Rebeca Araripe

Furtado Cunha

Naman Sharma

Zeno Toffano

François Dubois

Fuzzy Logic Behavior of Quantum-Controlled Braitenberg Vehicle Agents

Keywords: Quantum Robots, Fuzzy Logic, Quantum Gates, Braitenberg Vehicles, Emotion Analysis

The behavior of agents represented by Braitenberg vehicles is investigated in the context of the quantum robot paradigm. The agents are processed through quantum circuits with fuzzy inputs, this permits to enlarge the behavioral possibilities and the associated decisions for these simple vehicles. The logical formulation Eigenlogic, using quantum logical observables as propositions and eigenvalues as truth values is applied in this investigation. Fuzzy logic arises naturally in this formulation when considering input states that are not eigenvectors of the logical observables, the fuzzy membership being the quantum mean value of the logical observable on the input state. Computer simulations permits visualization of complex behaviors resulting from the multiple combination of quantum control gates. This allows the detection of new Braitenberg vehicle behavior patterns related to identified emotions and linked to quantum-like effects.

Introduction

With the recent improvements in quantum information, interest has been growing in the area of quantum robotics. This signifies using concepts from quantum computing to build and conceptualize systems capable of decision making. Simple robots capable of showing complex behavior have been introduced in Valentino Braitenberg's work on cybernetics proposing vehicle agents that show human-like emotions [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF]. Here we aim to implement Braitenberg Vehicles (BV) using quantum-like circuits. Paul Benioff who was the first to propose the idea of a quantum Turing machine in 1980 was also the proponent of the theoretical principle of a quantum robot [START_REF] Benioff | Quantum Robots and Environments as a first step towards a quantum mechanical description of systems that are aware of their environment and make decisions[END_REF].

A quantum-like implementation of BVs has been undertaken in a previous research work [START_REF] Raghuvanshi | Quantum robots for teenagers. 37th International Symposium on Multiple-Valued Logic ISMVL[END_REF], where Raghuvanshi et al. used the reversible quantum logical gate structure in building prototype BVs with Lego blocks. In the work presented here, in order to provide flexibility to the vehicle's behavior, we design fuzzy logic quantum-like circuits based on the quantum models proposed in [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF].

Our goal here is to test the multiple combinations of quantum gates used in the control of BV by analyzing their complex behavior. For this purpose, we have developed a visual simulation tool illustrating BV's behaviors. This allows us to investigate new emotional behavior patterns not expected by a simple theoretical analysis used for for the description of BV in [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF].

Mathematical models and simulations of individual and swarm automaton agents in response to environmental stimuli have attracted much interest for the understanding of complex behaviors of a group of animals. In [START_REF] Kagan | Probabilistic Control and Swarm Dynamics in Mobile Robots and Ants[END_REF], Kangan et al. developed a probabilistic control model (denominated ANIMAT) of mobile agents with biologically-inspired navigation, with the goal to mimic an ant colony. In this research it was made clear that the agent changes its state as a response to environmental stimuli and/or as a result of its own action on the environment, this observation can lead to path planning and intelligent-like emergent behavior of a group of agents. So agent behavioral models should take into account the uncertainty and the dynamics due to their environment.

A theoretical description of the model proposed here for the interaction of BV quantum robots with their environment is given in Section 2. This section details the mechanisms of transformation from the sensor input stimuli states to the control signals for the wheels of the BV (see Fig. In section 3 we provide a brief outline of the logical models used in the design of the BV quantum robots. These models are based on the Eigenlogic approach proposed in [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] and permit to characterize behaviors associated to fuzzy logic control.

We finish by illustrating the simulation results in Section 4. In order to test the accuracy of our simulations, we first verify the behaviors predicted by the thought experiments proposed by Valentino Braitenberg [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF]. This is done by observing how our simulator reacts to logical operators specifically designed for representing the vehicle's emotions of Fear, Aggression, Love and Exploration. After testing the BVs with these well established emotions, we test other Eigenlogic control operators such as i.e. the implication operator (see Dubois and Toffano in [4]). Finally using different operators in a variety of combinations gives us an interesting insight into the global emotional behavior of the vehicles.

Modeling of the vehicle

Introduction to Braitenberg Vehicles

In his book "Vehicles: Experiments in Synthetic Psychology" [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF] Valentino Braitenberg describes various thought experiments using simple machines (Braitenberg Vehicles BV) that consist of sensors, motors and wheels. Similarly in our approach the quantum robot agents are represented by two controlled wheels at the rear of the BV, with two sensors at the front. The sensors detect light produced by surrounding sources. The sensors can be be connected in different combinations to the wheels, and may have a positive or negative relation with the strength of the stimuli. These simple changes in configuration can lead to complex and surprising results in the agent behavior. Braitenberg terms this the "law of uphill analysis and downhill invention". According to this principle it is far easier to create machines that exhibit complex behavior based on simple connective structures than to try to derive their structures from behavioral observations and interpretations.

Quantum approach for BV decision making

To model the system we consider BVs with two sensors: SL in the upper left corner and SR in the upper right corner (see Fig. 2). The sensors detect light intensity and transform it into an input state vectors for the quantum circuit. The system then delivers as an output a fuzzy logical measure that is translated into wheel control by motors.

The computational block is composed of matrix operators designed using the quantum-like Eigenlogic method [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] discussed in section 3. We use two operators F L and F R . Each one takes the inputs and delivers signals to the respective motors (left or right). This approach avoids to physically permute the wiring connecting sensors and motors in order to obtain different behaviors, it all can be handled by the computing device. The diagram in Fig. 1 resumes the inputoutput process. The equivalent quantum formulation of the system consists in defining the input state vector presented in equation [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF],

|x = |x L ⊗ |x R = |x L x R ( 1
)
where the ket |x corresponds to the combined input state of the bilinear system, formed by the Kronecker product of the individual input states |x L and |x R corresponding respectively to the left and right sensors. These vectors can be understood as qubits with a given orientation in Hilbert space. The output signal is obtained by the quantum mean value (Born rule) of the logical projection observable on the compound input state |x . It is the control function for the wheels. The quantities µ L and µ R , for left and right wheel control, are:

µ L = x| F L |x (2) µ R = x| F R |x (3)
These quantities can also be interpreted as fuzzy membership functions [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF]. In the following section we will show that these values are proportional to the angular speed of the wheels.

Output states and vehicle's motion

Once the vehicle is submitted to the input state |x f , it feeds it into the computation block as presented in the diagram of Fig. 1. The mean values for left and right are obtained as in equations ( 2) and ( 3), depending directly in the logical control operators F L and F R . The wheel angular speeds ω L and ω R are calculated by applying the following relations:

ω L = K • µ L (4) ω R = K • µ R ( 5
)
where K is a constant which allows us to tune the sensitivity of the vehicle to the stimuli. The x and y components of the vehicle speed vector (cf. Fig. 2) are determined by the following differential equations:

ẋ = v x = 1 2 R (ω L + ω R ) cos(θ) (6) ẏ = v y = 1 2 R (ω L + ω R ) sin(θ) (7) θ = 1 d R (ω R -ω L ) (8)
where R represents the BV wheel radius. The period of time between every stimuli update is considered the unit of time of our system. Thus the vehicle position frame update equations are given by:

x(t n+1 ) = x(t n ) + v x (t n ) • (t n+1 -tn) = x(t n ) + v x (t n ) (9) y(t n+1 ) = y(t n ) + v y (t n ) • (t n+1 -tn) = y(t n ) + v y (t n ) (10)

Topics in Eigenlogic

Eigenlogic, [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] is a new quantum-like formulation in logic associating linear algebra and propositional logic. In this picture logical connectives are represented by observables and the truth values of the propositions are the eigenvalues of these observables. This permits to make a straightforward correspondence with quantum observables. As explained previously in Section 2, the computational block of our simulated vehicles is composed of two-argument logical observables.

The Eigenlogic operators

Because of logical completeness an n-argument (arity-n) binary logical system has a total of 2 2 n logical connectives. In Eigenlogic this is related to the number of possible compatible projection operators, which are commuting observables. For example, for a two qubit system defined in a 4-dimensional space, the number of possible logical connectives is 2 2 2 = 16. In this case, we have to consider 2 2 = 4 different distinct input cases corresponding to the 4 eigenstates of the logical system: |00 , |01 , |10 and |11 . For each propositional case (eigenstate) we can assign a truth value which corresponds to one of the eigenvalues {0, 1}. For a given logical connective, we derive the corresponding logical observable by matrix interpolation methods using the eigenvalues and the associated projection operators as discussed in [START_REF] Toffano | Interpolating Binary and Multivalued Logical Quantum Gates[END_REF]. In the case of two logical input arguments A and B, the basic logical projector operators and its negations are given in equation ( 11), where I is the identity operator.

F A = |1 1| ⊗ I F B = I ⊗ |1 1| F Ā = (I -|1 1|) ⊗ I = |0 0| ⊗ I F B = I ⊗ (I -|1 1|) = I ⊗ |0 0| (11)
In general every logical connective can be expressed as a linear combination of the preceding observables using the conjunction connective. In logic this corresponds to the canonical form: disjunction of conjunctions SOP (Sum Of Products). In a 2-argument Eigenlogic system, one will consider the 4 conjunction observables corresponding to the rank-1 projection operators

F A•B = F A • F B , F Ā•B , F A• B and F Ā• B .
The scalar coefficients of the linear combination are the truth-values of the logical connective one wants to express. The linear interpolation used for the "implication" logical observable is given by:

F A =⇒ B = 1 F A•B + 1 F Ā•B + 0 F A• B + 1 F Ā• B = I -F A + F A•B (12)

Incorporation of Fuzzy Logic

In order to better represent real situations, we consider that the stimuli of the BV's light sensors correspond to continuous variations of the light intensity.

In our quantum formulation the vector eigenspace of a 2-argument Eigenlogic observables is the 2-qubit canonical basis {|00 , |01 , |10 , |11 }. State |0 , corresponds to no light at all and state |1 , corresponds to maximum luminosity. Fuzzy logic deals with truth values that may be any number between 0 and 1, where the truth of a proposition may range between completely false and completely true. In quantum mechanics one can always express a state-vector as a decomposition on an orthonormal basis. To make a correspondence with fuzzy characteristics we will consider not only eigenstates as input states but in general linear combinations of eigenstates, as shown in (13).

|x f = α 00 |00 + α 01 |01 + α 10 |10 + α 11 |11 (13)
The coefficients of the development can be considered as the weight of a particular logical state. The square module of each coefficient corresponds to the probability of being in that state and the sum of these probabilities will add up to one bacause of the othonormalization of the vector states. In our case, the compound fuzzy input state |x f is the Kronecker tensor product of the vectors |x L and |x L as defined in eq. ( 1). So we can write the state vectors:

|x L = 1 -p L |0 + √ p L |1 = √ 1 -p L √ p L (14) |x R = 1 -p R |0 + √ p R |1 = √ 1 -p R √ p R ( 15
)
where the coefficients are function of the probabilities

p L = | x L |1 | 2 and p R = | x R |1 | 2 .
The input state of the compound fuzzy system is then written as:

|x f = |x L x R =     (1 -p L )(1 -p R ) (1 -p L )p R p L (1 -p R ) p L p R     1 2
(16)

BV quantum robot simulation results

We performed the simulation with different configurations of logical observables in the vehicle's computational unit. In this paper we illustrate only a small subset of the rich variety of possible combinations of observables. The graphical simulation interface allows the user to easily select the desired control operator corresponding to each wheel, as well as to add light sources and new vehicles in any time slot and location on the canvas ot the running simulation.

