AD EF : AN I TERATIVE A LGORITHM TO C ONSTRUCT A DVERSARIAL D EFORMATIONS
A BSTRACT
While deep neural networks have proven to be a powerful tool for many recognition and classiﬁcation tasks, their stability properties are still not well understood.
In the past, image classiﬁers have been shown to be vulnerable to so-called adversarial attacks, which are created by additively perturbing the correctly classiﬁed image.
In this paper, we propose the ADef algorithm to construct a different kind of adversarial attack created by iteratively applying small deformations to the image, found through a gradient descent step.
We demonstrate our results on
MNIST with convolutional neural networks and on ImageNet with Inception-v and ResNet-101.
I NTRODUCTION
In a ﬁrst observation in Szegedy et al.
(2013) it was found that deep neural networks exhibit unstable behavior to small perturbations in the input.
For the task of image classiﬁcation this means that two visually indistinguishable images may have very different outputs, resulting in one of them being misclassiﬁed even if the other one is correctly classiﬁed with high conﬁdence.
Since then, a lot of research has been done to investigate this issue through the construction of adversarial examples: given a correctly classiﬁed image x , we look for an image y which is visually indistinguishable from x but is misclassiﬁed by the network.
Typically, the image y is constructed as y
= x + r , where r is an adversarial perturbation that is supposed to be small in a suitable sense (normally, with respect to an (cid:96) p norm).
Several algorithms have been developed to construct adversarial perturbations, see Goodfellow et al.
(2014); Moosavi Dezfooli et al.
; Kurakin et al.
(2017b)
; Madry et al.
; Carlini & Wagner (2017b) and the review paper Akhtar & Mian (2018).
Even though such pathological cases are very unlikely to occur in practice, their existence is relevant since malicious attackers may exploit this drawback to fool classiﬁers or other automatic systems.
Further, adversarial perturbations may be constructed in a black-box setting (i.e., without knowing the architecture of the DNN but only its outputs)
(Papernot et al., 2017; Moosavi-Dezfooli et al., 2017) and also in the physical world (Kurakin et al., 2017b; Athalye & Sutskever, 2017; Brown et al., 2017; Sharif et al., 2016).
This has motivated the investigation of defenses, i.e., how to make the network invulnerable to such attacks, see Kurakin et al.
(2017a)
; Carlini & Wagner (2017a); Madry et al.
; Tram`er et al.
; Wong & Kolter (2018); Raghunathan et al.
; Athalye et al.
; Kannan et al.
In most cases, adversarial examples are artiﬁcially created and then used to retrain the network, which becomes more stable under these types of perturbations.
Most of the work on the construction of adversarial examples and on the design of defense strategies has been conducted in the context of small perturbations r measured in the (cid:96) ∞ norm.
However, this is not necessarily a good measure of image similarity: e.g., for two translated images x and y , the norm of x − y is not small in general, even though x and y will look indistinguishable if the translation is small.
Several papers have investigated the construction of adversarial perturbations not designed
In this work, we build up on these ideas and investigate the construction of adversarial deformations .
In other words, the misclassiﬁed image y is not constructed as an additive perturbation
y = x
+ r , but as a deformation y = x ◦ (id + τ ) , where τ is a vector ﬁeld deﬁning the transformation.
In this case, the similarity is not measured through a norm of y − x , but instead through a norm of τ , which quantiﬁes the deformation between y and x .
We develop an efﬁcient algorithm for the construction of adversarial deformations, which we call
ADef.
It is based on the main ideas of DeepFool (Moosavi Dezfooli et al., 2016), and iteratively constructs the smallest deformation to misclassify the image.
We test the procedure on MNIST (LeCun) (with convolutional neural networks) and on ImageNet (Russakovsky et al., 2015) (with
Inception-v3 (Szegedy et al., 2016) and ResNet-101 (He et al., 2016)).
The results show that ADef can succesfully fool the classiﬁers in the vast majority of cases (around 99%) by using very small and imperceptible deformations.
