AIU Online Deep Learning and Fuzzy Logic Python Coding Task


 I intend to have a python coding on Deep learning (CNN) and Fuzzy Logic. 
 implement a CNN with a fuzzy layer in it using python 

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arXiv:2003.00880v1 [cs.CV] 21 Feb 2020
Published in 2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).
DOI: 10.1109/FUZZ-IEEE.2019.8858790
Introducing Fuzzy Layers for Deep Learning
Stanton R. Price
Steven R. Price
Derek T. Anderson
U.S. Army Engineer
Research and Development Center
Vicksburg, MS USA
Department of Electrical Engineering
Mississippi College
Clinton, MS USA
Department of Electrical Engineering
and Computer Science
University of Missouri
Columbia, MO, USA
Abstract—Many state-of-the-art technologies developed in recent years have been influenced by machine learning to some
extent. Most popular at the time of this writing are artificial
intelligence methodologies that fall under the umbrella of deep
learning. Deep learning has been shown across many applications
to be extremely powerful and capable of handling problems that
possess great complexity and difficulty. In this work, we introduce
a new layer to deep learning: the fuzzy layer. Traditionally,
the network architecture of neural networks is composed of an
input layer, some combination of hidden layers, and an output
layer. We propose the introduction of fuzzy layers into the
deep learning architecture to exploit the powerful aggregation
properties expressed through fuzzy methodologies, such as the
Choquet and Sugueno fuzzy integrals. To date, fuzzy approaches
taken to deep learning have been through the application of
various fusion strategies at the decision level to aggregate
outputs from state-of-the-art pre-trained models, e.g., AlexNet,
VGG16, GoogLeNet, Inception-v3, ResNet-18, etc. While these
strategies have been shown to improve accuracy performance
for image classification tasks, none have explored the use of
fuzzified intermediate, or hidden, layers. Herein, we present a
new deep learning strategy that incorporates fuzzy strategies
into the deep learning architecture focused on the application of
semantic segmentation using per-pixel classification. Experiments
are conducted on a benchmark data set as well as a data set
collected via an unmanned aerial system at a U.S. Army test site
for the task of automatic road segmentation, and preliminary
results are promising.
Index Terms—fuzzy layers, deep learning, fuzzy neural nets,
semantic segmentation, fuzzy measure, fuzzy integrals
Artificial intelligence (AI) has emerged over the past decade
as one of the most promising technologies for advancing
mankind in a multitude of ways, from medicine discovery
and disease diagnostics to autonomous vehicles, semantic
segmentation, and personal assistants. For many years there
has been a desire to create computer algorithms/machines that
are able to replace or assist humans in signal understanding
for tasks such as automatic buried explosive hazard detection,
vehicle navigation, object recognition, and object tracking. AI
has grown to loosely encompass a number of state-of-the-art
This work was partially supported under the Maneuver in Complex Environments R&D program to support the U.S. Army ERDC. This effort is also based
on work supported by the Defense Threat Reduction Agency/Joint ImprovisedThreat Defeat Organization (DTRA/JIDO). Any use of trade names is for
descriptive purposes only and does not imply endorsement by the U.S.
Government. Permission to publish was granted by Director, Geotechnical
and Structures Laboratory, U.S. Army ERDC. Approved for public release;
distribution is unlimited.
technologies across many fields, e.g., pattern recognition, machine learning (ML), neural networks (NNs), computational
intelligence, evolutionary computation, and so on. Recently,
much buzz has surrounded deep learning (DL) for its ability to
provide desirable results for a number of different applications.
AI has been shown to have great potential for finding optimized solutions to various problems across multiple domains.
This technology has been heavily researched for computer
vision applications [1], [2], speech translations [3], [4], and
optimization tasks [5], [6]. Herein, the focus is on an extremely
relevant and popular branch of AI: deep learning. DL has
achieved much success in recent years on computer vision
applications and has benefited from the surge of attention
being given to advance its theories to being extremely generalizable for many problems. Part of DL’s recent resurgence
is the implementation of its network architecture being well
suited for processing on powerful, highly parallelized GPUs.
