glioma with varying
pathological features, which is challenging to visualize by only interpreting
the MRI images as the MRI contrast is quite uniform within the tumor boundary.
Hence, the aggressiveness and increased potential of the sub-tumor tissues
cannot be adequately determined.

Functional imaging modalities such as PET and SPECT are better equipped to
differentiate between the sub-regions of glioma by visualizing the functional
information like glucose metabolism and the extent of CBF or perfusion
activity in these sub-regions. The innermost part of glioma is the necrotic
core that contains dead tissues and there is no possibility of CBF and/or
glucose metabolism due to which it has very dark PET features. The region that
surrounds necrotic tissues is called an enhancing tumor which is filled with
inflammation fluid and the blood-brain barrier leaks in this region due to
cancerous angiogenesis. Hence, this region has glucose metabolism with blood
supply to support the cancerous cell growth resulting in lighter dark PET
features. The necrotic core and enhancing tumor region constitute the bulk of
glioma that clinicians want to visualize using fused MRI and PET features and
performs a precise resection of these two regions. The non-enhancing solid
tumor is the outermost region of glioma which is generally not resected since
the blood-brain barrier is still intact in these tissues. The bright regions
in the PET image resemble healthy tissues with a high level of blood perfusion
and normal glucose metabolism.

###  5.2 Clinical test examples

The clinical test examples were acquired from Harvard Whole Brain Atlas
database ref74  with the combination of MRI-T2 and PET/SPECT images from
patients suffering from different types of glioma. The test examples were
disjoint to the training dataset and the network has not seen such clinical
pathology during its training. The first clinical MRI-PET image pair is shown
in the first four columns of Figure 4. The scans are of a patient who was
suffering from Anaplastic Astrocytoma, a rare and malignant brain lesion
classified under the category of high-grade glioma. A lesion in the right and
the left side of the brain is visible in the MRI image and has bright
grayscale intensities. On the other hand, bright regions in the PET image
convey normal blood flow while very dark regions suggest no blood flow to the
necrotic tissues. In the second clinical MRI-PET image pair as shown in the
last four columns in Figure 4, the patient had a long history of tobacco usage
and was originally suffering from Metastatic Bronchogenic Carcinoma, which is
a type of lung cancer. The patient began having headaches and the scans
revealed brain metastases that occurred due to the spread of cancer cells
present in the lungs to the brain resulting in the diagnosis of glioma. A
lesion in the right side of the brain with bright features is visible in the
MRI image while very dark PET features reveal no blood flow in the necrotic
region.

###  5.3 Visualization requirements

As the anatomical MRI image cannot precisely estimate the sub-tumor boundaries
within glioma and the functional PET/SPECT image is unable to model the
overall extent of the tumor; therefore, one of the goals of MRI-PET image
fusion is to better delineate the sub-peripheries of glioma by envisioning the
metabolic and/or CBF characteristics of its sub-regions. However, it is
challenging to interpret the suitability of these fused images for the
mentioned clinical application, if the results are not supported by
supplementary visualization tools. This secondary visual analysis becomes
critically important when several fusion methods provide similar results.
Additionally, the recent image fusion methods are developed using unsupervised
learning based deep neural networks that are non-transparent (i.e. a ‘black
box’) due to its difficulty to retrace the fusion decision in light of huge
parameter space. This shortcoming is especially problematic in sensitive
medical domains like MRI-PET image fusion where understanding and explaining
fusion results obtained from neural networks is required for a robust clinical
decision. Therefore, the following visual tools are required to understand the
suitability of a fusion approach for usage in the clinical application:

  * •

A fusion approach should assist clinicians in visualizing the extent of hyper
dark PET regions resembling necrotic core with no blood flow being
superimposed on the bright anatomical boundary of the whole tumor mass. This
information is important for clinicians to estimate the extent up to which a
tumor resection is required. For example, in the first and the fifth column of
Figure 4, the principle pixel in this very dark PET region was chosen for
visual analysis.