We also test our adversarial attacks on adversarially trained networks for MNIST.
Our implementation of the algorithm can be found at https://gitlab.math.
ethz. [Equation] /tandrig/ADef . The results of this work have initially appeared in the master’s thesis Gauksson (2017), to which we refer for additional details on the mathematical aspects of this construction.
While writing this paper, we have come across Xiao et al.
(2018), in which a similar problem is considered and solved with a different algorithm.
Whereas in Xiao et al.
the authors use a second order solver to ﬁnd a deforming vector ﬁeld, we show how a ﬁrst order method can be formulated efﬁciently and justify a smoothing operation, independent of the optimization step.
We report, for the ﬁrst time, success rates for adversarial attacks with deformations on ImageNet.
The topic of deformations has also come up in Jaderberg et al.
(2015), in which the authors introduce a class of learnable modules that deform inputs in order to increase the performance of existing DNNs, and Fawzi & Frossard (2015), in which the authors introduce a method to measure the invariance of classiﬁers to geometric transformations.
A DVERSARIAL DEFORMATIONS 2. A DVERSARIAL PERTURBATIONS
Let K be a classiﬁer of images consisting of P pixels into L ≥ 2 categories, i.e. a function from the space of images X = R cP , where c = 1 (for grayscale images) or c = 3 (for color images), and into the set of labels L = { 1 , . . .
, L } .
Suppose x ∈ X is an image that is correctly classiﬁed by K and suppose y ∈ X is another image that is imperceptible from x and such that K ( y ) (cid:54) =
K ( x ) , then y is said to be an adversarial example .
The meaning of imperceptibility varies, but generally, proximity in (cid:96) p -norm (with 1 ≤ p ≤ ∞ ) is considered to be a sufﬁcient substitute.
Thus, an adversarial perturbation for an image x ∈ X is
a vector r ∈ X such that K ( x + r ) (cid:54) =
K ( x ) and (cid:107) r (cid:107) p is small, where
[Equation]
Given such a classiﬁer K and an image x , an adversary may attempt to ﬁnd an adversarial example y by minimizing (cid:107) x − y (cid:107) p subject to K ( y ) (cid:54)
=
K ( x ) , or even subject to K ( y ) =
k for some target label k
(cid:54) =
K ( x ) .
Different methods for ﬁnding minimal adversarial perturbations have been proposed, most notably FGSM (Goodfellow et al., 2014) and PGD (Madry et al., 2018) for (cid:96) ∞ , and the
DeepFool algorithm (Moosavi Dezfooli et al., 2016) for general (cid:96) p -norms.
2. D EFORMATIONS
Instead of constructing adversarial perturbations, we intend to fool the classiﬁer by small deformations of correctly classiﬁed images.
Our procedure is in the spirit of the DeepFool algorithm.
Before we explain it, let us ﬁrst clarify what we mean by a deformation of an image.
The discussion is at ﬁrst more intuitive if we model images as functions ξ : [0 , 1] 2 → R c (with c = 1 or c = 3 ) instead
Original ‘ ∞ : 1.
Translation by ( − 2 , 1) ‘ ∞ : 0.
Rotation by 10 ◦ ‘ ∞ : 1.
Deformation w.r.t.
τ
T : 2.
T : 3.
T : 1. of discrete vectors x in R cP . In this setting, perturbing an image ξ : [0 , 1] 2 → R c corresponds to adding to it another function ρ : [0 , 1] 2 → R c with a small L p -norm. While any transformation of an image ξ can be written as a perturbation ξ + ρ , we shall restrict ourselves to a particular class of transformations. A deformation with respect to a vector ﬁeld τ : [0 , 1] 2 → R 2 is a transformation of the form ξ (cid:55)→ ξ τ , where for any image ξ : [0 , 1] 2 → R c , the image ξ τ : [0 , 1] 2 → R c is deﬁned by
[Equation] for all u ∈ [0 , 1] 2 , extending ξ by zero outside of [0 , 1] 2 .
Deformations capture many natural image transformations.