Simulation environment

As a result of the principle of the"Law of uphill analysis and downhill invention" [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF], the behavior of these vehicles can become very complex. It is thus difficult to truly understand the response of these vehicles by the means of only thought experiments. Consequently, the natural method to analyze these vehicles is through computer simulations. The environment stimuli are light sources placed at different positions on the canvas. We opt for non punctual sources, where the intensity of each source is 1.0 inside the circle that delimits its border and decreases with the square of the distance to the border. The sources are considered as incoherent since they don't interact with each other, the total intensity at a given position is the sum of the intensities. Every vehicle has two sensors, one attached to its left and the other to its right side. The distance between both sensors is sufficient for the vehicle to distinguish the light intensities, in our model proportional to p L and p R , in each of the corners. Different intensities will result in a change of the angle of the vehicle's speed vector. To each considered behavior corresponds a characteristic truth table given by the logical operator mean values on the eigenstates.

We will now describe the basic behaviors associated to BVs.

Illustration of different emotional behaviors

Fear: F L = F A F R = F B In this configuration, the quantum control gates simply connect the sensor readings to the wheel on the same side. Here, we see that the value of µ R (resp. µ L ) corresponds to p R (resp. p L ), because the logical control connective corresponds to the logical input. In Fig. 3, we clearly see the vehicle trying to avoid the stimulus in the case where the source is placed at a skew angle. In the case where the source is directly in front of the vehicle, the vehicle moves towards this source. The vehicle then comes to a rest in an area with almost no stimulus. This is the expected behavior for a BV possessing the emotion Fear.

Aggression: F L = F B F R = F A This configuration is similar to Fear with the difference that the quantum control gates connect the sensor output to the motor of the wheel on the opposite side. In Fig. 3, we observe a vehicle trying to collide with the source, regardless of the position of the source with respect to the vehicle. Also in this case the vehicle comes to a rest in an area with almost no stimulus. This is the expected behavior for BV possessing the emotion Aggression.

Passion F L = F B =⇒ A F R = F A =⇒ B
According to simulations, as illustrated in Fig. 3, this vehicle remains at rest in the absence of stimulus. When it detects a light source, it moves towards it with increasing speed. The vehicle eventually hits the source and then comes to a rest. Observing the behavior of this vehicle, we compared this BV to the emotion Love [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF]: the vehicle goes towards the light source and stops when it reaches the source. But in the present case it also shows aggressiveness because the speed increases as it gets closer to the source. Hence, we associate this BV emotion with Passion.

Boredom: F L = F A =⇒ B F R = F B =⇒ A
In a thought experiment, one can expect that switching (L ↔ R) the Eigenlogic operators of the Passion vehicle should lead to a result similar to that performed by the Explore vehicle described by Braitenberg [START_REF] Braitenberg | Vehicles -Experiments in Synthetic Psychology[END_REF], but with its own peculiarities. As one can see in Fig. 3, the vehicle remains at rest in the absence of light. In the presence of light it accelerates for a short period of time but then abruptly decelerates and comes to a stop before it actually hits the source. In the case another stronger source suddenly appears, the agent moves towards this source and then occupies a position at half-distance from both sources and facing away from the sources.

Here, we interpret this BV emotion with Boredom.

Worship: F L = F HH F R = F B Here, the control logical operator is the normalized projector version of the double-Hadamard quantum gate H ⊗ H:

F HH = 1 2 (I -H ⊗ H) (17)
The inputs of the vehicle can be represented by the truth-table given in Table 1 proportional to the respective fuzzy measures on the control logical operators The vehicle keeps rotating around its own center in the absence of light. In the presence of light, it goes towards the source and starts to rotate around the source (or multiple sources when they are close together). This behavior can be seen in Fig. 3. We associate this vehicle emotion with Worship since the vehicles immediately start revolving around the light source once it senses it.

µ L = x| F HH |x and µ R = x| F B |x :

Doubt: F

L = F B =⇒ A F R = F XOR
The projective version of the exclusivedisjunction (XOR) self-inverse Eigenlogic operator Z ⊗ Z [START_REF] Dubois | Eigenlogic: A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] is used here:

F XOR = 1 2 (I -Z ⊗ Z) (18)
This operator provides a property that makes the vehicle turn around in circles, regardless of the presence or absence of stimuli. It is the other wheel control operator that completes the overall behavior of this vehicle. In the case considered here, we use the implication operator F B =⇒ A as control operator for the left wheel. The truth table for this vehicle is given in Table 2. This agent keeps rotating around itself in the absence of light. However, once it senses stimuli, it starts to roll towards it. Once it gets close to the source, it rotates outwards again and goes away from the source in a circular orbit. In some cases, the agent hits the source and comes to a stop (Fig. 3(6.a)). In particular cases, it starts rotating in a circle whose center itself is moving in a circle around the stimuli (Fig. 3(6.b)). We associate this vehicle with the emotional feeling of Doubt. The BV is unable to decide if it prefers light or not, and hence keeps coming to and going away from sources of light. In certain conditions, it decides to orbit around the source. In the case of sources on its both sides, it is unable to decide between the two and keeps switching from one to the other.

Quantum Wheel of Emotions

The concept of "wheel of emotions" introduced by Plutchik et al. [START_REF] Plutchik | The Nature of Emotions[END_REF] pictures the idea that a complex emotional state is the composition of elementary emotions. This picture can be interpreted in a quantum-like way using the quantum state vector |ψ . Each qubit (2-dimensional) quantum state can be mapped to a point on the surface of the Bloch unit sphere:

|ψ = cos( θ 2 ) |0 + e -i φ 2 sin( θ 2 ) |1 (19)
where φ and θ are the spherical angles. In order to simplify interpretation, the coefficients (associated with the degree of truth) multiplying the base states are taken as real numbers. The points of the vector are thus placed on a circle corresponding to a quantum wheel of emotions. We summarized our simulation results by associating the different observed behaviors to a sector in the wheel as shown in Figure The quantum wheel of emotions thus allows a continuous set of emotional states. A small perturbation in the angle of the input state |ψ due to environmental factors, even if still inside the same emotional sector, will correspond to small changes in the vehicle's behavior. The measurement of the input state implies the collapse of |ψ to a specific point of the wheel, and thus we can say that, in this aspect, the vehicle behaves as a quantum-like system. Furthermore, the fuzzy aspect of the system arouses naturally since the collapse can involve any state belonging to the continuous surface of the wheel. These observations can be compared to the observed similarities between neural network models and quantum systems. In particular, it has been suggested that it is possible to implement quantum learning algorithms dedicated to fuzzy qubits [START_REF] Hannachi | Emulating Quantum Interference and Quantum Associative Memory Using Fuzzy Qubits[END_REF] where the weighted sums of inputs of a neuron correspond to the superposition of quantum states at the input of a quantum circuit and the quantum function collapse corresponds to the threshold activation of a neuron.

Discussion and Conclusion

The purpose of this research is to show the multiplicity of behaviors obtained by using fuzzy logic along with quantum logical gates in the control of simple Braitenberg Vehicle agents. The number of cases becomes intractable in simple theoretical approaches with increasing complexity. A computer simulation is mandatory and allows us to abstract the complexity by observing the motion of the vehicles and use it for illustrative purposes. At the same time, we see that by changing and combining different quantum control gates we can tune small changes in the vehicle's behavior, and hence get specific features around the main basic BV emotions of Fear, Aggression, Love and Explore. By tweaking these quantum gates, one can also obtain a vehicle that has a mixture of multiple emotions.

Further extensions to this project can be imagined. Currently, when the vehicles collide, their respective control operators could change in order to reflect a quantum-like entanglement behavior due to interaction. It would be interesting to entangle the vehicles so that the behavior of one vehicle depends upon the current state of the environment from the perspective of other vehicles even after they separate after the collision. It could also be interesting to explore the Braitenberg vehicles using different types of stimuli (instead of only light) and sensors. Also a formalization of the quantum BV components as quantum neural networks could lead to new investigation strategies and could benefit researches in machine learning algorithms related to emotion analysis.

  Fig.2).

Fig. 1 .

 1 Fig. 1. Simplified diagram of the control circuit of a BV connected to the left and right input sensors SL, SR and to the output left and right wheel motors ML, MR.

Fig. 2 .

 2 Fig. 2. Braitenebrg Vehicle orientation at speed v showing at the rear the left and right wheels WL, WR and at the front the light sensors SL, SR spaced at distance d.

Fig. 3 .

 3 Fig. 3. Simulation screen-shots for (1)Fear, (2)Aggression, (3)Passion, (4)Boredom, (5)Worship, (6)Doubt

Fig. 4 .

 4 Other emotions not presented in Fig. Fig. 3 have been simulated such as for example: Interest, Curiosity, Distraction, Fear, Worship and Sadness. The two latter ones have been obtained using a circuit that combines the standard quantum gates H and C NOT .

Fig. 4 .

 4 Fig. 4. Quantum Wheel of Emotions. For a given emotion, we associate quantum operators controlling the speed of the left and the right wheel respectively. (Anger) FB , FA ; (Passion) F B =⇒ A , F A =⇒ B ; (Love) F Ā, F B ; (Interest) F B , F Ā ; (Curiosity) F A =⇒ B , F B =⇒ A ; (Distraction)F A =⇒ B , FA XOR B ; (Apprehension) F B =⇒ A , FA XOR B ; (Worship) H ⊗ H, FB ; (Sadness) C NOT , C NOT ; (Fear) FA, FB .

Table 1 .

 1 Worship emotion truth table

	|x f	µL µR	Behavior
	|00 0.25	0 turns to the right slowly
	|01 0.75	1	turns to the left slowly
	|10 0.75	0	turns to the right
	|11 0.25	1	turns to the left

Table 2 .

 2 Doubt emotion truth table

	|x f	µL µR	Behavior
	|00	0	0	no movement
	|01	1	1	goes straight
	|10	0	1 turns to the left
	|11	0	0	no movement

Credits and Acknowledgements

The first coauthor is under the Brafitec scholarship, CAPES Foundation, Ministry of Education of Brazil, Brasilia 70.040-020, Brazil.

We want to thank the AFSCET (Association Française de Science des Systèmes) for permitting us to present the idea of this work at the WOSC (World Organisation of Systems and Cybernetics) 2017 Congress in Rome Italy.