  * •

A fusion approach should preserve the very bright PET features which convey
high blood perfusion and normal metabolism in healthy brain tissues as it
helps clinicians in visualizing the regions with high brain activity due to
external stimuli at a particular time. For example, in the third and seventh
column of Figure 4, the principal pixel in the bright PET region was chosen
for visual analysis.

  * •

A fusion approach should be stable and less sensitive to changes in input
features from a clinically less significant modality. For example, the change
in grayscale MRI intensities within the necrotic core shall not highly
influence the fused grayscale intensities as it might corrupt the clinically
important dark PET features. For example, the MaskNet and DeepPedestrian
networks are less sensitive to changes in the MRI features which can be
visualized in the guidance MRI and guidance PET images of these networks shown
in Figure 5.

  * •

A fusion approach should be less sensitive to the changes in grayscale pixel
intensities located in one sub-region of glioma (say enhancing tumor) when the
principle pixel is located in the other regions of glioma (say necrotic core).
Therefore, a fusion method should have a negligible influence of the
neighborhood pixels exterior to a local feature with the principle pixel
interior to the local feature. For example, the fusion methods such as
Weighted Averaging and MaskNet have no or very low gradients in the
neighborhood pixels which are outside the very dark PET features resembling
necrotic core as shown in the jacobian images in Figure 7.

##  6 Results and discussion

In this section, we present a detailed visual analysis of the fusion based
neural networks using our FuseVis tool. First, we will analyze the fused
images from each of the fusion methods and discuss their clinical
significance. Then, we will discuss the visualization insights by performing
the saliency analysis and examine the feasibility of the fusion methods for
usage in the clinical application. Finally, we will present the timing results
of our FuseVis tool and show the tool’s real time capabilities.

Figure 4: The figure shows the fusion results for each of the fusion methods.
The zoomed image within the clinical region of interests are always placed on
the right of the unzoomed image.

###  6.1 Fused images

The fused images of the Weighted Averaging method as shown in the third row of
Figure 4 gives very high weightage to the bright input intensities in the
regions where intensities in the other modality are darker. Therefore, we can
perceive a reproduction of features from relatively brighter regions of the
MRI and PET images due to which Weighted Averaging method is unable to
preserve the clinically important dark PET features related to the necrotic
core but favorably preserves the bright PET features resembling healthy
tissues. The fused images of the FunFuseAn as shown in the fourth row of
Figure 4 are comparatively similar to the Weighted Averaging approach where it
is not able to preserve the very dark PET features. However, the fused images
from FunFuseAn look relatively dull compared to the fused images from Weighted
Averaging since the FunFuseAn network mixes both MRI and PET features even
though the boundary information about the necrotic region is lost.

The analysis of the fused images from MaskNet in the fifth row of Figure 4
shows a significant loss of anatomical edges from MRI such as the brain skull.
However, contrary to Weighted Averaging and FunFuseAn, MaskNet preserved the
PET features better in both the dark and bright regions resembling the
necrotic core and healthy tissues respectively. However, the dark PET features
are slightly blurred due to changes in the overall brightness of the fused
image. The fused image obtained from DeepFuse as shown in the sixth row of
Figure 4 has an even higher shift in grayscale intensities due to which the
method is unable to preserve the dark PET features even though the overall
anatomical MRI structures are well preserved. The change in brightness can be
explained by the fact that the $L_{\ell_{2}}^{MRI}$ and $L_{\ell_{2}}^{PET}$
converged at higher loss values compared to other networks due to which the
brightness component of SSIM was not properly optimized. Additionally, the
architecture of DeepFuse network has been crafted for adding exposure to
underexposed images by using the brightness component from each of the input
image modalities and employ it to generate very bright fusion results. The
fused image from DeepPedestrian as shown in the seventh row of Figure 4
clearly delineates the boundary of the necrotic core by preserving the very
dark PET features, which is of high clinical significance for medical
professionals. However, the anatomical edges from MRI are lost in the fused
image which is also one of the main artifacts in the MaskNet network. Another
important observation is that the bright PET features resembling healthy
tissues appear to be not well preserved due to an overall brightness shift.