For example, a translation of the image ξ by a vector v ∈ R 2 is a deformation with respect to the constant vector ﬁeld τ = v .
If v is small, the images ξ and ξ v may look similar, but the corresponding perturbation ρ = ξ v − ξ may be arbitrarily large in the aforementioned L p -norms.
Figure 1 shows three minor deformations, all of which yield large L ∞ -norms.
In the discrete setting, deformations are implemented as follows.
We consider square images of
W × W pixels and deﬁne the space of images to be X
= (cid:0) R W × W (cid:1) c .
A discrete vector ﬁeld is a function τ : { 1 , . . .
, W } 2 → R 2 .
In what follows we will only consider the set T of vector ﬁelds that do not move points on the grid { 1 , . . .
, W } 2 outside of [1 , W ] 2 .
More precisely,
[Equation]
An image x ∈ X can be viewed as the collection of values of a function ξ : [0 , 1] 2 → R c on a regular grid { 1 / ( W +1) , . . .
, W / ( W +1) } 2 ⊆
[0 , 1] 2 , i.e. x s,t = ξ ( s / ( W +1) , t / ( W +1) )
for s, t = 1 , . . .
, W .
Such a function ξ can be computed by interpolating from x .
Thus, the deformation of an image x with respect to the discrete vector ﬁeld τ can be deﬁned as the discrete deformed image x τ in X by
[Equation]
It is not straightforward to measure the size of a deformation such that it captures the visual difference between the original image x and its deformed counterpart x τ .
We will use the size of the corresponding vector ﬁeld, τ , in the norm deﬁned by
[Equation]
as a proxy.
The (cid:96) p -norms deﬁned in (1), adapted to vector ﬁelds, can be used as well.
(We remark, however, that none of these norms deﬁne a distance between x and x τ , since two vector ﬁelds τ, σ ∈ T with (cid:107) τ (cid:107) T (cid:54) =
(cid:107) σ (cid:107) T may produce the same deformed image x τ = x σ .)
2. T HE ALGORITHM AD EF
We will now describe our procedure for ﬁnding deformations that will lead a classiﬁer to yield an output different from the original label.
Let F = ( F 1 , . .
. , F L ) :
X → R L be the underlying model for the classiﬁer K , such that
[Equation]
Let x ∈
X be the image of interest and ﬁx ξ : [0 , 1] 2 → R c obtained by interpolation from x .
Let l = K ( x ) denote the true label of x , let k ∈ L be a target label and set f = F k − F l .
We assume that x does not lie on a decision boundary, so that we have f ( x ) < 0 .
We deﬁne the function g : T → R , τ (cid:55)→ f ( x τ ) and note that g (0) =
f ( x 0 ) =
f ( x ) < 0 .
Our goal is to ﬁnd a small vector ﬁeld τ ∈ T such that g ( τ ) =
f ( x τ ) ≥ 0 .
We can use a linear approximation of g around the zero vector ﬁeld as a guide:
[Equation]
for small enough τ ∈ T and D 0 g : T → R the derivative of g at τ = 0 .
Hence, if τ is a vector ﬁeld such that
[Equation]
and (cid:107) τ (cid:107) T is small, then the classiﬁer K has approximately equal conﬁdence for the deformed image x τ to have either label l or k .
This is a scalar equation with unknown in T , and so has inﬁnitely many solutions.
In order to select τ with small norm, we solve it in the least-squares sense.
In view of (2), we have ∂x τ ∂τ | τ
[Equation]
s,
[Equation]
∈ R c × 2 .
Thus, by applying the chain rule to g ( τ ) =
f
( x τ ) , we obtain that its derivative at τ = 0 can, with a slight abuse of notation, be identiﬁed with the vector ﬁeld
[Equation]
where (cid:0) ∇ f ( x ) (cid:1) s,t ∈ R 1 × c is the derivative of f in x calculated at ( s, t ) .
With this, (D 0 g ) τ stands for (cid:80) Ws,t =1 D 0 g ( s, t ) · τ ( s, t ) , and the solution to (5) in the least-square sense is given by
[Equation]
Finally, we deﬁne the deformed image x τ ∈ X according to (2).