This has allowed for extremely complex and deep network
architectures to be developed that were infeasible to implement
on older compute technologies.
Fusion is a powerful technique used to combine information from different sources. These sources could be different
features extracted, decisions, sensor output, etc., as well as different combinations thereof. Fusion methodologies often strive
to improve system performance by combining information in
a beneficial way that enables more discriminatory power to the
system in some form. This could be through the realization of
more robust features that generalize well from one domain to
another. For the task of classification, fusion could be used to
combine multiple decision makers, e.g., classifiers, to improve
overall accuracy performance. Fusion is a very rich and
powerful technique that, when implemented appropriately, can
lead to major algorithm improvements. Fusion is commonly
associated with fuzzy techniques, as Fuzzy Logic [7], [8] naturally lends itself to gracefully considering data with varying
degrees of belief. Ensemble approaches [9] are also commonly
used for fusion tasks. Generally, most fusion strategies attempt
to properly assign weights that encode the significance, or
importance, to the different information sources, and these
weights are the driving mechanism behind the fused result.
Historically, weights used when fusing multiple information
sources are either human-derived or found via an optimization
function/strategy such as group lasso [10], [11]. However,
there has been little-to-no research done on utilizing DL for
optimizing fusion performance. Herein, we propose a new
strategy to explore the potential benefits of combining DL with
fuzzy-based fusion techniques. Specifically, we introduce fuzzy
layers to the DL architecture.
The remainder of this work is organized as follows. In
Section II, related works are presented that explore using
fusion strategies to improve the classification performance of
the outputs from different state-of-the-art DL models (fusion
strategies and DL have been compartmentalized in their utilization). Fuzzy layers are introduced in Section III along with
the intuition behind this new strategy and their integration into
the DL architecture. Experiments and results are detailed in
Section IV, and in Section V, we conclude the paper.
As noted previously, DL is becoming the standard approach
for classification tasks. However, the performances exhibited
by DL classifiers are often the result of exhaustive evaluation
of hyperparamenters and various network architectures for the
particular data set used in the work. Fusion techniques can
help alleviate the comprehensive evaluation by combining the
classification outputs from multiple DL classifiers and thus
taking advantage of different DL classifier strengths. That is, if
the strengths of multiple classifiers can be appropriately fused,
then finding the optimal solution may not require finding the
ideal parameters and architecture for a particular data set.
Recently, fusion strategies have been employed that aggregate pre-trained models for improved classification performance. The appropriate fusion strategy largely considers the
formats of classifier outputs. Traditionally, a classifier output
consists of a definitive label (i.e., hard-decision), and typically,
majority voting is the fusion strategy implemented. However,
if the classifier can generate soft membership measures (e.g.,
fuzzy measures), the fusion strategy implemented can vary
greatly [12]–[15].
Fusion strategies, notably those associated with fuzzy measures (FMs), are conventionally applied at the decision level
to aggregate outputs and improve performance. For example,
DL classifier outputs were fused in [16]–[18] to improve
classification performance in remote sensing applications by
deriving FMs for each class and then fusing the measures with
the classifiers’ outputs through either the Sugeno or Choquet
integral (ChI) [19]. Still, fusion strategies occurring at the
input level can also benefit classification performance.
Rather than attempt to perform fusion at either the output or
feature level, it is the attempt of this work to incorporate fusion
techniques (utilizing FMs) within the architecture of a DL
classification system. While efforts have developed techniques
applying fuzzy sets and fuzzy inference systems in NNs,
application of fuzzy strategies concerning DL architectures is
limited [20]. One recent approach to implementing fuzzy sets
in DL evaluated the use of Sugeno fuzzy inference systems as
the node in the hidden layer of an NN and therefore could be
extended to DL architectures by extending the concept [21].
In this section, we introduce our proposed fuzzy layer to
incorporate fuzzy-based fusion approaches directly into the DL
architecture. First, we briefly discuss the problem that is being
considered herein: semantic image segmentation using DL.
Semantic segmentation is the process of assigning class labels
(e.g., car, tank, road, building, etc.) to each pixel in an image.
