# A Closer Look at Fourier Spectrum Discrepancies
for CNN-generated Images Detection

Keshigeyan Chandrasegaran Ngoc-Trung Tran Ngai-Man Cheung
Singapore University of Technology and Design (SUTD)
{ keshigeyan, ngoctrung_tran, ngaiman_cheung } @sutd.edu.sg

###### Abstract

CNN-based generative modelling has evolved to produce synthetic images
indistinguishable from real images in the RGB pixel space. Recent works have
observed that CNN-generated images share a systematic shortcoming in
replicating high frequency Fourier spectrum decay attributes. Furthermore,
these works have successfully exploited this systematic shortcoming to detect
CNN-generated images reporting up to 99% accuracy across multiple state-of-
the-art GAN models.

In this work, we investigate the validity of assertions claiming that CNN-
generated images are unable to achieve high frequency spectral decay
consistency. We meticulously construct a counterexample space of high
frequency spectral decay consistent CNN-generated images emerging from our
handcrafted experiments using DCGAN, LSGAN, WGAN-GP and StarGAN, where we
empirically show that this frequency discrepancy can be avoided by a minor
architecture change in the last upsampling operation. We subsequently use
images from this counterexample space to successfully bypass the recently
proposed forensics detector which leverages on high frequency Fourier spectrum
decay attributes for CNN-generated image detection.

Through this study, we show that high frequency Fourier spectrum decay
discrepancies are not inherent characteristics for existing CNN-based
generative models—contrary to the belief of some existing work—, and such
features are not robust to perform synthetic image detection. Our results
prompt re-thinking of using high frequency Fourier spectrum decay attributes
for CNN-generated image detection. Code and models are available at
https://keshik6.github.io/Fourier-Discrepancies-CNN-Detection/

##  1 Introduction

With serious concerns over Deepfakes being widely used for malicious purposes
[18, 19, 30, 37, 2, 36, 9, 32], detection of deepfake multimedia content has
become an important research field. With substantial improvement of CNN-based
generative modelling in the recent years [22, 45, 23, 24, 21, 35, 8, 3, 46,
33, 27, 1, 4, 16, 38, 39], it is becoming more and more difficult to assess
the “fakeness” of such synthetic content in the RGB pixel space.

###  1.1 Fourier spectrum discrepancies in CNN-generated images

Recent research suggests that CNN-based generation methods are unable to
reproduce high frequency distribution of real images. Existing work tends to
conclude that this incompetency is an intrinsic property of CNN-based
generative models [12, 10, 25]. While Zhang et al.[44] and Wang et al.[42]
report that CNN generated images have frequency artifacts, Dzanic et al.[12]
and Durall et al.[10] suggest spectrum discrepancies in high frequency: CNN-
generated images at the highest frequencies do not decay as usually observed
in real images (Figure 1). In particular,

  * •

Dzanic et al.[12] analyze high frequency of real and deep network generated
images, and conclude that “deep network generated images share an observable,
systematic shortcoming in replicating the attributes of these high-frequency
modes”.

  * •

Durall et al.[10] observe that “CNN based generative deep neural networks are
failing to reproduce spectral distributions”, and “this effect is independent
of the underlying architecture”.

  * •

Dzanic et al.[12] take a step further and propose to exploit this frequency
discrepancies for detection of deep network generated images, claiming an
accuracy of up to 99.2% across multiple state-of-the-art GAN and VAE models.

Some works also propose different techniques to disguise these high frequency
discrepancies via post-processing the deep network generated images [12, 25],
or modifying the GAN training objective to avoid these discrepancies [10].

It should be noted that the cause of this discrepancy111The terms spectral
discrepancies and spectral inconsistency are used invariably where we refer to
high frequency spectral decay discrepancies. We also use the terms CNN-
generated and synthetic invariably. has not been agreed upon. Zhang et al.[44]
and Durall et al.[10] suggest that this discrepancy could be caused by
transposed convolution. As transposed convolution is used throughout the
generator architectures, it is difficult to replace them. Therefore, Durall et
al.[10] propose spectral regularization to counteract this throughout the GAN
training. Meanwhile, Dzanic et al.[12] attribute this discrepancy to the
linear dependencies in the spectrum of convolutional filters [25], which
hinder learning of high frequencies.

---
Figure 1: The curves show the average azimuthal integration over the power
spectrum. (See section 3). Top row shows the evaluation on DCGAN [34], LSGAN
[31], WGAN-GP [17]. Note the discrepancies at the highest frequencies, the
same as reported in recent works. Note also that these models use transpose
convolutions. The bottom row shows the evaluation after replacing the last
feature map scaling operation with nearest and bilinear interpolation methods.
Refer to table 1 for experiment codes. All evaluation are done using CelebA
[29] (128x128). We observe that spectral consistent GANs are obtained when
using nearest and bilinear interpolation methods for the last feature map
scaling operation.

###  1.2 CNN-generated Image Detectors

Many works have addressed the possibilities of creating detectors to identify
synthetic images apart from real images. Though synthetic image detectors
(CNN-based) worked reasonably with RGB inputs [42, 5], with spectral
discrepancies being observed, several works have proposed to use the
corresponding Fourier representation to train these detectors [44, 13]. In
particular, Frank et al.[13] showed that detectors using frequency domain
yielded better results compared to using the RGB counterpart.

Though observations corresponding to distinguishable frequency footprints
being left by CNN-generated images remained mostly qualitative, Dzanic et
al.[12] and Durall et al.[10, 11] studied the spectral decay attributes and
quantified this behaviour via averaging the power spectrum over frequencies
radially to represent them as 1D information. Through this, they observed high
frequency spectral decay inconsistencies in CNN-generated images. Furthermore,
Dzanic et al.[12] proposed to use a simple KNN classifier using high frequency
spectral attributes (3 features extracted per image) that surprisingly
obtained very high accuracy in identifying synthetic images only with very
small amount of training data. Similarly, Durall et al.[11] used the entire
1d-power spectrum (contains all frequency information) as features to perform
detection using Logistic regression, SVM and K-means clustering algorithms.
These detectors [12, 11] rely on the belief that high frequency decay
discrepancies are intrinsic in CNN-generated images.

