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## 1 Invariant Scattering Convolution Networks (1203)

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8954 | Distinctive image features from scale-invariant keypoints,”
- Lowe
- 2004
(Show Context)
Citation Context ...ttering networks, introduced in [23], [22], build translation invariant representations with average poolings of wavelet modulus coefficients. The output of the first network layer is similar to SIFT =-=[21]-=- or Daisy [33] type descriptors. However, this limited set of locally invariant coefficients is not sufficiently informative to discriminate complex structures over large-size domains. The information... |

6496 | LIBSVM: A Library for Support Vector Machines, 2001. Software available at http://www.csie.ntu.edu.tw/ cjlin/libsvm
- Chang, Lin
(Show Context)
Citation Context ...ize of the training set, for different representations and classifiers. The affine space selection of section 4.1 is compared with an SVM classifier using RBF kernels, which are computed using Libsvm =-=[9]-=-, and whose variance is adjusted using standard cross-validation over a subset of the training set. The SVM classifier is trained with a renormalization which maps all coefficients to [−1,1]. The PCA ... |

2831 | R.C.: Online learning with kernels
- Kivinen, Smola, et al.
- 2002
(Show Context)
Citation Context ...f this variability is due to rigid translations, rotations or scaling. This variability is often uninformative for classification and should thus be eliminated. In the framework of kernel classifiers =-=[31]-=-, metrics are defined as a Euclidean distance applied on a representation Φ(x) of signals x. The operator Φ must therefore be invariant to these rigid transformations. Non-rigid deformations also indu... |

1923 | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories,” - Lazebnik, Schmid, et al. - 2006 |

784 | Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories.
- Fei-Fei, Fergus, et al.
- 2007
(Show Context)
Citation Context ...rse than the uniform texture at the bottom. As a result the top image has second order scattering coefficients of larger amplitude than at the bottom. For typical images, as in the CalTech101 dataset =-=[10]-=-, Table 1 shows that the scattering energy has an exponential decay as a function of the path length m. As proved by (11), the energy of scattering coefficients converges to 0 as m increases and is be... |

520 | On Discriminative vs. Generative Classifiers: A comparison of logistic regression and naive Bayes,“
- Ng, Jordan
- 2001
(Show Context)
Citation Context ...digit recognition and for texture discrimination. 4.1 PCA Affine Scattering Space Selection Although discriminant classifiers such as SVM have better asymptotic properties than generative classifiers =-=[26]-=-, the situation can be inverted for small training sets. We introduce a simple robust generative classifier based on affine space models computed with a PCA. Applying a DCT on scattering coefficients ... |

495 |
Representing and recognizing the visual appearance of materials using three-dimensional textons
- Leung, Malik
- 2001
(Show Context)
Citation Context ...re well adapted to detect edges or sharp transitions but do not have enough frequency and directional resolution to discriminate complex directional structures. For texture analysis, many researchers =-=[19]-=-, [30], [28] have been using averaged wavelet coefficient amplitudes |x ⋆ ψλ| ⋆ φJ(u), but calculated with a complex wavelet ψ having a better frequency and directional resolution. A scattering transf... |

424 | A parametric texture model based on joint statistics of complex wavelet coefficients.
- Portilla, Simoncelli
- 2000
(Show Context)
Citation Context ...ted to detect edges or sharp transitions but do not have enough frequency and directional resolution to discriminate complex directional structures. For texture analysis, many researchers [19], [30], =-=[28]-=- have been using averaged wavelet coefficient amplitudes |x ⋆ ψλ| ⋆ φJ(u), but calculated with a complex wavelet ψ having a better frequency and directional resolution. A scattering transform computes... |

347 |
Multiresolution elastic matching,
- Bajcsy, Kovacic
- 1989
(Show Context)
Citation Context ...pplied on a representation Φ(x) of signals x. The operator Φ must therefore be invariant to these rigid transformations. Non-rigid deformations also induce important variability within object classes =-=[3]-=-, [15], [34]. For instance, in handwritten digit recognition, one must take into account digit deformations due to different writing styles. However, a full deformation invariance would reduce discrim... |

252 | What is the best multi-stage architecture for object recognition? - Jarrett, Kavukcuoglu, et al. - 2009 |