Figure 5: The figure shows the guidance MRI and guidance PET images for each
of the fusion methods in the clinical region of interests. The
$\gamma_{corr2}$ was fixed at 0.5 for all the guidance images.

Summary: The analysis of the fused images shows that even though SSIM was used
as a natural loss function for training the fusion networks in a common
training setup, each of the fusion methods displayed very different fusion
results. The MaskNet and DeepPedestrian methods provide clinically relevant
results where the boundaries of the necrotic core of glioma are perceivable
with the dark PET features clearly superimposed on the anatomical tumor
features of MRI in the fused image. However, these methods do not provide
loss-free fusion results as there are missing anatomical information in both
these approaches while the dark PET features are not entirely preserved in the
MaskNet network. Crucially, it is not clear by only looking into the fused
images of these two methods, that up to what extent the changes in the bright
MRI features can affect the final fusion results within the clinically
significant dark PET regions. This analysis is important for visualizing the
stability of these networks to changes in the MRI features which is a critical
visualization requirement. Hence, we will perform this analysis using the
visualization concepts in the subsequent sub-sections.

###  6.2 Guidance images

In this section, we analyze the fusion methods with respect to its sensitivity
to feature level changes in the input images by visualizing the Guidance MRI
and Guidance PET images in Figure 5 and Guidance RGB images in Figure 6 for
the two clinically relevant regions. We chose the first region as bright MRI
with very dark PET features since it is the region that contains the necrotic
core of glioma which clinicians are interested to operate for resection. The
second region of interest has dark MRI with bright PET features resembling
healthy tissues which is interesting for the visualization of high brain
activity due to external stimuli. Ideally, a guidance MRI image should have
low gradients meaning a fusion method should be less sensitive to changes in
both dark and bright MRI regions.

Weighted Averaging: For the region with bright MRI and dark PET intensities in
both the test examples, it can be observed that the guidance MRI image is
significantly brighter than the guidance PET image, revealing higher
sensitivity to changes in MRI features. This suggests that a small change in
the MRI pixel intensities will sharply modify the fused pixel intensities
while it requires a substantial alteration in the PET pixel intensities to
hold a considerable effect on the fused pixel intensities. This is an
undesired outcome as a fusion method should be stable to changes in the pixel
intensities of MRI. The guidance RGB image shows magenta color in these
regions as the red channel containing guidance MRI image has high gradients
and the blue channel containing the fused image have high pixel intensities
compared to the green channel containing the guidance PET image with low
gradients. On the other hand, the region with dark MRI and bright PET
intensities shows an adverse effect where the guidance PET image is relatively
brighter compared to the guidance MRI image since higher weightage is now
given to the bright PET grayscale values. The guidance RGB image in this
region shows cyan color since the green channel with guidance PET image has
high gradients while the blue channel with the fused image also has higher
intensities.

FunFuseAn: For the region with bright MRI and dark PET intensities, there are
higher gradients in the guidance MRI image compared to the guidance PET image.
This conveys that the fused pixels within the necrotic core have a higher
sensitivity to changes in the MRI pixels. The guidance RGB image also shows
red color within the necrotic region indicating only the guidance MRI have
high intensities compared to the images in other color channels. For the
region with dark MRI and bright PET intensities, the guidance MRI and guidance
PET images have low gradients. This means that the fused pixels in this region
are quite stable to changes in the MRI and PET pixel intensities. Hence, the
color in the guidance RGB image is predominantly blue as the fused pixel
intensities are higher than the gradient values in both the guidance images.
Therefore, it is challenging to interpret which of the two input modalities
has a higher influence on the fused image in this particular region.

DeepFuse: In the bright MRI and dark PET region resembling necrotic core,
there are relatively high gradient values in the guidance MRI image compared
to the guidance PET image. This is the reason for the shades of magenta in the
guidance RGB image. For the dark MRI and bright PET intensity region in the
first test example, the pattern is similar where there are higher gradients in
the guidance MRI image compared to the guidance PET image due to which magenta
color is visible in the guidance RGB image. For the second test example;
however, the guidance images from both MRI and PET have low gradients
indicating a low influence of change in input pixel intensities within this
region. Hence, the guidance RGB image also appears to be light blue in color
in this region.