One might like to impose some degree of smoothness on the deforming vector ﬁeld.
In fact, it sufﬁces to search in the range of a smoothing operator S : T → T .
However, this essentially amounts to applying S to the solution from the larger search space T .
Let α = S
(D 0 g ) =
ϕ ∗ (D 0 g ) , where
S denotes the componentwise application of a two-dimensional Gaussian ﬁlter ϕ (of any standard deviation).
Then the vector ﬁeld
[Equation]
also satisﬁes (5), since S is self-adjoint.
We can hence replace τ by ˜ τ to obtain a smooth deformation of the image x .
We iterate the deformation process until the deformed image is misclassiﬁed.
More explicitly, let x (0) =
x and for n ≥ 1 let τ ( n ) be given by (7) for x ( n − 1) .
Then we can deﬁne the iteration as x ( n ) =
x ( n − 1) ◦
(id + τ ( n ) ) .
The algorithm terminates and outputs an adversarial example y =
x ( n ) if K ( x ( n ) ) (cid:54)
=
l .
The iteration also terminates if x ( n ) lies on a decision boundary of K , in which case we propose to introduce an overshoot factor 1+ η on the total deforming vector ﬁeld.
Provided that the number of iterations is moderate, the total vector ﬁeld can be well approximated by τ ∗ =
τ (1) + · · · + τ ( n ) and the process can be altered to output the deformed image y =
x ◦ (id+(1+ η ) τ ∗ ) instead.
The target label k may be chosen in each iteration to minimize the vector ﬁeld to obtain a better approximation in the linearization (4).
More precisely, for a candidate set of labels k 1 , . .
, k m , we compute the corresponding vectors ﬁelds τ 1 , . . .
, τ m and select
[Equation]
The candidate set consists of the labels corresponding to the indices of the m smallest entries of
F − F l , in absolute value.
Algorithm ADef
Input: Classiﬁcation model F , image x , correct label l , candidate labels k 1 , . . .
, k m
Output: Deformed image y
Initialize y ← x while
K ( y ) =
l do for j = 1 , . . .
, m do α j ← S (cid:0)(cid:80) ci =1 (cid:0) ( ∇ F k j ) i − ( ∇ F l ) i (cid:1) · ∇ y i (cid:1) τ
j ← −
F kj ( y ) − F l ( y ) (cid:107) α j (cid:107) 2 (cid:96)
S
α j end for i ← arg min j =1 ,...,m (cid:107)
τ j (cid:107) T y ← y ◦ (id + τ i ) end while return y
By equation (6), provided that ∇ f is moderate, the deforming vector ﬁeld takes small values wherever ξ has a small derivative.
This means that the vector ﬁeld will be concentrated on the edges in the image x
(see e.g. the ﬁrst row of ﬁgure 2).
Further, note that the result of a deformation is always a valid image in the sense that it does not violate the pixel value bounds.
This is not guaranteed for the perturbations computed with DeepFool.
E XPERIMENTS 3. S ETUP
We evaluate the performance of ADef by applying the algorithm to classiﬁers trained on the MNIST (LeCun) and ImageNet (Russakovsky et al., 2015) datasets.
Below, we brieﬂy describe the setup of the experiments and in tables 1 and 2 we summarize their results.
MNIST: We train two convolutional neural networks based on architectures that appear in Madry et al.
(2018) and Tram`er et al.
(2018) respectively.
The network MNIST-A consists of two convolutional layers of sizes 32 × 5 × 5 and 64 × 5 × 5 , each followed by 2 × 2 max-pooling and a rectiﬁer activation function, a fully connected layer into dimension 1024 with a rectiﬁer activation function, and a ﬁnal linear layer with output dimension 10.
The network MNIST-B consists of two convolutional layers of sizes 128 × 3 × 3 and 64 × 3 × 3 with a rectiﬁer activation function, a fully connected layer into dimension 128 with a rectiﬁer activation function, and a ﬁnal linear layer with output dimension 10.