###  1.3 Our contributions

In this work, we take a closer look at the high frequency decay discrepancies
in CNN-generated images. Analysis of CNN-generated images is a daunting task:
a large number of different architectures, algorithms and objective functions
have been proposed to train generators. Instead, our study focuses on the last
layer of the generators. Our justification is as follows. Based on Sampling
Theorem [20], the frequency contents of the outputs of individual layers are
limited by the sampling resolution of the corresponding outputs. As previous
work has reported the discrepancies between real and CNN-generated images at
the highest frequencies, we hypothesize that inner generator layers (which
produce lower resolution outputs) are not directly responsible for the high
frequency discrepancies. Therefore, we focus on the last upsampling layer of
generative CNN models.

Due to our focus being localized at the last layer, we are able to pinpoint
the component that is related to this discrepancy across multiple GAN loss
functions, architectures, datasets and resolutions. Our candidate GANs are
similar to Durall et al.[10]: DCGAN [34], LSGAN [31], WGAN-GP [17] and we also
extend to StarGAN [7]. Our experiments suggest that the frequency
discrepancies can be largely avoided by simply modifying the feature map
scaling of the last layer. Importantly, using the same training algorithms,
objective functions and network architectures (except using a different
scaling in the last layer) as in standard GAN models, we are able to avoid the
spectral discrepancies. Therefore, our work provides counterexamples to argue
that high frequency discrepancies are not intrinsic for CNN-generated images.
Furthermore, we are able to successfully bypass the synthetic image detector
proposed by Dzanic et al.[12] with only such change in the last scaling step,
showing that such approach may not be reliable for detection of deep network
generated images. The key takeaway from our work is:

  * •

High frequency spectral decay discrepancies are not intrinsic for CNN-
generated images. Therefore, we urge re-thinking in using such features for
CNN-generated image detection.

##  2 Related Work

Dzanic et al.[12] show that CNN-generated images (GANs and VAEs) demonstrate
different Fourier spectrum decay characteristics. Since the spectra of natural
images tend to behave following the power law [41], Dzanic et al.[12] show
that the Fourier modes of deep network generated images at the highest
frequencies did not decay as seen in real images, but instead stayed
approximately constant. Furthermore, they propose to exploit these
discrepancies to detect CNN-generated synthetic images by fitting a decay
function to the reduced spectra, and using the parameters of the fitted decay
function to build a simple kNN classifier.

Durall et al.[10] show that popular GAN image generators fail to approximate
the spectral distributions of real data, and they attribute this to the use of
transpose convolutions for upsampling. They show that this effect is
independent of the underlying architecture using 1-dimensional spectral
characteristics of images generated from DCGAN [34], LSGAN [31], WGAN-GP[17]
and DRAGAN [26]. Since transpose convolutions are used in the entire generator
models, the propose to counteract their effects by adding a spectral
regularization term to the Generator, thereby penalizing the generator for
spectral distorted samples.

Khayatkhoei and Elgammal [25] suggest the presence of a systematic bias in
GANs against learning high frequencies. They specifically show that for a
given kernel size, as resolution increases, the correlation/ dependency of the
kernel’s spectrum increases thereby systematically preventing GANs to learn
high frequencies without affecting the adjacent frequencies. To alleviate this
shortcoming, they propose frequency shifted generators whose frequencies are
shifted towards specific high frequencies.

##  3 Background

The 2D discrete Fourier transform $F(k_{x},k_{y})$ of a $M\times N$ 2D image
$f(x,y)$ can be written as,

$F(k_{x},k_{y})=\frac{1}{MN}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}f(u,v)e^{-i2\pi(\frac{k_{x}u}{M}+\frac{k_{y}v}{N})}$
(1)

for $k_{x}\in\\{0,1,2,...,M-1\\}$, $k_{y}\in\\{0,1,2,...,N-1\\}$. We follow
the convention in [12, 10] and compute the azimuthally average of the
magnitude of Fourier coefficients over radial frequencies to obtain the
reduced spectrum and normalize it. The reduced spectra indicates the strength
of the signal with respect to different spatial frequencies. Since our study
focuses on high frequency Fourier attributes, we pay attention to the last
$25\%$ of the spatial frequencies (0.75 - 1.0 normalized spatial frequencies)
similar to [12].

The Sampling Theorem ([20], chapter 4.2) states that: A bandlimited image
$f(x,y)$ with bandwidths $\xi_{x0},\xi_{y0}$ can be recovered without error
from the sample values provided the sampling rate is greater than the Nyquist
rate: $2\xi_{x0},2\xi_{y0}$. The image $f(x,y)$ with bandwidths
$\xi_{x0},\xi_{y0}$ means that there is no frequency content outside a bounded
region in the frequency plane defined by $\xi_{x0},\xi_{y0}$. Details and
mathematical proofs can be found in [20] chapter 4.2.

##  4 Last Upsampling Operation and Fourier Discrepancies

Based on last section, the maximum frequency represented by a discrete 2D
signal is constrained by the spatial resolution (sampling) of the signal.
Previous work has consistently reported discrepancies in the highest
frequencies [10, 12]. Therefore, we hypothesize that inner generator layers
that produce lower resolution outputs may not be directly responsible for the
high frequency discrepancies. Therefore, we focus on the last upsampling step.
In particular, we split the last step into 2 operations, 1) Feature map
scaling and 2) Convolution.

###  4.1 Feature Map Scaling

Feature map scaling is a non-parametric operation that scales the input in
both dimensions by some factor (Usually 2 in most GAN architectures). In this
work, we focus on 3 common feature map scaling techniques. 1) Zero-insert
scaling inserts zero between every row and column, scaling the input in both
dimensions which is also used by transpose convolutions. 2) Nearest
interpolation scales the input by inserting nearest neighbour values. 3)
Bilinear interpolation scales the input by inserts new values by taking the
weighted average of adjacent values.

From frequency perspective, zero-insertion introduces the largest amount of
high frequency content as it replicates the low frequency spectrum for the
high frequencies [44], followed by billinear and nearest interpolation. We
focus more on the “spectral trend” than the frequency values, and show a
schematic example of these upsampling effects on the normalized reduced
spectra in Figure 2 by upsampling a reference image of 128x128 from CelebA.
[29]

Figure 2: Example showing normalized spectral effects of upsampling an image.
Vertical line at 88 shows the maximum radial frequency of reference image.