228 | Learning Mid-Level Features For Recognition. In:
- Boureau
- 2010
(Show Context)
Citation Context ...ted as a pooling function in the context of convolution networks. The averaging by φ2J at the output is also a pooling operator which aggregates coefficients to build an invariant. It has been argued =-=[7]-=- that an average pooling loses information, which has motivated the use of other operators such as hierarchical maxima [8]. The high frequencies lost by the averaging are recovered as wavelet coeffici... |

224 | On contraction analysis for non-linear systems. - Lohmiller, Slotine - 1998 |

195 | Unsupervised learning of invariant feature hierarchies with applications to object recognition. In
- Ranzato, Huang, et al.
- 2007
(Show Context)
Citation Context ...mprised of at most 60,000 training samples and 10,000 test samples. If the training dataset is not augmented with deformations, the state of the art was achieved by deep-learning convolution networks =-=[30]-=-, deformation models [17], [3], and dictionary learning [27]. These results are improved by a scattering classifier. All computations are performed on the reduced cosine scattering representation desc... |

173 | Texture classification: Are filter banks necessary?
- Varma, Zisserman
- 2003
(Show Context)
Citation Context ...ases, and to variability due to illumination, rotation, scaling, or perspective deformations when textures are mapped on surfaces. Texture classification is tested on the CUReT texture database [21], =-=[36]-=-, which includes 61 classes of image textures of N = 200 2 pixels. Each texture class gives images of the same material with different pose and illumination conditions. Specularities, shadowing and su... |

126 | Daisy: An efficient dense descriptor applied to wide baseline stereo,”
- Tola, Lepetit, et al.
- 2009
(Show Context)
Citation Context ...ks, introduced in [23], [22], build translation invariant representations with average poolings of wavelet modulus coefficients. The output of the first network layer is similar to SIFT [21] or Daisy =-=[33]-=- type descriptors. However, this limited set of locally invariant coefficients is not sufficiently informative to discriminate complex structures over large-size domains. The information lost by the a... |

103 |
From model selection to adaptive estimation.
- Birge, Massart
- 1996
(Show Context)
Citation Context ...roximation in the affine space Ad,k than in affine spaces Ad,k ′ of other classes k′ = k. This is a model selection problem, which requires to optimize the dimension d in order to avoid over-fitting =-=[5]-=-. The invariance scale 2 J must also be optimized. When the scale 2 J increases, translation invariance increases11 but it comes with a partial loss of information which brings the representations of... |

90 | Exploring strategies for training deep neural networks.
- Larochelle, Bengio, et al.
- 2009
(Show Context)
Citation Context ...s it is 2.6% if the supremum is normalized. The scattering transform is stable but not invariant to rotations. Stability to rotations is demonstrated over the MNIST database in the setting defined in =-=[16]-=-. A database with 12000 training samples and 50000 test images is constructed with random rotations of MNIST digits. The PCA affine space selection takes into account the rotation variability by incre... |

86 | Task-driven dictionary learning
- Mairal, Bach, et al.
- 2012
(Show Context)
Citation Context ...t samples. If the training dataset is not augmented with deformations, the state of the art was achieved by deep-learning convolutional networks [29], deformation models [15], and dictionary learning =-=[25]-=-. These results are improved by a scattering classifier. All computations are performed on the reduced cosine scattering representation described in Section 3.3, which keeps the lower-frequency half o... |

85 | A statistical approach to material classification using image patch exemplars.
- Varma, Zisserman
- 2009
(Show Context)
Citation Context ...idual stochastic variability which decays as J increases and to variability due to illumination, rotation and perspective effects. Texture classification is tested on the CUReT texture database [19], =-=[35]-=-, which includes 61 classes of image textures of N = 200 2 pixels. Each texture class gives images of the same material with different pose and illumination conditions. Specularities, shadowing and su... |

79 | A.: Towards a coherent statistical framework for dense deformable template estimation
- Allassonnière, Amit, et al.
- 2007
(Show Context)
Citation Context ...hus deforms the image. The deformation gradient tensor ∇τ(u) is a matrix whose norm |∇τ(u)| measures the deformation amplitude at u. A small deformation is an invertible transformation if |∇τ(u)| < 1 =-=[2]-=-, [34]. Stability to deformations is expressed as a Lipschitz continuity condition relative to this deformation metric: ‖Φ(Lτx)−Φ(x)‖ ≤ C‖x‖ sup|∇τ(u)| , (2) u where ‖x‖ 2 = ∫ |x(u)| 2 du. This proper... |