We are grateful to Francesco Galofaro of LUB for fruitful discussions on logic and semantics and their link to quantum theory and for pointing out that the late Professor Valentin von Braitenberg was one of the academic founding members of LUB (Libera Università di Bolzano/Freie Universität Bozen, Italy).


Li Liu

Member, IEEE Jiasong Wu
email: jswu@seu.edu.cn

Dengwang Li
email: deng-wang@sdnu.edu.cn

Lotfi Senhadji
email: lotfi.senhadji@univ-rennes1.fr

Senior Member, IEEE Huazhong Shu

Fractional Wavelet Scattering Network and Applications

Keywords: Fractional Wavelet Scattering Network and Applications fractional wavelet transform (FRWT), scattering classification, histopathology image, gland segmentation

I. INTRODUCTION

I Nmost cases, when dealing with images, the conveyed information varies considerably over time and space. Extracting robust features and measuring similarities in order to get an effective image analysis is not straightforward [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. The presence of noise, deformations and non-stationary behavior make it difficult to segment [START_REF] Chen | Global minimum for a finsler elastica minimal path approach[END_REF]- [START_REF] Chen | Global minimum for a variant mumford-shah model with application to medical image segmentation[END_REF] or to classify [START_REF] Deng | A hierarchical fused fuzzy deep neural network for data classification[END_REF] signals.

In image processing, textural pattern, as one of robust features, is usually used to describe surface object properties and their relationships to the surrounding environment [START_REF] Haralick | Textural features for image classification[END_REF]- [START_REF] Ren | Enhanced local gradient order features and discriminant analysis for face recognition[END_REF].

The Wavelet Transform (WT) is one of the most widely used tools for transient and non-stationary signals and image texture analysis. It is able to simultaneously describe signals in both time and frequency domains although some limitations have been pointed out [START_REF] Gargour | A short introduction to wavelets and their applications[END_REF]. To overcome these difficulties, fractional Fourier transform (FRFT) has been proposed, generalizing the Fourier transform (FT) and extending the timefrequency plane to the time-fractional-frequency plane [START_REF] Ozaktas | Fractional fourier domains[END_REF]. However, the fractional Fourier representation fails in locating the occurrence of the FRFT spectral content at a particular time due to its global kernel [START_REF] Tao | Short-time fractional fourier transform and its applications[END_REF], which is indispensable when analyzing non-stationary signals. A modification, called short time FRFT [START_REF] Stanković | Time-frequency signal analysis based on the windowed fractional fourier transform[END_REF], is developed to try to overcome this time location problem. But the technique is limited by the fundamental uncertainty principle in its application [START_REF] Shinde | An uncertainty principle for real signals in the fractional fourier transform domain[END_REF]. Like the FRFT, the fractional WT (FRWT) derived in [START_REF] Mendlovic | Fractional wavelet transform[END_REF] is aimed at representing the fractional spectrum. Constructed on the FRFT, it also fails in obtaining the local information of the signal. Many other FRWTs are also developed to jointly display time and FRFD (fractional Fourier domain)frequency information of a signal [START_REF] Shi | Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform[END_REF]. More general FRWT methods are proposed based on fractional convolution [START_REF]A novel fractional wavelet transform and its applications[END_REF], [START_REF] Dai | A new fractional wavelet transform[END_REF]. The FRWT reported in [START_REF]A novel fractional wavelet transform and its applications[END_REF] is easy to implement and has low computational complexity. The FRWT described in [START_REF] Dai | A new fractional wavelet transform[END_REF] depicts more mathematical properties. In this study, we choose the FRWT presented in [START_REF]A novel fractional wavelet transform and its applications[END_REF] to construct the fractional wavelet scattering network (FrScatNet).

For classification purpose, a feature vector must be set. Wavelet coefficients are often used. Because WTs are not invariant to translation, a scattering operator that is invariant to translation and rotation is built from the wavelet coefficients [START_REF] Mallat | Group invariant scattering[END_REF]. The translation invariant representation is computed by cascading WTs and modulus pooling operators, which averages the amplitude of iterated wavelet coefficients [START_REF] Bruna | Invariant scattering convolution networks[END_REF], [START_REF] Mallat | Group invariant scattering[END_REF], [START_REF]Recursive interferometric representation[END_REF]. Due to the WT limitations in the time-frequency domain, if the energy concentration of a signal is not optimal in the frequency domain, the wavelet coefficients are not the best representation of its energy distribution. So, the detected scattering coefficients are not the most discriminative features for signal classification. In this study, the fractional scattering coefficients are computed by cascading FRWT and averaging the amplitude of iterated fractional wavelet coefficients in order to address this issue. The obtained FrScatNet generalizes the classical wavelet scattering network (ScatNet) from the scattering domain to the fractional scattering domain. The added fractional order parameter is then able to analyze the signal in the fractional scattering domain. We test the efficiency of the FrScatNet on two-dimensional images in this study. Besides, we also use it to achieve the gland segmentation in colon histology images of tissue slides stained with Hematoxylin and Eosin (H&E).

Let us briefly summarize the state-of-the-art in this very challenging research area. Histopathological biopsy evaluation is considered as the gold standard for colon, prostate, and breast cancer diagnosis, malignancy confirmation and grading [START_REF] Andrion | Malignant mesothelioma of the pleura: interobserver variability[END_REF], [START_REF] Reis | Automated classification of breast cancer stroma maturity from histological images[END_REF]. Colorectal adenocarcinoma is the most common form of colon cancer. It is of fundamental importance to achieve good intra-observer and inter-observer reproducibility in the cancer grading. An automated method with the capability to detect the morphological information can be used as computer assistant for cancer grading. There are plenty of difficulties existing in quantifying the morphology of gland automatically. The structure, morphology, size and location of a gland vary significantly. Especially, cancer progresses may cause changes in the component organization, and also lead to tissue degeneration. In addition, some glands are even touching to each other leading to the coalescence problem [START_REF] Xu | Gland instance segmentation using deep multichannel neural networks[END_REF]. Various approaches have been proposed to achieve gland segmentation, such as texture based methods [START_REF] Farjam | An image analysis approach for automatic malignancy determination of prostate pathological images[END_REF], [START_REF] Tosun | Graph run-length matrices for histopathological image segmentation[END_REF] ,and structure based methods [START_REF] Gunduz-Demir | Automatic segmentation of colon glands using object-graphs[END_REF], [START_REF] Sirinukunwattana | A stochastic polygons model for glandular structures in colon histology images[END_REF]. The drawbacks of all these methods are that they solely use pixel-level color information while assuming a regular architecture of glandular structures. Recently, convolutional neural networks have been considered. Kainz et al [START_REF] Kainz | Semantic segmentation of colon glands with deep convolutional neural networks and total variation segmentation[END_REF] combined deep convolutional neural network and total variation. Li et al [START_REF] Li | Gland segmentation in colon histology images using hand-crafted features and convolutional neural networks[END_REF] used handcrafted features and convolutional neural networks to recognize glands. Xu et al [START_REF] Xu | Gland instance segmentation using deep multichannel neural networks[END_REF] presented a multichannel learning approach to extract region, boundary and location cues. In the MICCAI 2015 gland segmentation challenge contest [START_REF] Sirinukunwattana | A stochastic polygons model for glandular structures in colon histology images[END_REF], the convolutional neural networks were used and achieved impressive performance. Chen et al [START_REF] Chen | Dcan: Deep contour-aware networks for accurate gland segmentation[END_REF] presented a novel deep contour-aware network. Sirinukunwattana et al [START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF] proposed a multipath convolutional neural network segmentation algorithm. Ronneberger et al [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] applied a u-shaped deep convolutional network. Despite the good results obtained by these neural networks, the involved cascaded nonlinearities make their properties and optimal configurations not clear. The scattering networks address these questions from a mathematical and algorithmic perspective by concentrating on a particular class of deep convolutional networks [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. Because the scattering network is learning-free due to the fixed wavelet basis, it can be implemented easily with less parameters and hardware resource.

In this paper, we propose a generalized FrScatNet based on the FRWT and use it to classify signals and solve the challenging problem in gland segmentation. Section II describes the data sets onto which our experiments are conducted and our proposed fractional scattering network. Section III illustrates the results achieved so far and compares our approach to other reference methods. A discussion is also provided before concluding (Section IV).

II. METHODS

The used public data sets are firstly sketched and then a full presentation of the FrScatNet is given.

A. Data Sets

The medical dataset Warwick-QU is provided by the Colon Histology Images Challenge Contest held at MICCAI2015 [START_REF] Sirinukunwattana | A stochastic polygons model for glandular structures in colon histology images[END_REF], [START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF]. The dataset consists of 165 images derived from 16 H&E stained colon histological slides of stage T3 or T42 colorectal adenocarcinoma. The slides are digitally scanned at 20 magnifications by Zeiss MIRAX MIDI Slide Scanner with a pixel resolution of 0.465m. The whole-slide images are subsequently rescaled to a pixel resolution of 0.620m. Each section belongs to a different patient. The dataset is grouped into 85 training images, 60 testing images (test A) and 20 testing images (test B). The annotations are regarded as ground truth. The dataset exhibits high inter-subject variability in both stain distribution and tissue architecture.

The public texture database created in [START_REF] Lazebnik | A sparse texture representation using local affine regions[END_REF] includes surfaces with significant viewpoint changes and scale differences within each class, whose texture is due mainly to albedo variations, 3D shapes, and a mixture of both. The database is also chosen as the benchmark for texture recognition. It contains 25 texture classes and each class has 40 examples. The resolution of the samples is 640 × 480 pixels.

B. The Proposed Fractional Scattering Network

In this section, the formulation of the FrScatNet is described. We start by introducing the definition of the FRWT. Then, a FRWT-based scattering network is proposed. In the following, the convolution network structure is described.

1) FRWT Definition: According to the definition in [START_REF] Tao | Short-time fractional fourier transform and its applications[END_REF], [START_REF] Sejdić | Fractional fourier transform as a signal processing tool: An overview of recent developments[END_REF], the FRFT can provide signal representation in the fractional Fourier domain, but it fails in obtaining the local structures of the signal. FRWT is able to handle the problem and display signal information in the fractional wavelet domain.

For the continuous-time finite energy signal x(t) (i.e. signal belonging to L 2 (R), the WT is defined as follows:

W x (a, b) = 1 √ a R x(t) ψ t -b a (1)
where ψ(t) is the mother wavelet. The ψ denotes complex conjugate of ψ. The compressed or dilated wavelet along the t domain. The different scale WT intrinsically behaves as frequency band-pass filters, which describes the signal in the time-frequency plane. FRWT has been proposed as an extension of WT to analyze the time-varying FRFT spectra [START_REF] Shi | Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform[END_REF], [START_REF] Dai | A new fractional wavelet transform[END_REF]. Different definitions of fractional convolution have been proposed in the literature. According to the definition introduced in [START_REF] Shi | Multichannel sampling and reconstruction of bandlimited signals in fractional fourier domain[END_REF], the fractional convolution of two signals x(t) and h(t) ∈ L 2 (R) is expressed as follows:

xΘ α y = e -j 2 t 2 cot(θ) [x(t)e j 2 t 2 cot(θ) * h(t)] (2)
where Θ α denotes the fractional convolution operator. The parameter α is the fractional order and θ = απ/2 represents the rotation angle.