###  4.2 Convolution

The subsequent convolution operation learns kernels in order to satisfy the
optimization objective. Convolutional kernels are capable of suppressing/
amplifying high frequencies. A Gaussian kernel suppresses high frequencies and
a bilateral kernel amplifies high frequencies. So when designing upsampling
blocks in GANs, the general intuition is that irrespective of the feature map
scaling method, the kernels will learn to manipulate the scaled feature maps
to satisfy the objective function.

##  5 Experiments

Here we discuss the main experiments. Additional experiments and analysis can
be found in Supplementary.

In order to investigate the effects of these 2 operations, we design a
rigorous test bed that can address feature map scaling, kernel size and number
of kernels independently. Our aim is to isolate the effects of these 2
operations and understand their roles (if any) in causing the high frequencies
discrepancies. We use celebA [29] dataset at 128x128 resolution and use 3 GANs
with identical architectures but different loss functions namely 1) DCGAN
[34], 2) LSGAN [31] and 3) WGAN-GP [17]. All baseline models consist of
transpose convolutions with kernel size 4 identical to most out of the box GAN
architectures including CycleGAN [46], StarGAN [7] and VQ-VAE [40].

Proposed Test Bed. Table 1 summarizes our test bed. All baseline experiments
using transpose convolutions of kernel size 4 are indicated by the experiment
code Baseline and our handcrafted experiments are indicated using the 3
character code: An experiment code of X.Y.Z indicates X type of feature map
scaling (Possible values are Z : Zero-insertion, N: Nearest interpolation, B:
Bilinear interpolation), Y number of convolutional blocks and Z sized
convolutional kernels for the last upsampling step. A code of N.1.5 indicates
nearest interpolation feature map scaling with a single convolutional block of
5x5 kernel for the last upsampling step. Note that we focus on up scaling by a
factor of 2 as used in most GAN models. Our test bed summary is shown in table
1 and it contains experiments addressing the following factors:

Feature Map scaling. In order to investigate the effect of feature map scaling
on high frequency Fourier attributes, we use experiments Z.1.5, N.1.5 and
B.1.5. We explicitly conduct experiments using zero-insert feature map scaling
as sanity check experiments to verify the effects of transpose convolutions.
Do note that we had to use odd size kernel to maintain the spatial size after
scaling, and hence we use kernels of size 5 for the last convolutional block.

Kernel Size. We use experiments x.1.3, x.1.5 and x.1.7 to investigate the
kernel size effects on high frequency Fourier behaviour. Here x refers to Z, N
or B for different types of scaling as discussed.

Number of kernels. Further, we use experiments x.1.5 and x.3.5 to study the
effect of number of kernels on the high frequency Fourier behaviour.

Code Details Baseline Transpose convolution (4x4 kernel) N.1.5 Nearest
Upsampling + 1 conv block of 5x5 kernel Z.1.5 Zero insert Upsampling + 1 conv
block of 5x5 kernel B.1.5 Bilinear Upsampling + 1 conv block of 5x5 kernel
N.1.3 Nearest Upsampling + 1 conv block of 3x3 kernel N.1.7 Nearest Upsampling
+ 1 conv block of 7x7 kernel Z.1.3 Zero insert Upsampling + 1 conv block of
3x3 kernel Z.1.7 Zero insert Upsampling + 1 conv block of 7x7 kernel B.1.3
Bilinear Upsampling + 1 conv block of 3x3 kernel B.1.7 Bilinear Upsampling + 1
conv block of 7x7 kernel N.3.5 Nearest Upsampling + 3 conv blocks of 5x5
kernel Z.3.5 Zero insert upsampling + 3 conv blocks of 5x5 kernel B.3.5
Bilinear upsampling + 3 conv blocks of 5x5 kernel

Table 1: Test Bed to study the effect of feature map scaling and convolution
in the last upsampling step of the generator, for different GANs: DCGAN,
LSGAN, WGAN-GP, StarGAN. “Baseline” refers to public released code of the GAN
model, which uses transpose convolution of $4\times 4$ kernel. For other
models, we replace the last transpose convolution in Baseline with the
corresponding configurations shown. We emphasize that we only modify the last
step specified as above; the algorithms, learning objectives and architectures
(except the last step) are kept identical as the public released code for
different GAN models.

##  6 Metrics

Spectral behaviour of synthetic images have been analysed only qualitatively,
not quantitatively in previous works [12, 10, 11]. Defining a spectral
consistency metric is non-trivial and to be consistent, we will qualitatively
analyse the high frequency spectral distribution in our experiments. Synthetic
images are spectral consistent if they demonstrate power spectrum decay
behaviour similar to their training data, and if not, vice versa. We use 4000
real and GAN images to generate spectral distribution curves. To ensure that
our setups were trained properly, we use FID scores to assess the quality of
samples and make sure that they are consistent with the FID scores from
baseline experiments.

##  7 Results

###  7.1 Effect of Feature Map Scaling Methods

Figure 3 illustrates the resulting spectral distributions when using different
feature map scaling methods. We observe that images from Z.1.5 and Baseline
experiments are spectral inconsistent for all 3 GANs. Nearest and bilinear
interpolation methods are able to replicate the spectral distribution of real
data reasonably across all 3 GAN models. Since the only change in our models
is the feature map scaling method in the last layer while the algorithms,
objective functions and majority of the network architectures are identical to
public released code, these results qualify to support our thesis that high
frequency Fourier discrepancies are not inherent to GANs.

---
|  |
Figure 3: Feature Map Scaling Results. We observe that experiments using
nearest and bilinear interpolation methods in the last step are able to
produce spectral consistent GANs. Refer to table 1 for experiment details.

###  7.2 Effect of Kernel Size

From Figure 4, we observe that smaller kernel sizes result in more turbulent
spectral behaviour for N.1.3 and B.1.3 LSGAN experiments. Apart from this
observation, experiments using nearest and bilinear interpolation methods are
able to reproduce spectral distributions of real data reasonably well, and
zero-insertion based methods always result in high frequency spectral
distortions. Further, from Figure 5, we observe that when using larger
kernels, all experiments using nearest and bilinear interpolation methods are
able to replicate spectral behaviour of real data, and even with larger
kernels zero-insertion based methods produce high frequency spectral
distortions (Z.1.7).

---
|  |
Figure 4: Smaller Kernel (3x3) Results. We observe that smaller kernels do not
substantially deteriorate spectral consistent GANs except for some turbulent
behaviour observed in LSGAN for N.1.3 and B.1.3 experiments. Refer to table 1
for experiment details.
---
|  |
Figure 5: Larger Kernel (7x7) Results. We observe that larger kernels do not
substantially manipulate the discrepancies in Z.1.7 experiments. Refer to
table 1 for experiment details.