Figure 6: The figure shows the guidance RGB images for each of the fusion
methods.

MaskNet & DeepPedestrian: For the necrotic region with bright MRI and dark PET
intensities in the first test example, the guidance PET image is relatively
brighter than the guidance MRI image for both MaskNet and DeepPedestrian
networks. This conveys that both the networks are highly sensitive to the
changes in very dark PET intensities of the necrotic region compared to the
changes in MRI intensities. Since both the guidance MRI and the fused image
have low intensities compared to the guidance PET image in this region, the
guidance RGB image exhibits green color. However, for the second test example,
both the guidance images have very low gradients in the same region,
showcasing the stability of the networks for pixel intensity changes in this
region. Since all the color channels in the guidance RGB image have low
intensities, the guidance RGB image does not have a fixed color scheme and is
dark. For the region with dark MRI and bright PET intensities; however, the
behavior is similar in both the test examples where both the guidance images
have low gradients due to which it is challenging to differentiate the
influence of each of the modalities while the MaskNet network has learned the
bright PET features better leading to blue color in the guidance RGB image.

Summary: The guidance images provided a static overview of the influence of
input principle pixel on the fused principle pixel and assisted in visualizing
which of the two modalities has higher influence in the clinical region of
interest. Therefore, it provided new insights related to the stability of
fusion networks in these regions which were not perceivable by looking at the
fused images. The guidance images clearly show that MaskNet and DeepPedestrian
networks performed very differently compared to other fusion methods. For the
region with the dark PET features resembling necrotic core, all the methods
except MaskNet and DeepPedestrian were sensitive to changes in the bright MRI
intensities which is not suitable for a reliable analysis of the necrotic
tumor boundary since the dark PET features might not be properly preserved in
the fused image due to changes in the MRI features. However, both MaskNet and
DeepPedestrian preserved the dark PET features in the fused image and were
quite stable to the changes in the MRI pixel intensities due to which both
these methods are far more suitable for the clinical application than any
other fusion methods. But the guidance images don’t reveal the influence of
neighborhood pixels on the fused principle pixel which is important for
estimating a better fusion approach between the two of these networks.

Figure 7: The figure shows the jacobian MRI and jacobian PET images for each
of the fusion methods in the clinical region of interests. The
$\gamma_{corr1}$ was fixed at 0.3 for all the jacobian images.

###  6.3 Jacobian images

The jacobian images of the selected principle pixel is shown in Figure 7. The
analysis of the jacobian images will help in understanding the influence of
the neighborhood pixels located around the principle pixel in the clinically
relevant regions of the input image. It will also assist in visually comparing
the neighborhood influence of MRI pixels with PET pixels.

Weighted Averaging: The jacobian MRI and jacobian PET images within the region
with bright MRI and dark PET intensities for both the test examples show a
higher gradient of the principle pixel in the jacobian MRI image compared to
the jacobian PET image. For the dark MRI and bright PET intensities, there is
a relatively high gradient in the principle pixel of the jacobian PET image
showcasing sensitivity to changes in the bright PET pixel. One crucial
observation from each of the jacobian images is that there is no influence of
the neighborhood pixels on the prediction of the fused principle pixel since
Weighted Averaging performs the per-pixel computation of fused pixel
intensities.

FunFuseAn: After analyzing the jacobian MRI and jacobian PET images within the
region of bright MRI and dark PET intensities, it can be observed that similar
to the Weighted Averaging method, there is a higher gradient value for the
principle pixel of the jacobian MRI image. The immediate neighborhood pixels
in the jacobian MRI image also have significant gradient values meaning these
pixels also influence the outcome of fused principle pixel. This is clinically
useful since the change in intensities of pixels located within the necrotic
core should affect the fused principle pixel. Additionally, there are more
positive gradient values for the outer neighborhood pixels in the jacobian MRI
image than the jacobian PET image. For the combination of dark MRI and bright
PET intensities; however, there are low gradients for the principle pixel in
both jacobian MRI and jacobian PET images. This conveys that the fused
principle pixel is relatively stable to small changes in the bright PET and
dark MRI pixel intensities. Hence, visualizing principle pixel in the jacobian
images does not give much information about which input principle pixel has a
higher influence on the fused principle pixel. On the other hand, the
neighborhood pixel influence on the fused principle pixel is quite widespread
due to the large number of positive gradients in both the jacobian MRI and
jacobian PET images.