During training, the latter convolutional layer and the former fully connected layer of MNIST-B are subject to dropout of drop probabilities 1 / 4 and 1 / 2 .
We use ADef to produce adversarial deformations of the images in the test set.
The algorithm is conﬁgured to pursue any label different from the correct label (all incorrect labels are candidate labels).
It performs smoothing by a Gaussian ﬁlter of standard deviation 1 / 2 , uses bilinear interpolation to obtain intermediate pixel intensities, and it overshoots by η = 2 / 10 whenever it converges to a decision boundary.
[Table] [Figure] ImageNet: We apply ADef to pretrained Inception-v3 (Szegedy et al., 2016) and ResNet-101 (He et al., 2016) models to generate adversarial deformations for the images in the ILSVRC2012 validation set.
The images are preprocessed by ﬁrst scaling so that the smaller axis has 299 pixels for the Inception model and 224 pixels for ResNet, and then they are center-cropped to a square image.
The algorithm is set to focus only on the label of second highest probability.
It employs a Gaussian ﬁlter of standard deviation 1 , bilinear interpolation, and an overshoot factor η = 1 / 10 .
We only consider inputs that are correctly classiﬁed by the model in question, and, since τ ∗ =
τ (1) + · · · + τ ( n ) approximates the total deforming vector ﬁeld, we declare ADef to be successful if its output is misclassiﬁed and (cid:107) τ ∗ (cid:107) T ≤ ε , where we choose ε = 3 .
Observe that, by (3), a deformation with respect to a vector ﬁeld τ does not displace any pixel further away from its original position than (cid:107) τ (cid:107) T .
Hence, for high resolution images, the choice ε = 3 indeed produces small deformations if the vector ﬁelds are smooth.
In appendix A, we illustrate how the success rate of ADef depends on the choice of ε .
When searching for an adversarial example, one usually searches for a perturbation with (cid:96) ∞ -norm smaller than some small number ε > 0 .
Common choices of ε range from 1 / 10 to 3 / 10 for MNIST [Table] classiﬁers (Goodfellow et al., 2014; Madry et al., 2018; Wong & Kolter, 2018; Tram ` er et al., 2018;
Kannan et al., 2018) and 2 / 255 to 16 / 255 for ImageNet classiﬁers (Goodfellow et al., 2014; Kurakin et al., 2017a; Tram ` er et al., 2018; Kannan et al., 2018).
Table 1 shows that on average, the perturbations obtained by ADef are quite large compared to those constraints.
However, as can be seen in ﬁgure 2, the relatively high resolution images of the ImageNet dataset can be deformed into adversarial examples that, while corresponding to large perturbations, are not visibly different from the original images.
In appendices B and C, we give more examples of adversarially deformed images.
3. A DVERSARIAL TRAINING
In addition to training MNIST-A and MNIST-B on the original MNIST data, we train independent copies of the networks using the adversarial training procedure described by Madry et al.
That is, before each step of the training process, the input images are adversarially perturbed using the PGD algorithm.
This manner of training provides increased robustness against adversarial perturbations of low (cid:96) ∞ -norm.
Moreover, we train networks using ADef instead of PGD as an adversary.
In table 2 we show the results of attacking these adversarially trained networks, using ADef on the one hand, and PGD on the other.
We use the same conﬁguration for ADef as above, and for
PGD we use 40 iterations, step size 1 / 100 and 3 / 10 as the maximum (cid:96) ∞ -norm of the perturbation.
Interestingly, using these conﬁgurations, the networks trained against PGD attacks are more resistant to adversarial deformations than those trained against ADef.
3. T ARGETED ATTACKS
ADef can also be used for targeted adversarial attacks, by restricting the deformed image to have a particular target label instead of any label which yields the optimal deformation.
Figure 3 demonstrates the effect of choosing different target labels for a given MNIST image, and ﬁgure 4 shows the result of targeting the label of lowest probability for an image from the ImageNet dataset.