###  7.3 Effect of Number of Kernels

Figure 6 illustrates that increasing the number of kernels do not yield
spectral consistency by itself. Apart from N.3.5 DCGAN experiment, we observe
that all other experiments using nearest and bilinear interpolation methods
are able to approximate the spectral behaviour of real data. Also, GAN
objective functions do not impose any spectral requirements. Thus the kernels
do not have any direct incentive for imposing spectral consistency.

---
|  |
Figure 6: Increased number of kernels (3 conv blocks) Results. We see that
even with more number of kernels in the last upsampling step, Z.3.5 experiment
is not able to produce spectral consistent GANs. Refer to table 1 for
experiment details.

Key observations. Throughout all 39 rigorous experiments, we observe that zero
insertion based feature map scaling methods (including Baseline) are
consistently showing high frequency spectral discrepancies, and most
experiments (22/24) using bilinear and nearest interpolation methods are able
to avoid these high frequency spectral discrepancies. All these results
support our statement that high frequency spectral discrepancies are not
inherent characteristics to GANs. The FID scores for all experiments were
comparable with the baseline FID, and are included in the Supplementary E. We
show image samples from WGAN-GP for Baseline, N.1.5 and B.1.5 setups in Figure
7. Do note that for all experiments, we use the exact same discriminator
architecture as the Baseline experiment.

|  |
---|---|---
Figure 7: WGAN-GP samples for Baseline (Left), N.1.5 (Middle) and B.1.5
(Right) for CelebA [29]. We observe that the visual quality is comparable when
replacing the last transpose convolutions with nearest and bilinear methods.
More visual results in Supplementary D.

##  8 Fourier Synthetic Image Detector

Dzanic et al.[12] state that the spectral properties of real and deep network
generated images are fundamentally different, and proposed a synthetic image
detection method using a “simple” k-nearest neighbours (KNN) classifier to
emphasize the extent of these spectral differences. We follow the exact
procedure as the original authors to train these classifiers and details can
be found in Supplementary F.1.

###  8.1 Experiment Setup

With nearest and bilinear interpolation methods obtaining spectral consistent
GANs for the previous experiments, we question whether the classifier proposed
above would be robust enough to detect these samples as fake. To investigate
this we follow the following steps:

  1. 1.

We train 3 KNN classifiers, one for DCGAN, LSGAN and WGAN-GP respectively. For
GAN images, we use images generated from the Baseline experiment (using
transpose convolutions) as training data.

  2. 2.

We test these classifiers using GAN images generated from the setups in our
test bed to evaluate the robustness of the classifier.

  3. 3.

We also repeat the experiments using 50% data for training the classifier (The
original work used only 10%) to observe any improvements in accuracy.

###  8.2 Detection Results

The complete detection results are shown in table 2. We observe that all
setups corresponding to N.x.x and B.x.x experiments are able to easily bypass
the classifier. Even when 50% training data is used, we are able to bypass the
classifier with ease (Included in Supplementary F.2). The results clearly
demonstrate that the proposed classifier relying on high frequency Fourier
attributes to detect synthetic images, fails to detect images generated from
identical GAN models with last feature map scaling replaced by nearest or
bilinear interpolation methods. These results are consistent with the observed
spectral distributions. By combining these detection results with the
empirical finding that high frequency spectral discrepancies are not inherent
characteristics of CNN-generated images, we suggest re-thinking of using such
discrepancies to detect synthetic images.

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.09\pm 0.03$% $0.34\pm 0.08$% $0.14\pm
0.05$% Z.1.5 $84.82\pm 3.72$% $88.16\pm 3.98$% $99.75\pm 0.14$% B.1.5 $0\pm
0$% $0.1\pm 0$% $0.2\pm 0.12$% N.1.3 $0\pm 0$% $0.06\pm 0.05$% $0.24\pm 0.13$%
N.1.7 $0\pm 0$% $0\pm 0$% $0.06\pm 0.05$% Z.1.3 $98.73\pm 0.56$% $73.09\pm
3.5$% $97.94\pm 0.87$% Z.1.7 $97.23\pm 1.1$% $95.66\pm 1.93$% $99.94\pm 0.07$%
B.1.3 $0\pm 0$% $0.19\pm 0.1$% $0.07\pm 0.05$% B.1.7 $0\pm 0$% $0.1\pm 0$%
$0.17\pm 0.13$% N.3.5 $0.16\pm 0.05$% $0\pm 0$% $0\pm 0$% Z.3.5 $77.67\pm 6$%
$67.66\pm 11.9$% $99.9\pm 0.19$% B.3.5 $0.03\pm 0.05$% $0.48\pm 0.04$%
$0.13\pm 0.05$%

Table 2:  Detection results for the detectors proposed by Dzanic et al.[12],
using CelebA dataset. We follow exactly the procedure in [12] to train the
detector for each GAN model (10% data for training). Then, the images
generated by GAN models using different Setups are tested on the corresponding
detectors. The table shows the successful detection rates, and we highlight
the cases when the detection rates are inferior (less than $10\%$). The
results consistently show that when a GAN model is trained with the last
feature scaling method based on nearest or bilinear, a detector trained using
high frequency features such as [12] fails to detect GAN images (Consistent
with spectral plot observations). All reported detection rates are averaged
over 10 independent runs

##  9 Extended experiments

With observations that high frequency Fourier spectrum discrepancies are not
intrinsic characteristics of GANs, we extend our experiments to address
different dataset, image resolution and GAN objective function to further find
evidences to support our thesis statement. We select 3 setups from our test
bed Z.1.5, N.1.5 and B.1.5 to conduct extended experiments since we have
observed that kernel size/ number of kernels do not substantially manipulate
high frequencies compared to feature map scaling methods. Similar to previous
experiments, we analyze the resulting spectral distributions and evaluate the
robustness of the synthetic detector proposed by Dzanic et al.[12].