A kind of FRWT with simple structure and easy implementation is proposed in [START_REF] Shi | Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform[END_REF], [START_REF]A novel fractional wavelet transform and its applications[END_REF]. It is constructed based on the fractional convolution according to the relationship between WT and the classical convolution shown in Eq. [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. For a given signal x(t) ∈ L 2 (R), the α order FRWT is:

W α x (a, b) = x(t)Θ α 1 √ a ψ(- t a ) = e -j 2 b 2 cot(θ) R (e j 2 t 2 cot(θ) ) ψa,b (t)dt (3)
where the kernel function is defined by multiplying the classical wavelet ψ a,b (t)with the following chirp signal:

ψ α,a,b (t) = ψ a,b (t)e -j 2 (t 2 -b 2 ) cot(θ) (4)
We only consider the parameter 0 < θ < π, and it is easy to extend outside the interval (0, π). When α = 1, the FRWT defined in Eq. ( 3) reduces to conventional WT. The FRWT on different scales also behaves as a set of band-pass filters.

2) Fractional Scattering Wavelets: In this step, a scattering transform computes nonlinear invariants from fractional wavelet coefficients by modulus operator chosen as a nonlinear pooling operator since it has the capability to preserve the signal energy. For the complex signal x(t) = x r (t) + jx i (t), x r (t), x i (t) ∈ L 2 (R), the modulus operator is defined as |x(t)| = (|x r (t)| 2 + |x i (t)| 2 ) 1/2 . The wavelet-modulus coefficients, termed as the translation invariant coefficients, are built from the fractional wavelet by the modulus operator defined as:

U [λ]x = |xΘ α ψ λ | (5)
More fractional wavelet-modulus coefficients can be obtained by further iterating on the FRWTs and the modulus operator along any path. The scattering propagator U [p] for a given signal x(t) ∈ L 2 (R) is defined by cascading fractional wavelet-modulus operators:

U [p]x = U [λ m ] . . . U [λ 2 ]U [λ 1 ]x = | xΘ α ψ λ1 |Θ α ψ λ2 | . . . Θ α ψ λm |, (6)
where U [∅]x = x and ∅ denotes an empty set. A path is defined by the sequence p = (λ 1 , λ 2 , . . . , λ m ) with a length of m.

For classification purpose, the local descriptors are computed by defining a windowed fractional scattering transform with a scaled spatial window φ 2 J :

S[p]x = U [p]xΘ α φ 2 J = | xΘ α ψ λ1 |Θ α ψ λ2 | . . . Θ α ψ λm |Θ α φ 2 J (7)
where S[∅]x = xΘ α φ 2 J . The fractional scattering operator S performs a spatial averaging on a domain whose width is proportional to 2 J . The resulting windowed scattering is nearly invariant to a translation. 3) Fractional Scattering Network: For the path p = (λ 1 , λ 2 , . . . , λ m ), a convolution network is constructed by iterating the fractional scattering propagator W , which is built based on the fractional scattering transform to compute the fractional complex wavelet coefficient modulus and filter the lower frequency. In this section a convolution network with three layers is constructed by iterating the fractional scattering propagator W (Fig. 2). We generalize the classical WT to FRWT, so the fractional wavelet coefficients are obtained from different fractional wavelet domain. The built The scattering transform is non-expensive and preserves the signal norm by given conditions like in work [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. According to the Proposition 2.1 in [START_REF] Mallat | Group invariant scattering[END_REF], if the input signal f is complex and the following rule is obeyed:

∀J ∈ Z, | φ(2 J ω)| 2 + j<J,r∈G | ψ(2 j rω)| 2 = 1 (8)
Then W f 2 = f 2 is obtained by using Plancherel formula. The similar condition can be given. If there exists ε > 0, such that for ω ∈ R 2 ,

1 -ε ≤ | φ(ω)| 2 + ∞ j=0 r∈G | ψ(2 j rω)| 2 ≤ 1 (9)
Applying the Plancherel formula, we can deduce that if f is complex, then

W f satisfies (1-ε) f 2 ≤ W f 2 ≤ f 2 with W f 2 = f * φ 2 2 + r∈G f * ψ λ 2 [18]
. According to the fractional wavelet transform in Eq. ( 2). The norm of W x(u) =

xΘ α φ 2 J (u), xΘ α ψ λ (u), (P = {λ = 2 -j r : r ∈ G, j ≤ J}) satisfies W 2 = W f 2 with f = x(u)e jt 2 cot(θ)/2 . f is a complex function and f 2 = x 2 . So we can get that (1 -ε) x 2 ≤ W 2 ≤ x 2 .
We suppose that ε ≤ 1 and the fractional wavelet transform W is non-expansive. If ε = 0, then W is unitary and it is able to contain the norm of x. In this work, we use two kinds of wavelet transform to construct FrScatNet. For the Morlet wavelet, the value of parameter σ in φ is set as 0.7. The value of σ in ψ is 0.5. And Eq. ( 9) is satisfied with ε = 0.98. φ is the low pass filter and ψ is the wavelet based on Gabor filter which is widely used in image processing [START_REF] Zhang | Histogram of gabor phase patterns (hgpp): A novel object representation approach for face recognition[END_REF], [START_REF] Ren | Band-reweighed gabor kernel embedding for face image representation and recognition[END_REF]. For the dual-tree complex wavelet, we let φ = (1/(2 √ 2)φ 1 and ψ = (1/8)ψ 1 satisfying Eq. ( 9) with ε = 0.99. φ 1 and ψ 1 are defined in work [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF], [START_REF] Selesnick | The design of approximate hilbert transform pairs of wavelet bases[END_REF]. The fractional scattering propagator W x(u) = xΘ α φ 2 J (u), |xΘ α ψ λ (u)| λ∈P is obtained with a fractional wavelet transform W followed by a modulus, which are both non-expansive. So W is also non-expansive. Since S iteratively applies W , it is also non-expansive. If W is unitary, then W also preserves the signal norm W x 2 = x 2 . The FrScatNet is built layer by layer by iterating on fractional scattering propagator W . If W preserves the signal norm, then the signal energy is equal to the sum of the fractional scattering energy of each layer plus the energy of the last propagated layer [START_REF] Bruna | Invariant scattering convolution networks[END_REF], [START_REF] Mallat | Group invariant scattering[END_REF]:

x 2 = m m=0 p∈P m S[p]x 2 + p∈P m+1 U [p] 2 (10)
By letting the network depth m tend to infinity, it results that the FrScatNet preserves the signal energy [START_REF] Bruna | Invariant scattering convolution networks[END_REF], [START_REF] Mallat | Group invariant scattering[END_REF]: The output of the FrScatNet is obtained by cascading the fractional scattering coefficients of every layer. The fractional windowed scattering S is computed with a cascade of wavelet modulus operators, thus its properties depend upon the FRWT properties. The fractional scattering process is implemented along the frequency decreasing paths where most of the fractional scattering energy is concentrated. A frequency decreasing path p = (2 -j1 r 1 , . . . , 2 -jm r m ) satisfies 0 ≤ j k ≤ j k+1 ≤ J. The fractional scattering coefficients are the lower frequencies filtered by the low pass filter, due to the energy of the signal mainly distributed on the lower frequency region. The signal analysis capability of the classical WT is limited in the time-frequency plane. If the distribution of the signal energy is not most concentrated on the frequency domain, the feature vector extracted by the network is not the best for the signal discrimination. Therefore, we extend the classical WT to the FRWT, which is able to offer signal energy representations in the time-fractional-frequency plane, and try to find the appropriate fractional order for the signal representation.

x 2 = p∈P m S[p]x 2 = Sx 2 (11)
Image textures can be modeled as realization of stationary processes. The spectrum method is an important tool for texture analysis. The power spectrum only depending on the second-order moments is then not sufficient to discriminate image textures whose second-order moments are very similar. The scattering representation of the stationary processes depends upon second-order and normalized higher order moments. Thus, it can discriminate textures having the same second-order moments but different higher order moments [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. In the proposed method, the fractional windowed scattering is computed based on the fractional wavelet transform, which can represent the image energy distribution in the fractional wavelet scattering domain. Therefore, the scattering implementation can be achieved through the path where most of the signal energy is concentrated in the fractional wavelet scattering domain, and the discrimination capability of the first order fractional scattering coefficients is improved. We can see that the first-order scattering coefficients depicted in Fig. 1(b), estimated from each realization, are nearly the same in the ScatNet while their second-order scattering coefficients shown in Fig. 1(c) are different. However, in the FrScatNet both the first-and second-order fractional scattering coefficients displayed in Fig. 1(d,e) are clearly different and have the discriminatory ability.

C. Classification

A scattering transform eliminates the image variability due to translations and linearizes small deformations [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. Classification is then carried out with a Gaussian kernel SVM and a generative PCA classifier, respectively. The overall framework of the FrScatNet is illustrated in Fig. 2.

Firstly, a three-layer FrScatNet is constructed based on fractional wavelet transform. Secondly, all the normalized data are put into the FrScatNet. The characteristic matrix Q, of size of L × N × D, is obtained by concatenating the fractional scattering coefficients of all three layers. L is the length of the feature vector detected from each input signal. N denotes the number of the total input signals. D represents the number of the fractional orders. In the next step, the input data are randomly divided into training and testing sets. Therefore, the corresponding characteristic matrix Q is also divided into the training characteristic matrix Q T rain , the testing characteristic matrix Q T est , and termed as

Q = [Q T rain Q T est ].

III. EXPERIMENTS AND RESULTS

We test here the classification performance of the FrScatNet on two-dimensional signals and analyze its properties. We also propose an algorithm to achieve gland segmentation from the histology images.