MaskNet: For the combination of bright MRI and dark PET intensities, the
jacobian images in the first test example convey quite distinct
characteristics compared to the jacobian images in the second test example.
For the first test example, the jacobian PET image has a much higher gradient
for the principle pixel compared to the jacobian MRI image. This reveals a
high sensitivity of the fused principle pixel within the necrotic tumor core
with respect to small changes in the pixel intensity of the PET principle
pixel whereas the fused principle pixel is quite stable to changes in the MRI
principle pixel. Additionally, the influence of neighborhood pixels of PET
within the dark necrotic tumor core on the fused principle pixel is highly
intense compared to neighborhood pixels of MRI. This conveys that the changes
in the pixel intensities of the PET inside the dark necrotic core can greatly
affect the fused principle pixel while the PET pixels outside the necrotic
core boundary have a negligible affect on the fused principle pixel. However,
in the second test example, both the jacobian MRI and jacobian PET images have
low gradients for the principle pixel as well as in the neighborhood region
within the necrotic core. This demonstrates that the network requires quite
significant changes in these features to change the fusion output. For the
combination of dark MRI and high PET intensities in both the test examples,
the jacobian MRI and jacobian PET have similar properties like FunFuseAn where
there are low gradients for the principle pixel in both the images; therefore,
equally influencing the fused principle pixel. However, there are less number
of neighborhood pixels in the jacobian MRI and jacobian PET images that
influences the principle pixel when compared to the FunFuseAn network.

Figure 8: The figure shows the scatterplots between the gradients of the
guidance MRI and guidance PET images for each of the fusion methods. The green
scatter points are the gradients for the pixels in the zoomed region of
interest.

DeepFuse: The jacobian images in both the test examples, when visualized in
the regions of bright MRI and dark PET intensities show a significantly high
number of neighborhood pixels influencing the fused principle pixel while a
high gradient value for the principle pixel can be seen in the jacobian MRI
image compared to the jacobian PET image. Therefore, the fused principle pixel
is highly sensitive to changes in the MRI pixel while it is stable to changes
in the PET pixel. By analyzing the jacobian images for the combination of dark
MRI and bright PET intensities, it is observable that the gradient values at
the principle pixel are low; therefore, showcasing stability to the pixel
intensity changes from both the modalities while widespread neighborhood
pixels are influencing the fused principle pixel.

DeepPedestrian: The combination of bright MRI and dark PET intensities in the
first test example shows different jacobian results compared to the second
test example. It can be noticed that the gradient value of the principle pixel
in the jacobian MRI image is low while the gradient for the principle pixel is
high in the jacobian PET image. This suggests that the fusion network is very
stable to changes in the MRI pixel while sensitive to changes in the PET pixel
within the necrotic tumor core. There are also positive gradients for the
neighborhood pixels within the clinically important necrotic sub-region of
glioma in the jacobian PET images due to which these pixels also influence the
fusion principle pixel prediction. In the jacobian PET image, there is some
influence from the neighborhood pixels outside the necrotic tumor core but
they have low gradients compared to the gradients of the neighborhood pixels
within the necrotic core. The jacobian images for the bright MRI and dark PET
intensities in the second test example have very unique characteristics since
both the jacobian MRI and jacobian PET images have very low gradients for the
principle pixel as well as for the neighborhood pixels. Therefore, the network
is very stable to changes in pixel intensities in this specific region of
interest. For the combination of dark MRI and bright PET intensities, both the
test examples show similar results where the neighborhood pixel influence on
the fused principle pixel are relatively low in both the input modalities
while the gradients for the principle pixel are also low meaning that the
network does not get influenced by changes in pixels of each of the input
modalities.