###  9.1 LSUN Bedrooms Dataset

In this experiment we use a subset of LSUN Bedrooms Dataset [43] (128x128) to
train DCGAN [34], LSGAN [31] and WGAN-GP [17] identical to previous setups.
The spectral plots are shown in Figure 8. We observe identical results to
CelebA experiment (Figure 3). That is we observe N.1.5 and B.1.5 are producing
spectral consistent GANs and this further supports our statement that high
frequency spectral discrepancies are not inherent in GANs. We also evaluate
the synthetic image detector and observe that N.1.5 and B.1.5 samples can
easily bypass the detector. (See table 3)

---
|  |
Figure 8: LSUN results. We observe spectral plots identical to CelebA [29]
experiments. Refer to table 1 for experiment details

Setup DCGAN LSGAN WGAN-GP N.1.5 $1.5\pm 2.09\%$ $0.62\pm 0.43\%$ $0.03\pm
0.05\%$ Z.1.5 $97.88\pm 1.04$% $95.79\pm 2.07$% $99.87\pm 0.13$% B.1.5
$3.54\pm 6.15\%$ $0.07\pm 0.09\%$ $0.14\pm 0.13\%$

Table 3:  Detection results for the forensics classifiers proposed by Dzanic [12], using LSUN dataset (128x128). The table shows the successful detection rates (10% data for training).  Setup | N.1.5 | Z.1.5 | B.1.5
---|---|---|---
Accuracy |  $52.06\pm 3.77$% |  $64.3\pm 2.7$% | $0\pm 0\%$
Table 4:  Detection results for the forensics classifiers proposed by Dzanic
[12], using CelebA dataset (256x256). The table shows the successful detection
rates (10% data for training).

###  9.2 Image-to-Image Translation

We extend our experiments to Image-to-Image translation domain using StarGAN
[7]. Here we use resized CelebA [29] (256x256) used by the official StarGAN
[7] implementation to study whether spectral consistency can be achieved by
modifying the last feature map scaling operation. We train StarGANs for Z.1.5,
N.1.5 and B.1.5 setups. The spectral plots are shown in Figure 9. We observe
that only B.1.5 is able to produce spectral consistent GANs and N.1.5 produces
high frequency Fourier discrepancies. We would not ask ourselves why nearest
interpolation method behaves differently than bilinear, but rather confirm
that we are able to find bilinear interpolation results as more evidence to
support our statement that high frequency spectral discrepancies are not
inherent characteristics to GANs. We further evaluate the synthetic image
detector and observe that B.1.5 samples can bypass the classifier. (See table
4)

##  10 Spectral Regularization

The recent Spectral regularization (SR) by Durall et al.[10] proposed to add a
regularizer term to the generator loss to explicitly penalize the generator
for spectral distortions. Using SR, they were able to obtain spectral
consistency for DCGAN [34], LSGAN [31], WGAN-GP [17] and DRAGAN [26] using the
celebA [29] (128x128). This method encounters computational overhead due to
calculation of reduced spectrum for images during training. We show that by
modifying the last feature map scaling operation, we are able to achieve
spectral consistent GANs without SR. Importantly, we show that no change in
objective functions is needed. These results and more discussion on SR can be
found in Supplementary B.

##  11 Discussion

In this study, we investigated the validity of contemporary beliefs that CNN-
based generative models are unable to reproduce high frequency decay
attributes of real images. We employ a systematic study to design
counterexamples to challenge the existing beliefs. With maximum frequency
bounded by the spatial resolution, and Fourier discrepancies reported at the
highest frequencies, we hypothesized that the last upsampling operation is
mostly related to this shortcoming. With carefully designed experiments
spanning multiple GAN architectures, loss functions, datasets and resolutions,
we observe that high frequency spectral decay discrepancies can be avoided by
replacing zero insertion based scaling used by transpose convolutions with
nearest or bilinear at the last step. Note that we do not claim that modifying
the last feature map scaling method will always fix spectral decay
discrepancies in every situation, but rather the goal of our study is to
provide counterexamples to argue that high frequency spectral decay
discrepancies are not inherent characteristics of CNN-generated images.
Further, we easily bypass the recently proposed synthetic image detector that
exploits this discrepancy information to detect CNN-generated images
indicating that such features are not robust for the purposes of synthetic
image detection.

In Supplementary material, we provide more GAN models [6, 28] with no high
frequency decay discrepancies. We also investigate whether such high frequency
decay discrepancies are found in other types of computational image synthesis
methods (synthesis using Unity game engine222https://unity.com/) [15, 14]. To
conclude, through this work we hope to help image forensics research manoeuvre
in more plausible directions to combat the fight against CNN-synthesized
visual disinformation.

Figure 9: Spectral plots for StarGAN images. We observe that Bilinear feature
map scaling produces spectral consistent GANs.

Acknowledgments:  This project was supported by SUTD project PIE-SGP-
AI-2018-01. This research was also supported by the National Research
Foundation Singapore under its AI Singapore Programme [Award Number:
AISG-100E2018-005]. This work was also supported by ST Electronics and the
National Research Foundation (NRF), Prime Minister’s Office, Singapore under
Corporate Laboratory @ University Scheme (Programme Title: STEE Infosec - SUTD
Corporate Laboratory).

We also gratefully acknowledge the support of NVIDIA AI Technology Center
(NVAITC) for our research.

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Supplementary materials include research reproducibility details, additional
results and more experiments to support our thesis statement.

##  Supplementary A Standard Deviation of Experiments

Standard deviation of spectral distributions were not included in the paper to
allow better readability of graphs. In this section, we include the standard
deviation of experiments for Baseline, Z.1.5, N.1.5 and B.1.5 experiments. The
spectral distributions with standard deviations for CelebA [29], LSUN [43] and
StarGAN [7] experiments are shown in Figures 10, 11 and 21 respectively. We
also show the Standard deviations of Spectral regularization experiments in
Figure 12. In all cases, we observe that the standard deviations are within
acceptable range.

---
|  |
Figure 10: This figure shows spectral plots from Figure 3 in the paper, with
standard deviations indicated.
---
|  |
Figure 11: This figure shows spectral plots from Figure 8 in the paper, with
standard deviations indicated.
---
|  |
Figure 12: This figure shows spectral plots from Figure 9 in the paper, with
standard deviations indicated. “SR” refers to Spectral Regularization

##  Supplementary B Spectral Regularization

Results of SR experiments are shown in Figure 13. More specifically, SR
performs generator loss scaling as there are no gradients with respect to the
power spectrum difference between real and synthetic images. We were intrigued
by the question on how generator loss scaling can achieve spectral consistency
as claimed by [10] and noticed that the source code uses N.3.5 setup together
with SR333https://github.com/cc-hpc-itwm/UpConv. Analysing SR is out of scope
for this work, but we have showed that N.3.5 setup (Main paper) is sufficient
to achieve spectral consistency in identical setups.