DL for semantic segmentation is most commonly implemented
in what can be separated into two parts: (1) standard CNN
network with the exception of an output layer and (2) an upsampling CNN network on the back half that produces a perpixel class label output. Zeiler et al. introduced deconvolution
networks in [22] for the task of visualizing learned CNNs
to help bridge the gap of understanding what a CNN has
learned. Therein, Zeiler et al. defined a deconvolution network
that attempts to reconstruct a feature map that identifies
what a CNN has learned through unpooling, rectification,
and filtering. In [23], Noh et al. modified this approach for
semantic segmentation by using DL to learn the up-sampling,
or deconvolution, filters rather than inverting (via transposing)
the corresponding filters learned in the convolution network.
Herein, we implement a similar approach as Noh et al.,
utilizing DL strategies to learn the up-sampling filters rather
than performing true deconvolution to reconstruct the feature
maps at each layer. Additionally, this work is focused strictly
on road segmentation, i.e., each pixel is labeled as either
road or non-road. Representing the architecture of our learned
model as f (x, γ), where γ represents the parameters that are
learned by the network such that the error for an example xi
given its ground-truth yi is minimized and can be described
γ̂ = argmin
(L(f (xi , γ), yi ),
where N is the training data set and L is the sof tmax (crossentropy) loss.
As this paper is focused on the introduction of a new fuzzy
layer that can be injected directly into the DL architecture, a
defined network architecture is not presented. Rather, we explore different use cases of the fuzzy layers at different points
throughout the network architecture. For comparison, the fuzzy
layers are utilized either in the down-sampling (convolution
network), up-sampling (“deconvolution” network), or both
sections of the semantic segmentation network. To maintain
consistency in our exploration, a template network architecture
was used such that the only change in the network architecture
was the inclusion or removal of one or more fuzzy layers. The
details of the architecture template used are given in Table I,
with ‘*’ denoting the points in which a fuzzy layer might
be included in the architecture. Note: it is not required that a
fuzzy layer be incorporated after a rectified linear unit (ReLU);
this occurs in the results presented herein to maintain more
consistency across experiments in this exploratory work. It
would have been equally valid to implement a fuzzy layer after
any convolution or pooling layer (referencing layers utilized in
this architecture). The best way(s) to implement fuzzy layers
input data
conv1 1
conv1 2
conv2 1
conv3 1
Kernel Size


Output Size
512 × 512 × 3
512 × 512 × 64
512 × 512 × 64
512 × 512 × 64
512 × 512 × Nf
256 × 256 × 64
256 × 256 × 64
256 × 256 × 64
256 × 256 × Nf
128 × 128 × 64
128 × 128 × 64
128 × 128 × 64
128 × 128 × Nf
256 × 256 × 30
256 × 256 × 30
256 × 256 × Nf
512 × 512 × 30
512 × 512 × 30
512 × 512 × Nf
512 × 512 × Ncl
within the DL architecture is an open question and one that
requires additional research. Technically, a fuzzy layer can be
implemented anywhere in the DL architecture as long as it
follows the input layer and precedes the output layer.
A. Fuzzy Layer
Theoretically, the fuzzy layer can encompass any fuzzy
aggregation strategy desired to be utilized. Herein, we focus on the Choquet integral as the means for fusion. Let
X = {x1 , . . . , xN } be N sources, e.g., sensor, human, or
algorithm. In general, an aggregation function is a mapping
of data from our N sources, denoted by h(xi ) ∈ R, to data,
f (h(x1 ), . . . , h(xN ), Θ) ∈ R, where Θ are the parameters of
f . The ChI is a nonlinear aggregation function parameterized
by the FM. FMs are often used to encode the (possibly
subjective) worth of different subsets of information sources.
Thus, the ChI parameterized by the FM provides a way to
combine the information encoded in the FM with the (objective) evidence or support of some hypothesis, e.g., sensor data,
algorithm outputs, expert opinions, etc. The FM and ChI are
defined as follows.
Definition 1. (Fuzzy Measure) For a finite set of N information sources, X, the FM is a set-valued function g : 2X →
[0, 1] with the following conditions:
1) (Boundary Conditions) g(∅) = 0 and g(X) = 1
2) (Monotonicity) If A, B ⊆ X with A ⊆ B, then g(A) ≤
Note, if X is an infinite set, there is a third condition
guaranteeing continuity.