A. Image Classification, Application and Discussion

In the proposed framework, we generalize the Morlet wavelet and dual-tree complex wavelet to fractional wavelets according to Eq. ( 3). These two corresponding fractional wavelets are used to construct the FrScatNet and compute the fractional scattering representations, respectively. So two sets of characteristic matrix are created to test the performance of the proposed framework and analyze the signal energy distributions. The FrScatNet constructed by the fractional complex Morlet wavelets identified as FrameworkI is initialized with the finest scale 2 J = 16 and the total number of angles K = 8. The band-pass filters are constructed using the Gabor wavelets and the low frequency band is a Gaussian. The 2D fractional dual-tree complex wavelet based scattering network denoted as FrameworkII is initialized with five stages and six distinct orientations. In each direction, one of the two wavelets can be interpreted as the real part of a 2D complexvalued wavelet, the other as its imaginary part [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF], [START_REF] Selesnick | The design of approximate hilbert transform pairs of wavelet bases[END_REF]. The fractional orders for both frameworks are the same. For the two-dimensional wavelet, two fractional orders α 1 and α 2 are needed to determine the rotational angle. The angle is defined as θ = απ/2 ranging from 0 to π, so the fractional order α changes from 0 to 2. To save computation time, we fix one order as 1.00 and the other one changes within the range zero to two for computing the fractional scattering coefficients. The chosen values are 0.05, 0.10, 0.40, 0.70, 1.00, 1.30, 1.60, 1.90, and 1.95, respectively. The corresponding FRWT reduces to conventional WT when α 1 = α 2 = 1.00. Their performance is evaluated by comparing the classifier outputs.

The feature matrix obtained from FrameworkI is then put into two different kinds of classifiers named as PCA and SVM to achieve the classification, respectively. So does the feature matrix from FrameworkII. The dimensions of principal component are chosen as [START_REF] Ozaktas | Fractional fourier domains[END_REF][START_REF] Shi | Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform[END_REF][START_REF] Andrion | Malignant mesothelioma of the pleura: interobserver variability[END_REF][START_REF] Gunduz-Demir | Automatic segmentation of colon glands using object-graphs[END_REF][START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF][START_REF] Zhang | Histogram of gabor phase patterns (hgpp): A novel object representation approach for face recognition[END_REF][START_REF] Salah | Multiregion image segmentation by parametric kernel graph cuts[END_REF][START_REF] Lecun | Convolutional networks and applications in vision[END_REF]50,60,70, and 80 in the PCA classifier. For the SVM classifier, the best-performing set of parameters is identified by calculating the five-fold cross-validation error over the trainings for each combination of parameters. The parameter C is set to 2 0 , 2 4 , 2 8 and the parameter gamma is set to 2 -16 , 2 -12 , 2 -8 . The input data are divided into training and testing sets randomly. We set the ratio as 1:1 and 4:1 in our experiment. The classification results are averaged over 5 different random splits for each ratio. We construct the benign and background database (BD) from the 37 original pathological color images with benign glands. The corresponding annotated image is used as a mask to determine the benign gland area. Firstly, a k-mean based algorithm [START_REF] Achanta | Slic superpixels[END_REF] aggregates nearby pixels into super-pixels of nearly uniform size for each original images. Then, the original image with size of m × n × 3 is enlarged to obtain a new image with size of (m + 16) × (n + 16) × 3 by a mirror method. In the next, a sliding block window with size of 32 × 32 × 3 is positioned at the center of these super-pixels. If the overlapping area between the sliding window and the irregular benign area is more than 0.95, then the obtained sub-image is classified as belonging to benign database, or it is denoted as background database. In this way, we obtain a total of 7631 sub-images shown as the red rectangles from the benign areas and 12529 sub-images as the green rectangles from the background areas of 37 original pathological color images with benign gland in Fig. 3(a). The same process is also applied to each of the original images issued from the 48 original pathological color images with malignant glands to construct the malignant and background database (MD). There are 15857 sub-images illustrated as the yellow rectangles from the malignant areas and 10782 sub-images shown as the green rectangles from the background areas in Fig. 3(c). These two constructed sets are used for training.

All normalized sub-images from the databases are put into the FrScatNet. For the BD, the obtained feature matrix Q corresponding to FrameworkI and FrameworkII have respectively the size 681×20160×18 and 391×20160×18. For the MD, the obtained Q in the same situation corresponds respectively to 681×26639×18 and 391×26639×18. The classification result is shown in Table . I. For the BD, the classification error is minimum in the FrameworkI when the fractional orders are set to α 1 = 1.00 and α 2 = 0.40 by using the PCA classifier and the fractional orders α 1 = 1.60 and α 2 = 1.00 for the SVM classifier. In the FrameworkII , the minimum classification error happens when α 1 = 1.00 and α 2 = 0.40 with the PCA classifier and, with SVM, α 1 = 1.60 and α 2 = 1.00. We can see that the classification result is better in the fractional wavelet scattering domain than the wavelet scattering domain for the BD, and the performance from FrameworkI is better than that from the FrameworkII. The fractional scattering coefficients of images from the benign group (Fig. 4(a1-h1)) and the background group (Fig. 4(a2-h2)) of BD detected by the FrameworkI are shown in Fig. 4. The first-and secondorder fractional scattering coefficients (Fig. 4(a1-b1), (a2-b2), (g1-h1), (g2-h2)) from the FrScatNet are clearly different.

(a1) (a2) (a3) (a4) (a5) (a6) (a) (b1) (b2) (b3) (b4) (b5) (b6) (b) (c1) (c2) (c3) (c4) (c5) (c6) (c) (d1) (d3) (d4) (d5) (d6) (d) (e1) (e2) (e3) (e4) (e5) (e6) (e) (f1) (f2) (f3) (f4) (f5) (f6) (f) (g1) (g2) (g3) (g4) (g5) (g6) (g) (h1) (h2) (h3) (h4) (h5) (h6) (h)
For the MD, in the FrameworkI, the error is minimum when α 1 = 1.00 and α 2 = 0.40 when using PCA classifier and α 1 = 1.00 and α 2 = 1.00 (the classical wavelet transform) when applying SVM. In the FrameworkII , the best classification occurs when α 1 = 1.60 and α 2 = 1.00 with PCA and α 1 = 1.00 and α 2 = 1.60 with SVM. We get the conclusion that for most MD cases, the best classification is achieved in the fractional wavelet scattering domain. The fractional scattering coefficients of images belonging to the MD resulting from FrameworkI are also displayed in Fig. 4(a3-h3) and Fig. 4(a4-h4). The first-and second-order scattering coefficients (Fig. 4(c3-d3) and Fig. 4(c4-d4)) from the classical ScatNet are clearly different, and those fractional scattering coefficients (Fig. 4(g3-h3) and Fig. 4(g4-h4)) from the FrScatNet are also clearly different. . I. The best classification performance is obtained when α 1 = 1.00 and α 2 = 1.00, which means that the classification is better in thewavelet scattering domain than the fractional wavelet scattering domain. We display the scattering coefficients computed by the FrameworkI in (Fig. 4(a5-h5, a6-h6). By comparing the coefficients computed from the FrScatNet, it clearly appears that the first and second order scattering coefficients of ScatNet are different. Therefore, the classification performance is better in the wavelet scattering domain.

In conclusion, the classification errors are minimum when α 1 = 1.00 and α 2 = 0.40 for BD and MD, and α 1 = 1.00 and α 2 = 1.00 for TSD. This shows that the proposed FrScatNet based on different FRWT is able to represent images in the fractional wavelet scattering domain and to provide the best fractional order for classification.

B. Gland Segmentation in Colon Histology Images

We test the FrScatNet on the dataset provided in the MICCAI challenge contest [START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF] aiming at gland segmentation. The rules defining the training set and the test group are followed as specified in the challenge. . I and II, we see that the best classification is achieved when α 1 = 1.00 and α 2 = 0.40, Ratio = 0.50 for the benign images and α 1 = 1.00 and α 2 = 0.40, Ratio = 0.80 for the malignant images, which are used for the following gland segmentation.

For each of the test group images, we try to classify it into two classes: the target (benign or malignant) region and the background region. To achieve this goal, we firstly divide the original pathological color images into sub-images of the same size 32 × 32 × 3 in the training data. We apply a k-mean based algorithm [START_REF] Achanta | Slic superpixels[END_REF] to aggregate nearby pixels into superpixels of nearly uniform size. Their boundaries closely match true target boundaries. Then, the original image is divided into sub-images as described above. Each sub-image is entered into the FrScatNet to get the feature matrix in the fractional wavelet scattering domain including wavelet scattering domain (α 1 = 1.00 and α 2 = 1.00) as comparing group. The PCA classifier allows determining the class of super-pixels (the background is labeled as 1 and the target is labeled as 2 and the corresponding average approximation error for each instance and class pair. We get a fully labeled color image and the average approximation error for each class from this step.

2) Gland Segmentation: Some initially labelled images have misclassified areas within the glands and need to be corrected. In a first step, we use an erosion operator to reduce these misclassifications and one dilation step to keep the gland shape. The obtained result is regarded as the labelled image shown in Fig. 5. The gland boundaries are better delineated in this new labelled image but some areas of the central regions are lost mainly. This happens because some regions within the gland structures have similar features to the background region. But the complex gland frames are detected due to the fact that FrScatNet is mainly based on the texture information but not shape feature. The epithelial cells are lined up around a lumen to form glandular structures. Although the cancer leads to morphological and structural deviations, the texture information especially within the boundary regions is stable and effective. Besides, very close or even touching glands make also the segmentation difficult. The average error image is used to handle these problems in this study. We use the graph cut method [START_REF] Salah | Multiregion image segmentation by parametric kernel graph cuts[END_REF] to classify the average approximation error into two classes: the foreground region and the background region. The initial glands shown in Fig. 5 are used to locate the gland structures in the image. They are usually smaller than the corresponding annotated structure. A region growing algorithm, regarded as the fusion step in Fig. 5, is applied to increase the gland area in order to better fit the true gland regions and to keep touching glands separated. Then, a simple erosion operation is performed to delete isolated points followed by a dilation operation for preserving the gland shapes.

3) Evaluation: To visually evaluate the performance of the method, some segmentation results on the testing dataset are shown in Fig. 6. A comparison between FrScatNet and the ScatNet is also provided. As it can be seen in the third and the fourth rows, FrScatNet leads to a better segmentation in both benign and malignant cases from test A and test B. In the ScatNet based method, some touching glands cannot be separated while they are with FrScatNet. Besides, some overor undersegmentation problems can be observed. This is not surprising because the fractional transform makes it possible to separate different signals by successive rotations and filtering on the indicated side of the axes using some low or high-pass filter [START_REF] Bultheel | Recent developments in the theory of the fractional fourier and linear canonical transforms[END_REF]. In addition, the extra parameter obtained by rotating angles in the fractional transform gives an additional degree of freedom that can be used to optimize the performance of the network. The correlation peak within a certain class of signals can be made more prominent or sharper in an optimally chosen fractional scattering domain [START_REF] Bultheel | Recent developments in the theory of the fractional fourier and linear canonical transforms[END_REF], [START_REF] Lohmann | Synthesis of pattern recognition filters for fractional fourier processing[END_REF]. Last, the traditional wavelet transform detects too many edge details in histology image leading to unsmoothed boundaries and deviations from the ground truth while the fractional wavelet transform offers a better balance between these two issues.

However, these latter problems still remain when applying our method. Non-smoothed boundaries are mainly caused by the construction of the super-pixel regions. It is also challenging to recognize small background areas within a given target region as shown in Fig. 6(h). It happens that the windows used to detect the feature vectors contain several target regions and the graph cut method classifies those little regions as target regions.