---
|  |
Figure 13: We show the entire spectrum similar to [10]. “SR” refers to
Spectral Regularization. We observe that nearest and bilinear interpolation
methods produce similar spectral distributions comparing to those models
trained with SR. Refer to table 1 in main paper for experiment details.

##  Supplementary C Higher Resolution Experiments

In order to further investigate our thesis statement that high frequency decay
discrepancies are not inherent characteristics for CNN-generated images, we
extend our analysis to larger resolutions. We use image reconstruction as a
representative task to investigate these effects at higher resolutions (We use
512x512).

We select a subset of CelebA-HQ [21] dataset to train a standard autoencoder
for image reconstruction. Similar to experiments in the paper, we perform
experiments corresponding to Baseline, Z.1.5, N.1.5 and B.1.5 setups. We
observe that Baseline and Z.1.5 setups produce high frequency Fourier
discrepancies for reconstructed images, and N.1.5 and B.1.5 setups produce
spectral consistent reconstructed images. The spectral distributions are shown
in Figure 14. This further confirms that high frequency Fourier discrepancies
are not intrinsic for CNN-generated images.

Figure 14: Spectral plots for Image reconstruction at 512x512 for CelebA-HQ.
The plots are averaged over 1000 samples. We show the entire spectrum similar
to [10]. Similar to our other results, we observe that nearest and bilinear
interpolation methods for the last upsampling step allows to obtain spectral
consistent image reconstructions. We also show the standard deviation of
experiments in the plot in addition to the mean.

##  Supplementary D Samples

We show more samples and spectral distributions for important CelebA [29]
experiments in this section.

GAN Samples: We show extensive samples corresponding to Baseline, Z.1.5, N.1.5
and B.1.5 setups for DCGAN [34], LSGAN [31], WGAN-GP [17] in Figures 15, 16,
17 respectively.

Baseline |
---|---
Z.1.5 |
N.1.5 |
B.1.5 |
Figure 15: DCGAN [34] samples for CelebA [29]. Refer to Table 1 in paper for experiment codes. Baseline |
---|---
Z.1.5 |
N.1.5 |
B.1.5 |
Figure 16: LSGAN [31] samples for CelebA [29]. Refer to Table 1 in paper for experiment codes. Baseline |
---|---
Z.1.5 |
N.1.5 |
B.1.5 |
Figure 17: WGAN-GP [17] samples for CelebA [29]. Refer to Table 1 in paper for
experiment codes.

Image-to-Image Translation:  For StarGAN [7] experiments, we show an example
of a reference image with corresponding translated images for Baseline, Z.1.5,
N.1.5 and B.1.5 setups in Figures 18. We show the corresponding spectral
distributions in Figure 19.

|  |  |  |
---|---|---|---|---
Figure 18: Image Translation results using StarGAN. Original Image (leftmost),
Baseline (column 2), Z.1.5 (column 3), N.1.5 (column 4), B.1.5 (rightmost) for
attribute Blonde hair is shown. Corresponding high frequency spectral
distributions are shown in Figure 19. Refer to table 1 in paper for experiment
codes.
---
Figure 19: Spectral distributions for examples shown in 18. Real refers to
Original image. We observe that B.1.5 setup produces spectral consistent
images similar to observations recorded in the paper. Refer to table 1 in
paper for experiment codes.

Image Reconstruction:  We show image reconstruction results for a few CelebA-
HQ [21] examples in Figure 20

|  |  |  |
---|---|---|---|---
Figure 20: Reconstruction Results. Original Image (leftmost), Baseline (column
2), Z.1.5 (column 3), N.1.5 (column 4), B.1.5 (rightmost) for CelebA-HQ [21]
samples are shown. Refer to table 1 in paper for experiment codes. Figure 21:
This figure shows spectral plots from Figure 10 (StarGAN) in the paper, with
standard deviations indicated

##  Supplementary E FID scores

FID scores for all CelebaA [29] experiments are shown in table 5. We used 50k
real images and 50k GAN images to calculate each FID score. We observe that
the FID scores of nearest and bilinear interpolation methods are comparable or
better than the Baseline FID for all GAN setups.

Setup Code | DCGAN | LSGAN | WGAN-GP
---|---|---|---
Baseline | $88.6$ | $73.26$ | $60.6$
N.1.5 | $87.52$ | $70.69$ | $48.69$
Z.1.5 | $69.14$ | $60.29$ | $47.73$
B.1.5 | $84.65$ | $78.66$ | $52.18$
N.1.7 | $90.8$ | $73.09$ | $60.11$
Z.1.7 | $71.45$ | $59.55$ | $43.1$
B.1.7 | $79.92$ | $76.33$ | $55.28$
N.1.3 | $93.54$ | $74.06$ | $58.35$
Z.1.3 | $65.46$ | $61.45$ | $56.91$
B.1.3 | $76.04$ | $81.97$ | $58.55$
N.3.5 | $73.63$ | $78.31$ | $55.47$
Z.3.5 | $68.41$ | $66.27$ | $57.59$
B.3.5 | $80.89$ | $72.29$ | $54.84$
SR | $99.2$ | $86.16$ | $60.81$
Table 5:  FID scores of GAN images trained on CelebA [29] dataset. We include
the FID scores of Spectral Regularized GANs (indicated as SR) for comparison.

##  Supplementary F Fourier Synthetic Image Detector

We include more information and results for the synthetic image detector
proposed by Dzanic et al.[12]. All detection rates are averaged over 10
independent runs.

###  F.1 Classifier Implementation Details

The exact procedure used by Dzanic et al.[12] to implement the classifier is
shown below. For easier understanding, let us assume that we want to train a
classifier to detect between real and StyleGAN images

  1. 1.

Collect a repository of 1000 StyleGAN images and 1000 real images.

  2. 2.

Obtain the un-normalized reduced spectrum for every real and StyleGAN image
for the last 25% spatial frequencies (0.75 - 1.0).

  3. 3.

Fit the reduced spectrum using power law function for every real and GAN
image.

  4. 4.