Summary: The analysis of jacobian images helped to visually compare the
influence of the input neighborhood pixels from each of the input modalities
on the fused principle pixel. It was observed that most of the fusion based
neural networks have a significant influence on neighborhood pixels. This
visual analysis was not possible by only interpreting the guidance and fused
images due to which jacobian images provide additional interpretation towards
the suitability of a particular fusion method in a clinical setup. It was
observed that Weighted Averaging has no neighborhood influence which is
understandable by the fact that it is a per-pixel computation scheme. The
MaskNet network; however, provides clinically favorable results as it has a
high influence of neighborhood pixels within the necrotic core and very low
influence from the pixels outside the necrotic core boundary.

###  6.4 Scatterplots

The scatterplots shown in Figure 8 helps in understanding the relationship
between the gradients of guidance MRI and guidance PET images where each point
resembles a pixel with the gradient value in guidance MRI at the horizontal
axis and the gradient value in guidance PET at the vertical axis. A positive
correlation between these gradients would mean that both the input modalities
influence the fused pixel with equal strength. A negative correlation will
show that an increase in the influence of one modality will lead to a decrease
in the influence of the other modality. For the cases, where the gradients
from one of the modalities are constant and only the gradient from the other
modality varies, then this conveys that the feature in the input modality
where the gradient varies is localized only in a subset of the pixels. If
there is no correlation we cannot draw any conclusion from the scatterplot.

By examining the scatterplots for each of the fusion methods, no clear
relationship between the guidance MRI and guidance PET images can be observed,
except for the positive correlation between the low gradient values of
guidance MRI and guidance PET images in the FunFuseAn and DeepFuse networks as
well as a negative correlation between the gradients in the Weighted Averaging
method. However, by looking at the gradients within the zoomed region of
interests shown by green points, a correlation patterns for few of the fusion
networks can be determined. The scatterplot for Weighted Averaging shows a
negative correlation between the guidance MRI and guidance PET images.
Therefore, for bright MRI and dark PET intensity setups in each of the two
test examples, higher gradient values in guidance MRI causes the green scatter
points to tilt towards the higher end of the horizontal axis while for dark
MRI and bright PET intensity setup in the two test examples, the scatter
points tilts in the opposite direction. This behavior shows that an increase
in the influence of one modality led to a decrease in the influence of the
other and Weighted Averaging only prefers brightness as a feature which is
clinically not interesting. The scatterplot of FunFuseAn shows a positive
correlation between the lower gradient values of guidance MRI and guidance PET
and there is no correlation between the other gradient values of guidance MRI
and guidance PET images. However, by visualizing the green points for dark MRI
and bright PET intensities, we can observe a positive correlation between the
gradient values which means both input modalities equally influence the fused
pixel in this region. For the bright MRI and dark PET intensity setup, the
scatter points are distributed in such a way that gradients from guidance MRI
are high and constant while the gradients from the guidance PET varies.

The scatterplots for the MaskNet network in bright MRI and dark PET intensity
setup of the first test example show a positive correlation between the
gradient values. In the second test example though, a positive correlation is
seen only between the very low gradient values of guidance MRI and guidance
PET images, while for higher gradients it is difficult to interpret a
correlation. For the dark MRI and bright PET intensity setup, the gradients
show a positive correlation in both the test examples. The scatterplots for
DeepFuse does not show a clear correlation pattern between the gradients of
guidance MRI and guidance PET images even after interpreting the green scatter
points in the specific region of interests. For the second test example in the
bright MRI and dark PET features of DeepPedestrian, the gradients of guidance
PET image varies comprehensibly while the gradients of guidance MRI image
remains low and constant. In the dark MRI and bright PET intensity setup for
both the test examples, DeepPedestrian has gradients that are closely
clustered and it is challenging to interpret a correlation.