To quantitatively assess the performance of our method, we conduct comparisons with Fully Convolutional Network (FCN) [START_REF] Long | Fully convolutional networks for semantic segmentation[END_REF], dilated FCN (DFCN) [START_REF] Chen | Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs[END_REF], Deep Multichannel Neural Networks (DMNN) [START_REF] Xu | Gland instance segmentation using deep multichannel neural networks[END_REF], and some of the solutions proposed in the MICCAI challenge such as CUMedVision1 (CUM1), CUMedVision2 (CUM2) [START_REF] Chen | Dcan: Deep contour-aware networks for accurate gland segmentation[END_REF], Frerburg1 (Fre1), Frerburg2(Fre2) [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF], ExB1, ExB2, ExB3 , CVIP Dundee (Dund) and LIB [START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF]. All of them have used the datasets provided by the MICCAI 2015 Gland Segmentation Challenge Contest. The ground truth being known, the evaluation methods proposed in this competition include the accuracy of the detection of individual glands, the volume-based accuracy of the segmentation of individual glands and the boundary-based similarity between glands and their corresponding segmentation [START_REF] Sirinukunwattana | A stochastic polygons model for glandular structures in colon histology images[END_REF], [START_REF] Sirinukunwattana | Gland segmentation in colon histology images: The glas challenge contest[END_REF]. A metric for the gland detection is the F1-score, defined by

F 1score = 2 • P • R P + R , P = T P T P + F P , R = T P T P + F N (12)
where T P is the number of true positives, F P is the number of false positives, and F N is the number of false negatives. The ground truth for each segmented glandular object is the glandular object in the manual annotation that shares maximum overlap with the segmented glandular object. A segmented object is considered as a true positive if it has more than 0.50 area overlap with its ground truth, or it is considered as a false positive. An object of the ground truth is regarded as a false negative if it has no corresponding prediction or has less than 0.50 area overlap with its predicted glandular object.

Two sets of pixels are defined as G representing the ground truth and S denoting the segmented objects. The Dice index is used for the evaluation of the segmentation on the whole image D(G, S) = 2(|G∩S|)/(|G|+|S|). An object-level Dice index can also be defined as:

D object (G, S) = 1 2 n S i=1 ω i D(G i , S i ) + n G i=1 ωi D( Ĝi , Ŝi ) , (13)
ω i = |S i |/ n S j=1 |S j |, ωi = | Ĝi |/ n G j=1 | Ĝj |. ( 14
)
where n G denotes the total number of ground truth objects in image and n S the total number of segmented objects in an image. S i is the ith segmented object in an image, and G i is the ground truth object that maximally overlaps S i . Ĝ denotes the ith ground truth object in an image, and Ŝ denotes a segmented object that maximally overlaps Ĝ in the image.

The Hausdorff distance between G and S is often used to evaluate the shape similarity defined as

H(G, S) = max{sup x∈G inf y∈S f x -y , sup y∈S inf x∈G f x -y } (15)
We can also measure the shape similarity between all segmented objects by using the object-level Hausdorff distance:

H object (G, S) = 1 2 n S i=1 ω i H(G i , S i ) + n G i=1 ωi H( Ĝi , Ŝi ) , (16)
ω i = |S i |/ n S j=1 |S j |, ωi = | Ĝi |/ n G j=1 | Ĝj |. (17)
For the overall results, the final score is the summation of all rankings (RS) from test set A and test set B based on these three criteria. Smaller final ranking stands for a better segmentation performance. Another index, the weighted rank sum (WRS), is used based on the weighted average of three evaluation criteria on the two test sets and is defined as [START_REF] Xu | Gland instance segmentation using deep multichannel neural networks[END_REF]: The final ranking score (RS) and the weighted ranking score (WRS), combining region, location and edge information, show that the FrScatNet and the DMNN methods lead to almost similar results and perform well better than all other solutions. Several observations can be made on these results. If FrScatNet is able to well detect the gland texture features, to correctly locate the glands and to separate them when they touch each other, it may have difficulties in capturing their high variabilities. The deep convolution networks usually have learning filters to take into account these unknown variabilities [START_REF] Lecun | Convolutional networks and applications in vision[END_REF]. The FrScatNet is only able to provide the first two layers of such network and eliminate translation or rotation variability. But the FrScatNet is learning-free due to the fact that the wavelet bases are fixed. The second layer fractional scattering coefficients provide important complementary information with a small computational and memory cost. FrScatNet achieves a better performance than ScatNet, the fractional scattering coefficients being more stable and discriminative.

W eightedRS =
The performance of FrScatNet obtained on test B is encouraging because this set contains more cancerous glands with more complicated shapes and size variabilities. For malignant cases, cancer progresses may cause changes in the component organization, and also lead to tissue deviations from their normal appearances. Our method mainly utilizes texture information to detect the margin of glands and to overcome the unclear boundary problem. The DMNN method and also the CUMedVision1 and CUMedVision2 approaches integrate edge information and location context to improve the results on both test sets.

IV. CONCLUSION

In this paper, the FrScatNet has been proposed and implemented. It extends the traditional ScatNet to the fractional scattering domain and provides signal representation in the time-fractional-frequency plan to improve the signal classification and segmentation performance. An automated method for gland segmentation in histological images was also proposed based on the FrScatNet. The graph cut method was used to process the average approximation error image and to locate the glands. We compared the proposed method with those reported in the MICCAI 2015 Gland Segmentation Challenge. Experimental analysis showed stable and comparable results. Significantly better results for malignant objects were obtained. They could be further improved by integrating edge information. The proposed approach is flexible and generic enough to be considered for organ/tissue segmentation in multimodal medical image.

Fig. 1 .

 1 Fig. 1. (a) Realizations of two stationary processes X(µ). T op: Brodatz texture. Bottom: Gaussian process. (b) First-order scattering coefficients S[λ 1 ]x from the ScatNet [1] are nearly the same (α 1 = 1.00, α 2 = 1.00). (c) Second-order scattering coefficients S[λ 1 , λ 2 ]x from the ScatNet [1] are clearly different (α 1 = 1.00, α 2 = 1.00). (d) First-order fractional scattering coefficients S[λ 1 ]x from the FrScatNet are clearly different (α 1 = 1.00, α 2 = 0.70).(e) Second-order fractional scattering coefficients S[λ 1 , λ 2 ]x from FrScatNet are also clearly different (α 1 = 1.00, α 2 = 0.70).

Fig. 2 .

 2 Fig. 2. An illustration of classification application of the FrScatNet.

  FrScatNet has the ability to analyze the image in fractional scattering domain. The fractional scattering propagator W is applied to the input signal x to compute the first layer of fractional wavelet coefficient modulus U [λ 1 ]x and output its local average S[∅]x. Applying W to all propagated signals U [p]x of the mth layer outputs fractional scattering signals S[p]x and computes all propagated signals on the next layer.

  Fig.3. Sub-images obtained by the proposed method. Green rectangles represent the sub-images from the background regions. Red rectangles denote the sub-images from the benign areas. Yellow rectangles denote the sub-images from the malignant areas.

1 )

 1 Warwick-QU Dataset: The training set Warwick-QU includes 85 original pathological color images and their corresponding annotated binary images. The annotation is shown in Fig. 3(b) and 3(d). The training set is combined by 37 color images with gland type labelled as Benign and 48 color images 1 Database 2 F ramework 3 Classif ier 4 Ratio with gland type identified as Malignant. Two sets of databases are generated from the training set to test the properties of the proposed framework.

Fig. 4 .

 4 Fig. 4. Fractional scattering coefficients of two-dimensional images from the FrameworkI in different fractional orders. (a1-h1) are of a random benign image from the DB. (a2-h2) are of a random background image from the DB. (a3-h3) are from a random malignant image from the MB. (a4-h4) are from a random background image from the MB. (a5-h5) are computed from a random image in T01 bark1 of TSD. (a6-h6) are computed from a random image in T08 granite of TSD; (a) are first-order fractional scattering coefficients when α 1 = 1.00, α 2 = 0.40. (b) are second-order fractional scattering coefficients when α 1 = 1.00, α 2 = 0.40. (c) are first-order scattering coefficients when α 1 = 1.00, α 2 = 1.00. (d) are second-order scattering coefficients when α 1 = 1.00, α 2 = 1.00. (e) are first-order fractional scattering coefficients when α 1 = 1.00, α 2 = 1.60. (f) are second-order fractional scattering coefficients when α 1 = 1.00, α 2 = 1.60. (g) are first-order fractional scattering coefficients when α 1 = 1.60, α 2 = 1.00. (h) are second-order fractional scattering coefficients when α 1 = 1.60, α 2 = 1.00.

Fig. 5 .

 5 Fig. 5. The framework structure of the proposed gland segmentation algorithm.

Fig. 6 .

 6 Fig. 6. Segmentation results of test cases (from left to right): benign, benign, malignant, malignant, benign, benign, malignant, malignant cases. (from top to bottom): original images, ground truth, segmentation results by FrScatNet, and segmentation results by ScatNet (different colors individual gland objects).

TABLE I CLASSIFICATION

 I ERRORS IN TERMS OF FRSCATNET IN TWO-DIMENSIONAL DATABASES.