Extract 3 features $b_{1},b_{2},b_{3}$ from the fitted spectrum where $b_{1}$:
start value of the fitted spectrum, $b_{2}$: decay value of the fitted
spectrum, $b_{3}$: end value of the fitted spectrum. Though the authors
mention only $b_{1},b_{2}$ in their paper, their official implementation
contained $b_{3}$ as well. This difference is acceptable since $b_{3}$ is
linearly dependent on $b_{1},b_{2}$ under the assumption that the power law is
a good fit.

  5. 5.

Train/ apply a binary KNN classifier (with k=5) using the 3 features extracted
per image to predict if the GAN images are real or fake. The authors use 100
real and GAN images each to train the classifier and use the remaining 900
samples to test.

###  F.2 Additional Results

Additional detection results when using 50% data to train (authors used 10%
and these results are shown in the main paper) for CelebA [29], LSUN [43] and
StarGAN [7] experiments are shown in table 6, 7, 8 respectively. We also
conduct experiments using the reconstructed images and the detection results
are shown in 9.

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.1\pm 0$% $0.28\pm 0.04$% $0.2\pm 0$% Z.1.5
$82.18\pm 0.26$% $86.05\pm 0.43$% $99.7\pm 0$% B.1.5 $0\pm 0$% $0.1\pm 0$%
$0.29\pm 0.03$% N.1.3 $0\pm 0$% $0\pm 0$% $0.3\pm 0$% N.1.7 $0\pm 0$% $0\pm
0$% $0.08\pm 0.04$% Z.1.3 $98.43\pm 0.13$% $71.77\pm 0.48$% $97.79\pm 0.03$%
Z.1.7 $96.55\pm 0.07$% $94.59\pm 0.09$% $99.9\pm 0$% B.1.3 $0\pm 0$% $0.12\pm
0.04$% $0.1\pm 0$% B.1.7 $0\pm 0$% $0.1\pm 0$% $0.15\pm 0.08$% N.3.5 $0.2\pm
0$% $0\pm 0$% $0\pm 0$% Z.3.5 $74.07\pm 0.68$% $62.17\pm 1.05$% $99.97\pm
0.05$% B.3.5 $0\pm 0$% $0.49\pm 0.03$% $0.12\pm 0.04$%

Table 6:  Detection results for the detectors proposed by Dzanic et al.[12],
using CelebA dataset (50% data for training). We follow exactly the procedure
in [12] to train the detector for each GAN model. The table shows the
successful detection rates, and we highlight the cases when the detection
rates are inferior (less than $10\%$). The results are consistent with
observations in the spectral plots.

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.3\pm 0\%$ $0.6\pm 0\%$ $0\pm 0\%$ Z.1.5
$98.41\pm 0.15$% $94.84\pm 0.07$% $99.93\pm 0.05$% B.1.5 $0.1\pm 0\%$ $0\pm
0\%$ $0.06\pm 0.05\%$

Table 7:  Detection results for the detectors proposed by Dzanic et al.[12],
using LSUN Bedrooms [43] dataset (50% data for training). The table shows the
successful detection rates, and we highlight the cases when the detection
rates are inferior (less than $10\%$).

Setup N.1.5 Z.1.5 B.1.5 Accuracy $53.7\pm 0.15$% $64.61\pm 0.37$% $0\pm 0\%$

Table 8:  Detection results for the forensics classifiers proposed by Dzanic [12], using CelebA [29] dataset (256x256) for StarGAN (50% data for training). We observe that B.1.5 samples easily bypasses the classifier.  Setup | 10% train data | 50% train data
---|---|---
N.1.5 | $0\pm 0\%$ | $0\pm 0\%$
Z.1.5 |  $94.8\pm 1.75$% |  $95.44\pm 0.26$%
B.1.5 | $0\pm 0\%$ | $0\pm 0\%$
Table 9:  Detection results for the forensics classifiers proposed by Dzanic
[12], using reconstructed images. We observe that N.1.5 and B.1.5 samples can
easily bypass the classifier.

##  Supplementary G Stronger Classifiers

###  G.1 SVM and MLP classifiers

We perform additional experiments using SVM and MLP classifiers (exact same
setup as [12], only change in classifiers). The results are shown in Table 10
and Table 11. Our results are consistent: even with SVM/MLP classifiers, we
can bypass them by replacing zero insertion last layer with nearest (N) or
bilinear (B). For all experiments, we use 10% data for training, and the
reported results are averaged over 10 runs. We also observe similar detection
rates when using 50% and 80% data for training.

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.1\pm 0$% $0.31\pm 0.06$% $0.23\pm 0.16$%
Z.1.5 $82.22\pm 1.98$% $87.33\pm 2.77$% $99.45\pm 0.21$% B.1.5 $0\pm 0$%
$0.11\pm 0.09$% $0.25\pm 0.17$% N.1.3 $0.01\pm 0.03$% $0.07\pm 0.05$% $0.35\pm
0.22$% N.1.7 $0\pm 0$% $0\pm 0$% $0.05\pm 0.05$% Z.1.3 $98.3\pm 0.45$%
$72.13\pm 2.21$% $96.81\pm 1.63$% Z.1.7 $95.81\pm 0.93$% $95.55\pm 1.23$%
$99.24\pm 0.43$% B.1.3 $0\pm 0$% $0.25\pm 0.12$% $0.15\pm 0.15$% B.1.7 $0\pm
0$% $0.11\pm 0.03$% $0.3\pm 0.27$% N.3.5 $0.1\pm 0$% $0\pm 0$% $0\pm 0$% Z.3.5
$74.27\pm 3.32$% $65.37\pm 6.5$% $93.82\pm 0.6$% B.3.5 $0.04\pm 0.07$% $0.5\pm
0.05$% $0.21\pm 0.14$%

Table 10:  Detection rates using SVM (RBF kernel) using same features as
Dzanic et al.[12].

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.1\pm 0$% $0.77\pm 0.15$% $1.53\pm 0.32$%
Z.1.5 $81.14\pm 2.9$% $83.88\pm 2.59$% $99.77\pm 0.09$% B.1.5 $0.04\pm 0.1$%
$0.87\pm 0.46$% $3.03\pm 0.82$% N.1.3 $0.18\pm 0.04$% $0.05\pm 0.13$% $1.4\pm
0.2$% N.1.7 $0\pm 0$% $0.04\pm 0.05$% $0.67\pm 0.18$% Z.1.3 $97.54\pm 0.41$%
$72.65\pm 2.64$% $98.11\pm 0.44$% Z.1.7 $94.53\pm 0.97$% $93.07\pm 1.6$%
$99.97\pm 0.05$% B.1.3 $0.03\pm 0.09$% $1.6\pm 0.54$% $2.79\pm 0.5$% B.1.7
$0.01\pm 0.03$% $0.42\pm 0.29$% $4.63\pm 1.01$% N.3.5 $0.17\pm 0.05$% $0\pm
0$% $0.37\pm 0.27$% Z.3.5 $74.88\pm 2.79$% $71.22\pm 4.46$% $99.8\pm 0$% B.3.5
$0.28\pm 0.14$% $1.89\pm 0.45$% $3.66\pm 1.19$%

Table 11:  Detection rates using MLP (2 hidden layers of size 10 with sigmoid
activation) using same features as Dzanic et al.[12].