Original: ﬁg
Deformed: grocery store
T : 0.
Vector ﬁeld ‘ ∞ : 0.
Perturbation
Original: ﬁg
Deformed: gazelle
T : 2.
Vector ﬁeld ‘ ∞ : 0.
Perturbation
An image from the
ILSVRC2012 validation set deformed to the labels of second highest (ﬁrst row) and lowest (second row) probabilities (out of 1,000) for the original image. The vector ﬁelds and perturbations have been ampliﬁed for visualization.  C ONCLUSION In this work, we proposed a new efﬁcient algorithm, ADef, to construct a new type of adversarial attacks for DNN image classiﬁers.
The procedure is iterative and in each iteration takes a gradient descent step to deform the previous iterate in order to push to a decision boundary.
We demonstrated that with almost imperceptible deformations, state-of-the art classiﬁers can be fooled to misclassify with a high success rate of ADef.
This suggests that networks are vulnerable to different types of attacks and that simply training the network on a speciﬁc class of adversarial examples might not form a sufﬁcient defense strategy.
Given this vulnerability of neural networks to deformations, we wish to study in future work how ADef can help for designing possible defense strategies.
Furthermore, we also showed initial results on fooling adversarially trained networks.
Remarkably, PGD trained networks on MNIST are more resistant to adversarial deformations than
ADef trained networks.
However, for this result to be more conclusive, similar tests on ImageNet will have to be conducted.
We wish to study this in future work.
A CKNOWLEDGMENTS
The authors would like to thank Helmut B¨olcskei and Thomas Wiatowski for fruitful discussions.
A D ISTRIBUTION OF VECTOR FIELD NORMS
Figures 5 and 6 show the distribution of the norms of the total deforming vector ﬁelds, τ ∗ , from the experiments in section 3.
For networks that have not been adversarially trained, most deformations fall well below the threshold of (cid:15) = 3 .
Out of the adversarially trained networks, only MNIST-A trained against PGD is truly robust against ADef.
Further, a comparison between the ﬁrst column of ﬁgure 5 and ﬁgure 6 indicates that ImageNet is much more vulnerable to adversarial deformations than MNIST, also considering the much higher resolution of the images in ImageNet.
Thus, it would be very interesting to study the performance of ADef with adversarially trained network for
ImageNet, as mentioned in the Conclusion.
[Table]
B S MOOTH DEFORMATIONS
The standard deviation of the Gaussian ﬁlter used for smoothing in the update step of ADef has signiﬁcant impact on the resulting vector ﬁeld.
To explore this aspect of the algorithm, we repeat the experiment from section 3 on the Inception-v3 model, using standard deviations σ = 0 , 1 , 2 , 4 ,(where σ = 0 stands for no smoothing).
The results are shown in table 3, and the effect of varying σ is illustrated in ﬁgures 7 and 8.
We observe that as σ increases, the adversarial distortion steadily increases both in terms of vector ﬁeld norm and perturbation norm.
Likewise, the success rate of
ADef decreases with larger σ .
However, from ﬁgure 8 we see that the constraint (cid:107) τ ∗ (cid:107) T ≤ 3 on the total vector ﬁeld may provide a rather conservative measure of the effectiveness of ADef in the case of smooth high dimensional vector ﬁelds.
C A DDITIONAL DEFORMED IMAGES C. MNIST
Figures 9 and 10 show adversarial deformations for the models MNIST-A and MNIST-B, respectively.
The attacks are performed using the same conﬁguration as in the experiments in section 3.
Observe that in some cases, features resembling the target class have appeared in the deformed image.
For example, the top part of the 4 in the ﬁfth column of ﬁgure 10 has been curved slightly to more resemble a 9.
C. I MAGE N ET
Figures 11 – 15 show additional deformed images resulting from attacking the Inception-v3 model using the same conﬁguration as in the experiments in section 3.
Similarly, ﬁgures 16 – 20 show deformed images resulting from attacking the ResNet-10 model.