Definition 2. (Choquet Integral) For a finite set of N
information sources, X, FM g, and partial support function
h : X → [0, 1], the ChI is
h◦g =
wi h(xπ(i) ),
where wi = (Gπ(i) − Gπ(i−1) ), G(i) = g({xπ(1) , . . . , xπ(i) }),
Gπ(0) = 0, h(xi ) is the strength in the hypothesis from source
xi , and π(i) is a sorting on X such that h(xπ(1) ) ≥ . . . ≥
h(xπ(N ) )
The FM can be obtained in a number of ways: human defined, quadratic program, learning algorithm, S-Decomposable
measure (e.g., Sugeno λ-fuzzy measure), etc. Herein, we
define the FM to be five known OWA operators and one
random (but valid) OWA operator. Specifically, the more well
known operators used are max, min, average, soft max, and
soft min. The top 5 sources (i.e., convolution/deconvolution
filter outputs) were sorted based on their entropy value and
fused via the ChI. Therefore, the fuzzy layer accepts the output
from the previous layer as its input, sorts the images (sources)
by some metric (entropy used herein), and performs the ChI
for each of the defined FMs resulting in six fused outputs
(we have six different FMs) that are passed on to the next
layer in the network. An example of a potential fuzzy layer
implementing the ChI as its aggregation method is shown in
Figure 1.
B. Why Have Fuzzy Layers?
As the architecture of DL continues to grow more complex, there is a need to help alleviate the ill-conditioning
that is prone to occur during learning due to the weights
approaching zero. Additionally, a well-known problem when
training deep networks is the internal-covariate-shift problem
[24]. This results in difficulty to optimize the network due
to the input distributions changing at each layer over iterations during training with the changes in distribution being
amplified through propagation across layers. While there are
other approaches that seek to help with this (e.g., batch
normalization [24]), fusion poses itself as a viable solution to
aiding with this problem. One example of the potential benefit
of this is fusion can take 10s, 100s, etc. of inputs (outputs of
previous layers) and condense that information into a fraction
of images. For example, if the output of a convolution, ReLU,
or pooling layer had 256 feature maps, a fuzzy layer could be
utilized to fuse these 256 feature maps down to some arbitrary
reduced number of feature maps, e.g., 30, that capture relevant
information in unique ways from all, or some subset of the 256
feature maps (dependent on the FMs used as well as the metric
used for sorting). Thus, this alone has two potential major
benefits: (1) reduced model complexity and (2) improving the
utilization of the information learned at the previous layer in
the network.
This section first describes the dataset and implementation
details. Next, we present and analyze the results for various
Fig. 1. Illustration of the fuzzy layer. In this example, the layer feeding into the fuzzy layer is a convolution layer. The feature maps are passed as inputs to
the fuzzy layer where they are then sorted, as required for the ChI, based on some metric (entropy used herein). The ChI is computed for six different FMs,
producing six ChI fused resultant images. These six images are then passed on to the next layer in the network.
network configurations as we investigate the implementation
of fuzzy layers.
A. Dataset
The dataset was collected from an UAS with a mounted
MAPIR Survey2 RGB camera. Specifically, the sensor used is
a Sony Exmor IMX206 16MP RGB sensor, which produces
a 24-bit 4,608×3,456 pixel RGB image. The UAS was flown
at an altitude of approximately 60 meters from the ground.
The dataset was captured by flying the UAS in a grid-like
pattern over an area of interest at a U.S. Army test site.
The dataset used in this work comes from a single flight
over this area, which contains 252 images, 20 of which were
selected as training data. The imagery was scaled to size (i.e.,
512×512 pixels) using bilinear interpolation to make them
more digestible by the DL algorithms implemented on the
computer system used herein. As is common with training
for DL algorithms (and ML algorithms in general), data
augmentation strategies are employed in part to increase the
amount of training data available during learning, but also to
lead to more robust solutions. Herein, each image from the
training data set is rotated 90°, 180°, and 270° to provide a
total of 80 images used for training (example shown in Figure
2). Finally, the image road mask for each of the 252 instances
were annotated by hand (see Figure 3).