Setup DCGAN LSGAN WGAN-GP N.1.5 $0.12\pm 0.04$% $1.01\pm 0.29$% $0.08\pm
0.06$% Z.1.5 $95.91\pm 2.43$% $98.29\pm 0.85$% $96.62\pm 1.61$% B.1.5 $0.09\pm
0.03$% $0.24\pm 0.11$% $0\pm 0$% N.1.3 $0.26\pm 0.07$% $0.9\pm 0.2$% $0.08\pm
0.04$% N.1.7 $0\pm 0$% $0.09\pm 0.12$% $0.12\pm 0.04$% Z.1.3 $97.72\pm 1.86$%
$88.44\pm 2.55$% $91.7\pm 3.25$% Z.1.7 $98.43\pm 1.43$% $99.29\pm 0.5$%
$97.09\pm 2.8$% B.1.3 $0\pm 0$% $1.11\pm 0.3$% $0\pm 0$% B.1.7 $0.01\pm 0.03$%
$0.16\pm 0.1$% $0.17\pm 0.05$% N.3.5 $0\pm 0$% $10.85\pm 7.05$% $0\pm 0$%
Z.3.5 $97.32\pm 1.24$% $97.79\pm 1.4$% $98.69\pm 1.77$% B.3.5 $0.23\pm 0.05$%
$1.09\pm 0.17$% $0.02\pm 0.04$%

Table 12:  Detection rates using classifier proposed by Durall et al.[10].
Following [10], we use entire 1D spectrum as features.

###  G.2 Using entire spectrum as features

We followed similar setup as Durall et al.[10] to train a SVM classifier using
entire spectrum as features (88 dimensional features for 128x128 images). The
finding is consistent: we observe that the features are still non-separable
when using nearest (N) and bilinear (B) for the last upsampling step. The
results are shown in Table 12.

##  Supplementary H Virtual KITTI

GANs belong to a larger family of computational image synthesis algorithms. In
this section, we investigate the high frequency decay attributes of data
created entirely using Unity Game Engine. We compare the spectral behaviour
between the Official KITTI tracking benchmark [15] (Real images) and the
Virtual KITTI [14] (Synthesized images). Virtual KITTI [14] recreates real-
world videos from the KITTI tracking benchmark [15] inside
Unity444https://unity.com/ game engine. We show some samples in Figure 25. We
show the entire power spectrum in Figure 22 and we observe that the images
synthesized from the game engine do not have high frequency spectral
discrepancies.

Figure 22: This figure shows the spectral plots for all the frequencies for
KITTI [15] and Virtual KITTI [14] datasets. All the images were center cropped
to 370x370. We observe that the Virtual KITTI images do not have high
frequency discrepancies.

##  Supplementary I CRN/ IMLE

We also observe that high frequency decay discrepancies are not seen in some
out-of-the-box GAN models. Specifically, we observe that CRN [6] and IMLE [28]
GANs do not have such discrepancies. We show the entire power spectrum for CRN
[6] and IMLE [28] GANs in Figure 23 and 24 respectively. Do note that both
these models are pre-trained on GTA game data (Another instance of data
synthesized from game engines). This further helps to confirm that such
discrepancies are not intrinsic.

Figure 23: This figure shows the spectral plots for all the frequencies for
GTA (Real) images and CRN [6] synthesized images. All the images were center
cropped to 256x256. We observe that the CRN generated images do not have high
frequency discrepancies. Figure 24: This figure shows the spectral plots for
all the frequencies for GTA (Real) images and IMLE [28] synthesized images.
All the images were center cropped to 256x256. We observe that the IMLE
synthesized images do not have high frequency discrepancies.

##  Supplementary J Dataset Details

For CelebA [29] experiments, we use the officially released train subset
consisting of 162, 770 images. For LSUN [43] experiments, we select a random
subset of 200, 000 images for training. For StarGAN experiments, we follow the
official implementation. For autoencoder experiments, we select a random
subset of 1000 images from CelebA-HQ [21].

Figure 25: Samples of real images from KITTI tracking benchmark [15] dataset
and the recreated images using Unity game engine obtained from Virtual KITTI
[14] dataset (Right)

.

##  Supplementary K Implementation Details

For GAN experiments, we use Adam optimizer with $\beta_{1}=0.5$,
$\beta_{2}=0.999$ and batch size 64. For all CelebA [29] experiments, we used
an initial learning rate = $2\times 10^{-4}$. The learning rate was reduced
based on FID scores for all experiments. For DCGAN, LSGAN and WGAN-GP
experiments, we use the GitHub code used by the Spectral Regularization
paper[10] 555https://github.com/LynnHo/DCGAN-LSGAN-WGAN-GP-DRAGAN-Pytorch. For
StarGAN [7], we use the official implementation
666https://github.com/yunjey/stargan with default hyper-parameters. For
Spectral Regularization [10] experiments, we use the officially released code
777https://github.com/cc-hpc-itwm/UpConv.

For all autoencoder experiments, we use Adam optimizer with $\beta_{1}=0.9$
and $\beta_{2}=0.999$, batch size 128, initial learning rate $2.5\times
10^{-3}$ and learning rate decay scheme that scales the learning rate by 0.9
when reconstruction error plateaus.

For Fourier synthetic detector [12] experiments, we use the officially
released matlab code
888https://github.com/tarikdzanic/FourierSpectrumDiscrepancies for feature
extraction and use our own script to perform KNN classification. For FID
calculation, we used the open-source Pytorch FID implementation
999https://github.com/mseitzer/pytorch-fid.

More details on hyper-parameters and research reproducibility can be found in:
https://keshik6.github.io/Fourier-Discrepancies-CNN-Detection/