However, in order to increase variability in the output labels, we perform a targeted attack, targeting the label of 50th highest probability.
Soup bowl
Original
Espresso σ =0.
Espresso σ =2.
Espresso σ =8.
T : 0. T : 0.
T : 1.
‘ ∞ : 0. ‘ ∞ : 0.
‘ ∞ : 0.
Porcupine
Original
Colobus σ =0.
Colobus σ =2.
Colobus σ =8.
T : 0.
T : 1.
T : 2.
[Figure] [Figure] [Figure]
[Figure] [Figure]
Recreational vehicle
Original
Mobile home σ =0.
Mobile home σ =2.
Mobile home σ =8.
T : 1. T : 1.
T : 3.
‘ ∞ : 0. ‘ ∞ : 0.
‘ ∞ : 0.
Harvester
Original
Thresher σ =0.
Thresher σ =2.
Thresher σ =8.
T : 0. T : 0.
T : 1.
‘ ∞ : 0. ‘ ∞ : 0.
‘ ∞ : 0.
O r i g i n a l
T-norm: 1.
D e f o r m e d
T-norm: 1.
T-norm: 1.
T-norm: 1. T-norm: 0. T-norm: 1.
T-norm: 0. T-norm: 0.
T-norm: 0.
T-norm: 1.
O r i g i n a l
T-norm: 1.
D e f o r m e d
T-norm: 1.
T-norm: 0.
T-norm: 1. T-norm: 1. T-norm: 1.
T-norm: 1. T-norm: 1.
T-norm: 0.
T-norm: 2.
O r i g i n a l
T-norm: 1.
D e f o r m e d
T-norm: 1.
T-norm: 1. T-norm: 1.
T-norm: 0.
T-norm: 1.
T-norm: 0.
T-norm: 0. T-norm: 0.
T-norm: 1.
O r i g i n a l
T-norm: 1.
D e f o r m e d
T-norm: 1.
T-norm: 0.
T-norm: 1. T-norm: 1. T-norm: 1.
T-norm: 1. T-norm: 1.
T-norm: 0.
T-norm: 1.
Original: water bottle
T : 0.
Deformed: oil ﬁlter
Original: sea cucumber
T : 0.
Deformed: chiton
Original: reﬂex camera
T : 0.
Deformed: projector
Original: bison
T : 0.
Deformed: ox
Original: minibus
T : 0.
Deformed: police van
Original: red wolf
T : 1.
Deformed: dam
Original: water tower
T : 2.
Deformed: chime
Original: pelican
T : 0.
Deformed: bath towel
Original: reﬂex camera
T : 0.
Deformed: Polaroid camera
Original: pretzel
T : 0.
Deformed: chain
Original: garbage truck
T : 0.
Deformed: trailer truck
Original: ice lolly
T : 1.
Deformed: spatula
[Figure]
Original: sewing machine
T : 1.
Deformed: paper towel
Original: starﬁsh
T : 1.
Deformed: electric ray
Original: freight car
T : 1.
Deformed: triceratops
Original: parachute
T : 1.
Deformed: golfcart
Original: Christmas stocking
T : 0.
Deformed: crane
Original: sock
T : 1.
Deformed: ice cream
Original: black stork
T : 1.
Deformed: hippopotamus
Original: worm fence
T : 1.
Deformed: viaduct
Original: chain mail
T : 0.
Deformed: ski mask
Original: mousetrap
T : 0.
Deformed: perfume
Original: pillow
T : 0.
Deformed: electric guitar
Original: pickup
T : 0.
Deformed: projector
Original: forklift
T : 1.
Deformed: minibus
Original: dingo
[-If x is twice continuously differentiable in a neighborhood of K and (cid:107) x (cid:107) X < ∞ , then Θ x : U → X is Fr ´ echet differentiable at 0 and its Fr ´ echet derivative at 0 , D 
-]
[-One may also limit the number of iterations to some n max ≥ 1 and the magnitude of the total deformation (cid:80) n − 1 m =
-]
T : 0.
Deformed: lynx