B. Implementation Details
We based the template network architecture shown in Table
I on the VGG-16 framework [25]. There are modifications to
Fig. 2. Example of data set augmentation used (image rotation). Starting at
the top left and going clockwise: 0°, 90°, 180°, 270°.
the number of convolution layers and filters used throughout
the network; however, the VGG-16 framework served as the
motivation behind the defined template architecture implemented. Initially, we implemented the standard stochastic gra-
Std. Dev.
Fig. 3. Road masks shown (right column) for two sample images.
dient descent with momentum for optimization but achieved
poor results. The Adam algorithm [26] provided the best
optimization performance on this dataset and network design
and was used for all experiments reported, with the initial
learning rate, gradient decay rate, and squared gradient decay
weight set to 0.1, 0.9, and 0.999, respectively. Dropout [27]
is used after pooling with a dropout rate of 50%.
C. Evaluation
To measure network classification accuracy performance,
we utilize an evaluation protocol that is based on Intersection
over Union (IoU) between the ground-truth and predicted
segmentation. We report the mean and standard deviation
of the IoU scores for all test images for each approach
investigated. For clarity, we denote the different experiments
(i.e., different architecture configurations) as follows
• baseline– no fuzzy layers;
• conv-FLs– fuzzy layers are implemented after ‘relu1’,
‘relu2’, and ‘relu3’ in the convolution network (downsampling half);
• deconv-FLs– fuzzy layers are implemented after ‘relu4’
and ‘relu5’ in the “deconvolution” network (up-sampling
• conv-FLs+deconv-FLs– fuzzy layers are implemented
after every ReLU.
The quantitative results for these different architectures are
presented in Table II.
From these preliminary results, we see that the inclusion of
fuzzy layers shows promise for improving DL performance
(in terms of accuracy). In particular, these results indicate
that fuzzy layers are better utilized in the deconvolution phase
of the architecture. Example feature maps randomly selected
from one instance at each layer (the ReLU output is ommited
in the deconvolution network for compactness) are shown in
Figure 4. Looking specifically at the feature maps denoted
Fig. 4. Example feature maps and final segmentation for a randomly selected
image. Also, the feature maps were randomly chosen at each layer.
as ‘fuzzyLayer1’ and ‘fuzzyLayer2’, we see evidence of the
fuzzy layers’ aggregation strategy accumulating evidence of
road information. We note that ‘deconv-FLs’ only performs approximately 2% better than the baseline method, while having
a slightly higher standard deviation. Nevertheless, this helps
show fuzzy layers potential of improving classification performance. It is our conjecture that, for this problem, applying the
fuzzy layers during the convolution stage (results shown as
‘conv-FLs’) results in the loss of too much information from
prior layers (after each ReLU, we summarize 64 filters down
to 6– this is likely too extreme for such an early stage of
learning). Hence, we see a noticeable drop in performance
for both experiments that include fuzzy layers during the
convolution stage (‘conv-FLs’ and ‘conv-FLs+deconv-FLs’).
However, there are a number of factors involved that could
lead to improved performance during the convolution phase,
e.g., increased number of FMs, perhaps a different metric for
sorting should be used, different fuzzy aggregation method,
etc. It should also be noted that the inclusion of the fuzzy
layers had minimal impact on training time (total training time
increased by seconds to a few minutes at most).
We proposed a new layer to be used for DL: the fuzzy
layer. The proposed fuzzy layer is extremely flexible, capable
of implementing any fuzzy aggregation method desired, as
well as capable of being included anywhere in the network
architecture, depending on the desired behavior of the fuzzy
layer. This work was focused on the introduction and early
exploration of the fuzzy layer, and additional research is
needed to further advance the fuzzy layer for DL. For ex-
ample, future work should consider investigating the metric
used for sorting the information sources and its effect on
accuracy performance. Future work is planned to investigate
how the FM should be defined for aggregating via fuzzy
integrals. Additionally, where are the fuzzy layers best utilized
in the network architecture (problem dependent; however,
can a general guidance be developed)? These are but a few
questions that need to be addressed for the fuzzy layer and its
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