	D t	1	Fw 2	C f	3	R t	4	0.05	0.10	0.40	Fractional order (α 1 = 1, α 2 ) 0.70 1.00 1.30	1.60	1.90	1.95	0.05	0.10	0.40	Fractional order (α 2 = 1, α 1 ) 0.70 1.00 1.30	1.60	1.90	1.95
	BD	I II	PCA SVM PCA SVM	0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8	0.218 0.241 0.213 0.239 0.243 0.251 0.227 0.228	0.241 0.274 0.257 0.276 0.274 0.303 0.263 0.276	0.108 0.113 0.156 0.206 0.129 0.147 0.216 0.219	0.187 0.192 0.200 0.219 0.237 0.228 0.228 0.228	0.204 0.220 0.260 0.292 0.303 0.345 0.302 0.312	0.196 0.217 0.202 0.229 0.234 0.243 0.230 0.232	0.185 0.203 0.200 0.223 0.227 0.241 0.185 0.218	0.231 0.254 0.249 0.295 0.296 0.295 0.267 0.270	0.226 0.248 0.199 0.218 0.242 0.251 0.233 0.233	0.235 0.264 0.228 0.269 0.284 0.292 0.257 0.264	0.264 0.289 0.251 0.285 0.282 0.293 0.272 0.257	0.197 0.210 0.243 0.267 0.277 0.287 0.244 0.254	0.117 0.121 0.227 0.261 0.277 0.292 0.258 0.275	0.204 0.220 0.260 0.292 0.303 0.345 0.302 0.312	0.200 0.217 0.259 0.277 0.284 0.276 0.246 0.250	0.200 0.216 0.155 0.201 0.259 0.277 0.173 0.202	0.251 0.272 0.250 0.273 0.291 0.308 0.328 0.331	0.251 0.269 0.210 0.259 0.133 0.152 0.230 0.242
	MD	I II	PCA SVM PCA SVM	0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8	0.241 0.283 0.291 0.310 0.255 0.289 0.289 0.315	0.237 0.272 0.280 0.314 0.265 0.298 0.322 0.356	0.141 0.115 0.277 0.320 0.247 0.270 0.295 0.312	0.232 0.277 0.290 0.335 0.259 0.289 0.285 0.320	0.148 0.138 0.154 0.183 0.276 0.333 0.336 0.344	0.239 0.279 0.297 0.325 0.188 0.204 0.293 0.324	0.221 0.262 0.270 0.315 0.244 0.280 0.172 0.118	0.227 0.302 0.289 0.329 0.279 0.290 0.371 0.366	0.233 0.258 0.274 0.305 0.245 0.292 0.286 0.312	0.241 0.262 0.292 0.319 0.268 0.302 0.288 0.329	0.255 0.286 0.333 0.356 0.317 0.330 0.326 0.357	0.220 0.265 0.285 0.334 0.255 0.294 0.300 0.326	0.237 0.289 0.294 0.333 0.273 0.301 0.304 0.327	0.148 0.138 0.154 0.188 0.276 0.333 0.336 0.344	0.240 0.288 0.298 0.355 0.264 0.299 0.309 0.328	0.229 0.269 0.294 0.325 0.185 0.208 0.172 0.203	0.246 0.302 0.318 0.370 0.320 0.364 0.386 0.397	0.230 0.267 0.308 0.346 0.259 0.271 0.289 0.312
	TSD	I II	PCA SVM PCA SVM	0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8	0.440 0.353 0.415 0.333 0.545 0.501 0.475 0.432	0.401 0.369 0.401 0.332 0.498 0.455 0.435 0.400	0.442 0.418 0.405 0.333 0.544 0.497 0.461 0.410	0.487 0.460 0.458 0.380 0.498 0.551 0.511 0.480	0.161 0.086 0.190 0.120 0.281 0.178 0.276 0.133	0.487 0.445 0.453 0.385 0.577 0.548 0.525 0.447	0.423 0.382 0.409 0.365 0.526 0.492 0.453 0.402	0.394 0.331 0.389 0.310 0.483 0.460 0.443 0.380	0.453 0.359 0.415 0.332 0.542 0.488 0.460 0.418	0.363 0.282 0.367 0.312 0.498 0.406 0.451 0.400	0.396 0.305 0.394 0.340 0.548 0.502 0.495 0.413	0.412 0.347 0.377 0.348 0.518 0.517 0.432 0.405	0.405 0.338 0.423 0.352 0.562 0.495 0.491 0.445	0.161 0.086 0.190 0.120 0.281 0.178 0.276 0.133	0.408 0.344 0.413 0.333 0.549 0.505 0.490 0.423	0.398 0.336 0.389 0.327 0.545 0.505 0.456 0.450	0.411 0.346 0.381 0.340 0.506 0.426 0.479 0.427	0.363 0.275 0.363 0.303 0.528 0.457 0.483 0.397

TABLE II CLASSIFICATION

 II ERRORS IN TERMS OF FRSCATNET IN H&E TRAINING DATABASES.

										Fractional order (α 1 , α 2 )
	Data	Ratio	(0.4,	(0.7,	(1.3,	(1.6,	(0.4,	(0.7,	(1.3,	(1.6,	(0.4,	(0.7,	(1.3,	(1.6,	(0.4,	(0.7,	(1.3,	(1.6,
			0.4)	0.4)	0.4)	0.4)	0.7)	0.7)	0.7)	0.7)	1.3)	1.3)	1.3)	1.3)	1.6)	1.6)	1.6)	1.6)
	BD	0.5 0.8	0.249 0.231	0.248 0.239	0.228 0.218	0.244 0.237	0.236 0.249	0.247 0.245	0.243 0.236	0.250 0.232	0.208 0.219	0.238 0.229	0.227 0.235	0.245 0.276	0.224 0.252	0.232 0.247	0.255 0.248	0.278 0.267
	MD	0.5 0.8	0.229 0.259	0.239 0.261	0.284 0.264	0.263 0.255	0.280 0.269	0.245 0.242	0.239 0.234	0.310 0.291	0.292 0.271	0.292 0.249	0.286 0.276	0.297 0.293	0.288 0.282	0.296 0.286	0.304 0.305	0.292 0.284

  The test group is divided into Test A and Test B as indicated in SectionII.A. Test A contains 33 histological images with benign glands and 27 images with malignant glands. Test B includes 4 benign images and 16 malignant images. We estimate the best parameter on the training subset. According to the experiments shown in Table. I, the fractional orders (including 0.4 0.7 1.3 1.6) appear as promising for classification. Here, the obtained fractional orders are tested by cross validation to determine the best classification parameter for the H&E images. According to the results shown in Table. I, we select FrameworkI and PCA classifier. The validation results are shown in Table. II. By comparing the classification results in Table

1) Feature Detection:

  All the results are listed in Table. III. For the gland detection evaluation, the results of the F1 score are shown in the left columns in Table. III. The CUMedVision2 (CUM2) with a contour-aware component achieves the best results for test A but the performance of our method remains high. FrScatNet behaves better than all other methods on test B. For the gland segmentation evaluation, the results of the object-level Dice index (D object in Table. III) show that the Deep Multichannel Neural Networks (DMNN) algorithm achieves the best performance on test A and our method on test B. For the shape similarity evaluation (H object column in Table. III), the DMNN algorithm is also ranked first but the index value obtained with FrScatNet is very close.

	3/4	testARank + 1/4	testBRank
			(18)
	4) Results And Discussion:

ACKNOWLEDGMENT

We thank the MICCAI 2015 Gland Segmentation Challenge for providing histology dataset and the Lazebniks group for offering access to the texture database. We thank Mallat group for offering access to the ScatNet software.

This work was supported supported in part by the National Key R&B Program of China (2017YFC107900) and in part by the National Natural Science Foundation of China (Nos. 61201344, 61271312, 61401085, 31571001, 31640028, 31400842, 61572258, 11301074, 61471226), by the Qing Lan Project and the 333 project (No. BRA 2015288), by the Shortterm Recruitment Program of Foreign Experts (WQ20163200398), and by the Natural Science Foundation of Shandong Province (NO. JQ201516, 2018GGX101018).


Jiasong Wu

Shijie Qiu

Youyong Kong

Yang Chen

Lotfi Senhadji

Huazhong Shu

Keywords: Deep learning, convolutional neural network, PCANet, MomentsNet, Zernike moment

In this paper, we propose a new simple and learning-free deep learning network named MomentsNet, whose convolution layer, nonlinear processing layer and pooling layer are constructed by Moments kernels, binary hashing and block-wise histogram, respectively. Twelve typical moments (including geometrical moment, Zernike moment, Tchebichef moment, etc.) are used to construct the MomentsNet whose recognition performance for binary image is studied. The results reveal that MomentsNet has better recognition performance than its corresponding moments in almost all cases and ZernikeNet achieves the best recognition performance among MomentsNet constructed by twelve moments. ZernikeNet also shows better recognition performance on a binary image database than that of PCANet, which is a learning-based deep learning network.

Convolutional neural networks (CNNs) with backpropagation process have shown its great success in various image classification tasks [START_REF] Lecun | Gradientbased learning applied to document recognition[END_REF][START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Sermanet | Overfeat: Integrated recognition, localization and detection using convolutional networks[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF]. However, backpropagation process has very high computational complexity. To address this issue, convolutional networks without backpropagation have been proposed by many researchers. Chan et al. [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF] proposed a lightweight and learning-based convolutional network named principle component analysis network (PCANet), who extracts the feature of the input images by using cascaded PCA filter, binary hashing, and block-wise histogram successively. PCANet does not use the backpropagation process but works unexpectedly well in most image classification tasks. Due to the success of PCANet, many researchers try to modify and extend this network in recent years [START_REF] Ng | DCTNet: A simple learning-free approach for face recognition[END_REF][START_REF] Feng | DLANet: A manifold-learning-based discriminative feature learning network for scene classification[END_REF][START_REF] Yang | Canonical correlation analysis networks for two-view image recognition[END_REF][START_REF] Zeng | Color image classification via quaternion principal component analysis network[END_REF][START_REF] Qin | DeepFish: Accurate underwater live fish recognition with a deep architecture[END_REF]. Ng and Teoh [START_REF] Ng | DCTNet: A simple learning-free approach for face recognition[END_REF] proposed a discrete cosine transform network (DCTNet) for face recognition. Feng et al. [START_REF] Feng | DLANet: A manifold-learning-based discriminative feature learning network for scene classification[END_REF] proposed a discriminative locality alignment network (DLANet) for scene classification. Yang et al. [START_REF] Yang | Canonical correlation analysis networks for two-view image recognition[END_REF] proposed a canonical correlation analysis network (CCANet) for two-view image recognition.

On the other hand, image moments [START_REF] Teague | Image analysis via the general theory of moments[END_REF][START_REF] Mukundan | Image analysis by Tchebichef moments[END_REF][START_REF] Yap | Image analysis by Krawtchouk moments[END_REF][START_REF] Zhu | Image analysis by discrete orthogonal dual Hahn moments[END_REF][START_REF] Yap | Two-dimensional polar Harmonic transforms for invariant image representation[END_REF][START_REF] Hoang | Generic polar harmonic transforms for invariant image representation[END_REF][START_REF] Mukundan | Fast computation of Legendre and Zernike moments[END_REF][START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF][START_REF] Asli | Fast computation of Krawtchouk moments[END_REF][START_REF] Teh | On image analysis by the methods of moments[END_REF][START_REF] Liao | On image analysis by moments[END_REF] are widely used in binary image recognition due to their invariant representations of images. The image moments have a wide range of orthogonal and nonorthogonal basis functions and are simple to compute whatever the order requires. The image sampling can be either rectangular or polar, based on uniform or nonuniform lattices. The researchers conducted on various moment families which showed the efficiency of the moments. Geometric moments are the simplest moments but they are not orthogonal. Teague [START_REF] Teague | Image analysis via the general theory of moments[END_REF] introduced the continuous orthogonal moments (COTs) for which Legendre moments and Zernike moments are two typical representatives. To overcome the discretization error problem of the COTs, researchers proposed various discrete orthogonal moments, including Tchebichef moments [START_REF] Mukundan | Image analysis by Tchebichef moments[END_REF], Krawtchouk moments [START_REF] Yap | Image analysis by Krawtchouk moments[END_REF] and dual Hahn moments [START_REF] Zhu | Image analysis by discrete orthogonal dual Hahn moments[END_REF], etc. In order to get a more invariant image representation, Yap et al. [START_REF] Yap | Two-dimensional polar Harmonic transforms for invariant image representation[END_REF] produced a set of polar harmonic transforms (PHTs), including polar complex exponential transform (PCET), polar cosine transform (PCT), and polar sine transform (PST). Hoang and Tabbone [START_REF] Hoang | Generic polar harmonic transforms for invariant image representation[END_REF] then presented a set of generic PHTs (GPHTs), including generic PCET (GPCET), generic PCT (GPCT), and generic PST (GPST). Note that many fast algorithms for example, [START_REF] Mukundan | Fast computation of Legendre and Zernike moments[END_REF][START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF][START_REF] Asli | Fast computation of Krawtchouk moments[END_REF] are proposed for moments due to their learning-free (or independence of input data) properties, while deriving the fast algorithms for learning-based PCA is very difficult. For more references on moments, we refer to [START_REF] Shu | Moment-based approaches in imaging Part 1: basic features[END_REF][START_REF] Mukundan | Moment Functions in Image Analysis: Theory and Application[END_REF][START_REF] Flusser | Moments and Moment Invariants in Pattern Recognition[END_REF].