Summary: The scatterplots helped in interpreting the correlation between the
gradients of the guidance MRI and guidance PET images for each of the fusion
methods, which was difficult to estimate by visualizing only the jacobian and
the guidance images. These correlations are important to visualize since an
ideal fusion method should not have equal influence from both the input
modalities in the regions where there are more clinically important features
in one modality than the other. For example, in the region where there are
dark PET features resembling the necrotic core, the gradients from guidance
MRI should always be low and the scatterplot should not have a positive
correlation between the gradients. The scatterplots of MaskNet and
DeepPedestrian showed such characteristics but only for one of the test
examples.

###  6.5 Memory and Frame rates

Table 2 shows the timing results of the jacobian computations as well as the
frame rates of our FuseVis tool for the visualization of jacobian images. As
it can be seen from the results, the computation of jacobians is quite fast
due to the powerful GPU hardware used during the experiments. The timings are
averaged over 100 randomly selected principle pixels. The results unveil that
FunFuseAn and DeepFuse are the fastest among the fusion based networks. One
reason is that these networks have far fewer parameters to be optimized.
Networks such as MaskNet and DeepPedestrian have a high number of trained
parameters and dense hidden layers due to which the automatic differentiation
through backpropagation requires higher computational time. We also estimated
frame rates within a static framework where we iteratively saved the jacobian
images for each of the principle pixels without using our FuseVis tool. Then,
we displayed these images in the FuseVis tool during the mouseover operation.
For this, we measured a frame rate of almost 40 frames per second ($fps$)
which is expected considering there were no jacobian computations with
backpropagation heuristics involved for each change of principle pixel during
the mouseover operation. However, the disadvantage of such a setup is that it
takes around 3 $GB$ of memory to save jacobian images locally for each input
modality at 100 dots per inch ($DPI$) resolution whereas our tool requires
zero additional memory usage.

Table 2: The table shows the timing and frame rate results for each of the fusion based neural networks. Setting |  FunFuseAn |  MaskNet |  DeepFuse |  DeepPedestrian
---|---|---|---|---
Jacobian computations |  0.003 s |  0.004 s |  0.003 s |  0.005 s
FuseVis - Jacobian images |  20 fps |  15 fps |  20 fps |  10 fps
Guidance images |  198 s |  265 s |  195 s |  360 s

##  7 Conclusions

In this work, we presented a novel interactive approach to visually inspect
fusion networks. For achieving this, we developed an easy-to-use FuseVis tool
that enables the end-user to visualize jacobian images of the selected
principle pixel in real-time while performing a mouseover interaction. The
real-time generation of jacobian images using an interactive user interface
showcases the influence of pixels in the input images on a fused pixel. It is
an intuitive idea that helps to overcome the huge computational complexity of
generating per-pixel jacobian images that involves several backpropagation
heuristics. The FuseVis tool also enables the user to visualize guidance
images which estimate the sensitivity of image fusion networks to the changes
in input features. Therefore, the guidance images are a very useful technique
to get a concise overview of the huge information content in the jacobian
images by analyzing the influence of the input pixel to the fused pixel at the
same location for all pixels. We currently train different fusion
architectures in a similar environmental setting to foster a suitable
comparison of black box fusion algorithms for a critical application in
medical diagnosis. However, our first of its kind visualization tool can
easily be used to interpret any neural network in its original experimental
setup. By visually analyzing the neural networks for image fusion within a
specific clinical application, it was found that only MaskNet and
DeepPedestrian provided results that would be helpful to clinicians. However,
these methods do not preserve all clinically relevant features efficiently due
to which these methods cannot be used in a clinical setup yet. Additionally,
it is not trivial to attain fusion groundtruth of an ideal fused image based
on the specific clinical requirements. Therefore, there is a need to train
fusion based neural networks by using annotations provided by medical artists.
We expect that our work should enable further research in the field of visual
explanation and interpretation of fusion based neural networks as well as
neural networks for other image processing based applications.

This work was supported by the European Social Fund and the Free State of
Saxony under NeuroFusion grant (project no. 100312752) and SePIA grant
(project no. 100299506).

###### Acknowledgements.

Some of the computations were also performed on an HPC Cluster at the Center
for Information Services and High Performance Computing (ZIH) at TU Dresden.
The authors declare that there is no conflict of interest. References

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Figure 9: The figure shows different neural network architectures used in this
work.