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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

stantonprice@yahoo.com

Department of Electrical Engineering

Mississippi College

Clinton, MS USA

srprice1@mc.edu

Department of Electrical Engineering

and Computer Science

University of Missouri

Columbia, MO, USA

andersondt@missouri.edu

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

I. I NTRODUCTION

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.

II. R ELATED W ORK

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].

III. M ETHODOLOGY

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

as

N

X

γ̂ = argmin

(L(f (xi , γ), yi ),

(1)

γ

i=1

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

TABLE I

T EMPLATE ARCHITECTURE IN DETAIL . T HE ‘*’ REPRESENTS LOCATIONS

IN THE ARCHITECTURE THAT A FUZZY LAYER MIGHT BE INCLUDED

HEREIN . T HIS IS NOT A RESTRICTION . Nf AND Ncl REPRESENT THE

NUMBER OF FUSED OUTPUTS AT THAT LAYER AND THE NUMBER OF

CLASSES , RESPECTIVELY.

Name

input data

conv1 1

conv1 2

relu1

*

pool1

conv2 1

relu2

*

pool2

conv3 1

relu3

*

up-conv1

relu4

*

up-conv2

relu5

*

output

Kernel Size

Stride

5×5

5×5

2×2

5×5

2×2

5×5

6×6

6×6

–

1

1

2

1

2

1

2

2

–

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) ≤

g(B).

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

Z

N

X

h◦g =

wi h(xπ(i) ),

(2)

i=1

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.

IV. E XPERIMENTS & R ESULTS

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-

TABLE II

E VALUATION RESULTS ON TEST DATASET FOR ROAD SEGMENTATION .

Method

baseline

conv-FLs

deconv-FLs

conv-FLs+deconv-FLs

Mean

Std. Dev.

78.22%

62.43%

80.79%

68.76%

12.3%

21.5%

14.8%

20.7%

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

half);

• 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).

V. C ONCLUSION

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

implementation.

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