In this paper, we propose a novel simple and learning-free deep learning network named MomentsNet for extracting the features of the input images. Learning-free deep learning network means that all the three layers (the convolution layer, nonlinear processing layer and pooling layer) of the network are independent of input data.

The rest of the paper is organized as follows. In Section 2, various moments are briefly introduced. The architecture of MomentsNet is described in Section 3. The recognition performance of MomentsNet is analyzed with respect to various parameters and also compared to moments, and PCANet in Section 4. Section 5 concludes the paper.

The general definition of two-dimensional (n+m)th order moment is given by , where ( , ) nm x y and ( , ) nm r are moments kernels in Cartesian coordinates and in polar coordinates, respectively. f(x,y) and f(r, ) are image values in Cartesian coordinates and in polar coordinates, respectively. A summary of the moments used in this paper is given in Table 1.

The architecture of the proposed MomentsNet is summarized in Fig. 1. We then take the MomentsNet-2, in which the first two convolutional layers are constructed by moments, as an example to describe in details. The moments feature maps extracted from the second stage are binarized, weighted, and summed to reduce the redundancy features.

n k P x x k n k n k Zernike [13] 1 2 0 0 ( | | )/2 2 0 1 ( ) ( , ) , 0 | | , | | , ( )! ( ) ( 1) ![( 2 | |)/2]![( 2 | |)/2]! jm nm nm n m k n k nm k n Z E r e f r rdrd m n n m is even n k E r r k n k m n k m Tchebichef [14] 1 1 1 0 0 0 2 2 2 2 2 [ ( , ) ( , )] ( ) ( ) ( , ), 1 ( ) ! ( 1) , ( , ) ( 1)
( 2 ) ( ) / (2 1). N M nm n m x y n n k n k D nN mM t xt y f xy N k n k x t x n n k n k n N N N N N n n Krawtchouk [15] 1 1 1 2 0 0 2 1 2 1 0 ( ; , 1
N M nm n m x y n n n x N x n n k k k k k k Q Kx pN K y pM fx y K x p N K x p N w x p N n p N N p n w x p N p p n p N x p N K x p N F n x N p a b z F a b c z c k a ( 1)( 2)...( 1) ( ) / ( ). a a a a k a k a Dual Hahn [16] 1 1 ( ) ( ) ( ) ( ) 2
( ) 3 2 ˆ( , , ) ( , , ) ( , ), , 0,1,2,..., 1
( ) 1 ˆ( , , ) ( , , ) [ ( )], 2 ( 1 )( 1 ) ( , , ) ! ( , , 1; 1, 1;1), ( ) b b c c nm n m x a y a c c n n n c n n n W w x a b w y a bfx ym n N x w x a b w x a b x x d a b a c w x a b n F n a x a x a b a c x 2 1 2 3 3 2 1 2 3 1 2 1 2 0 ( 1 ) ( 1 ) , ( 1) ( ) ( 1) ( 1) ( 1 ) , !( 1)! ( )
( ) ( ) ( ) ( , , ; , ; ) , ( ) ( ) ! 1/ 2 , | | 1 , . n k k k k k k k a x c x x a b x b x x c a c n d n b a n b c n a a a z F a a a b b z b b k a b c a b a N PCET [17] 2 2 1 * 0 0 2 [ ( , )] ( , ) , | | | | 0,1, , , ( , ) ( ) ,
Firstly, each element in {Ii,j,h, i=1,2,…,N1; j=1,2,…,L1; h=1,2,…,L2} is binarized by using a modified Heaviside function defined as 0, ( ) Since different moment features can capture different variations of the original images, the binarized features should be weighted to form new single moments features as:

L L B T i i i i L f Bhist T Bhist T Bhist T
The feature vector is then sent to a SVM classifier [START_REF] Fan | LIBLINEAR: A library for large linear classification[END_REF] to get the final recognition results.

The database used in the following experiments is a set of binary images of 9 classes, including bird, camel, children, elephant, fork, hammer, key, ray and turtle. Each class contains 144 images and all the images are rotated from 0 to 330 with an increment of 30 . Some image samples are shown in Fig. 2. All the images are then scaled to 32×32 to reduce the computational complexity in the experiments. In this subsection, we focus on the selection of various parameters. We take ZernikeNet as an example to show the impact of the parameters in the following.

(a) (b) Fig. 3 The impact of threshold t. (a) the recognition rate versus the threshold t. (b) the "result" versus the threshold t. Note that the "result" denotes the recognition rate for the curve above and the percentage of 1 after binarization for the curve below.

The impact of threshold t:

From our experiments, the threshold t in modified Heaviside function has a major effect on the final recognition result. To discuss the effect of threshold t, we firstly set the number of filters, patch size, block size and overlapping ratio to 9, 11×11, 8×8, and 0.5, respectively. The experiments are carried out on the one-layer Zernike Moments networks (ZernikeNet-1) for simplicity. Fig. 3(a) shows the recognition rates vary with the change of threshold t. We can see that there is a dramatic change in the recognition rate as the threshold t changes and the recognition rate is relatively good when 0.1 t 0.2. A question is raised: Is there any other parameters that help to choose the threshold value t? Since t decides the proportion of the number of 1 and 0 in the image after the binarization, we thus additionally record the percentages of the number 1. Fig. 3(b) is the recognition result or the percentage of 1 after binarization vary with the threshold changes from 0 to 0.2 with step 0.01. From the figure, we can find that when the percentage of 1 is in the range of [0.3, 0.5], the thresholds correspond to good recognition results. This is in fact a good guidance for the choice of threshold value t not only for ZernikeNet-1 but also for ZernikeNet-2. For ZernikeNet-1, we set the threshold t to 0.1.

The impact of the number of filters Ll, patch size k1×k2, block size h1×h2, and overlapping ratio R:

Fig. 4 shows the recognition results as the parameters change. It is apparent that the recognition rate tends to be better with the increase of the number of filters L1, but the performance does not improve greatly when L1>13. The recognition rate tends to first rise and then fall with the increase of the patch size k1×k2 and the block size h1×h2. ZenikeNet-1 can get the best performance when the patch size and the block size are 11×11 and 4×4, respectively. Comparing with other parameters, the overlapping ratio R has little impact on the recognition rate. We simply set R=0.5. ZenikeNet-1 achieves the recognition accuracy of 96.72% with the aforementioned parameter setting. We compare the performance of the proposed MomentsNet with the moments and also the PCANet [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF]. The best recognition rates and their corresponding parameter settings are shown in Table 2. Note that the comparison results are averaged over 5 independent experiments. Note also that the accuracy of moments is obtained by extracting the features by using various moments and then feed the features into the SVM classifier [START_REF] Fan | LIBLINEAR: A library for large linear classification[END_REF].

From Table 2, we can see that MomentsNet has better recognition performance than its corresponding moments in almost all cases and ZernikeNet performs the best and GeometryNet performs the worst in terms of the recognition performance among MomentsNet constructed by twelve moments. Most learning-free MomentsNet shows better recognition performance than that of PCANet [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF], which is a learning-based deep learning network. We also find that the recognition performance of some MomentsNet-1 outperforms its corresponding MomentsNet-2, for example, TchebichefNet. We think the reason is that the descriptive powers of these moments tend to its upper limit when they are used in the construction of one-stage network, therefore, adding more stage is not helpful for improving the recognition performance. In this paper, we propose a simple and learning-free deep learning network named MomentsNet in which 12 different moments are explored in detail. The impact of various parameters on the recognition performance is analyzed in MomentsNet. MomentsNet has better recognition performance than its corresponding moments in almost all cases and also shows better recognition performance than that of PCANet in a typical binary image recognition task.

Fig. 1

 1 Fig. 1 The Architecture of MomentsNet-2

  …,N1; j=1,2,…,L1. The pixel values of Ti,j are integers in the range of use a block of size h1×h2 to slide each of the L1 images Ti,j, j=1,2,…,L1, with overlap ratio R. Then, the features are divided into B boxes and we compute the histogram of the decimal values for each box and denote it as hist(B)d, d=1,2,…,B. After this pooling process, we concatenate all the histograms of B boxes into one vector

Fig. 2

 2 Fig. 2 Some Samples of Binary Image Database We randomly pick up half of the images in each class as the training set and the others left as the testing set.

Fig. 4

 4 The recognition rates versus various other parameters. (a) the recognition rate versus the number of filters; (b) the recognition rate versus the patch size; (c) the recognition rate versus the block size; (d) the recognition rate versus the overlap ratio.

Table 1

 1

					. The definitions of various moments
	Moments		Definition	Moments								Definition
	Geometric	nm G	2	( , ) x y f x y dxdy n m	Legendre [13]	nm L	(2 1)(2 4 n	m	1)	1 1 1 1	( ) ( ) ( , ) m P x P y f x y dxdy n	,
						n	( )	1 2 n	n k	/2 0	( 1)	k	! (	( 2 2) ! ) ! ( 2) !	2 n k	.

Table 2 .

 2 The best recognition rates of different methods. Quintets (L1, k1, h1, R, t) whose elements denote the number of filters, patch size, block size, overlap ratio, threshold, respectively. Note that we always set L2=L1, k2=k1, and h2=h1, in this paper.

		The	The accuracy of	The accuracy of
	Moment	accuracy of	MomentsNet-1	MomentsNet-2
	s	Moments	(%)	(%)
		(%)
	Geometr	12.11±1.01	68.09±3.51	67.32±3.98
	y	(n+m=4)	(9, 13, 3, 0.5, 0)	(9, 13, 3, 0.5, 0)
	Legendr	35.03±1.54	90.97±2.55	91.98±2.62
	e	(n+m=16)	(11,11,3,0.6,-0.02)	(11, 11, 3, 0.5, 0)
	Zernike	80.48±2.39	±1.97
		(n+m=16)

This work was supported by the National Natural Science Foundation of China (No. 61201344, 61271312, 61401085, 31640028, 61572258, 11301074).