56 | Pop: Patchwork of parts models for object recognition.
- Amit, Trouvé
- 2007
(Show Context)
Citation Context ...ation x of each x. Variability due to rigid transformations is removed if is invariant to these transformations. Nonrigid deformations also induce important variability within object classes [17], =-=[3]-=-. For instance, in handwritten digit recognition, one must take into account digit deformations due to different writing styles [3]. However, a full deformation invariance would reduce discrimination ... |

54 | Deformation models for image recognition.
- Keysers, Deselaers, et al.
- 2007
(Show Context)
Citation Context ...d on a representation Φ(x) of signals x. The operator Φ must therefore be invariant to these rigid transformations. Non-rigid deformations also induce important variability within object classes [3], =-=[15]-=-, [34]. For instance, in handwritten digit recognition, one must take into account digit deformations due to different writing styles. However, a full deformation invariance would reduce discriminatio... |

53 |
Phase recovery, MaxCut and complex semidefinite programming.
- Waldspurger, d’Aspremont, et al.
- 2015
(Show Context)
Citation Context ...ude signal representation which measures the sparsity of wavelet coefficients. The loss of information does not come from the modulus that removes the complex phase of x ?sðuÞ. Indeed, one can prove =-=[37]-=- that x can be reconstructed from the modulus of its wavelet coefficients fjx ?sðuÞjg, up to a multiplicative constant. The information loss comes from the integration of jx ?sðuÞj, which removes a... |

49 |
From model selection to adaptive estimation. Festschrift for Lucien Lecam: Research Papers in Probability and Statistics.
- Birge, Massart
- 1997
(Show Context)
Citation Context ...better approximation in the affine space Ak than in affine spaces Al of other classes l 6 k. This is a model selection problem, which requires an optimization of the dimension d to avoid overfitting =-=[5]-=-. The invariance scale 2J must also be optimized. When the scale 2J increases, translation invariance increases, but it comes with a partial loss of information, which brings the representations of di... |

43 | Local geometry of deformable templates
- Trouvé, Younes
(Show Context)
Citation Context ... representation Φ(x) of signals x. The operator Φ must therefore be invariant to these rigid transformations. Non-rigid deformations also induce important variability within object classes [3], [15], =-=[34]-=-. For instance, in handwritten digit recognition, one must take into account digit deformations due to different writing styles. However, a full deformation invariance would reduce discrimination sinc... |

41 | Tangent distance kernels for support vector machines,
- Haasdonk, Keysers
- 2002
(Show Context)
Citation Context ... the class ˆ k which yields the best approximation space: ˆk(X) = argmin‖SJX −PAd,k k≤K (SJX)‖ . (17) The minimization of this distance has similarities with the minimization of a tangential distance =-=[12]-=- in the sense that we remove the principal scattering directions of variabilities to evaluate the distance. However it is much simpler since it does not evaluate a tangential space which depends upon ... |

36 | Rotation invariant texture classification using LBP variance with global matching,” - Guo, Zhang, et al. - 2010 |

27 | Group invariant scattering
- Mallat
- 2012
(Show Context)
Citation Context ...ress these questions from a mathematical and algorithmic perspective by concentrating on a particular class of deep convolutional networks, defined by the scattering transforms introduced in [24] and =-=[25]-=-. A scattering transform computes a translation invariant representation by cascading wavelet transforms and modulus pooling operators, which average the amplitude of iterated wavelet coefficients. It... |

26 |
Using basic image features for texture classifica- tion.
- Crosier, Griffin
- 2010
(Show Context)
Citation Context ...ification error of 17%. The lowest published classification errors obtained on this dataset are 2% for Markov Random Fields [36], 1.53% for a dictionary of textons [15], 1.4% for Basic Image Features =-=[11]-=- and 1% for histograms of image variations [6]. A PCA classifier applied to a Fourier power spectrum estimator also reaches 1% error. The power spectrum is estimated with windowed Fourier transforms c... |

24 |
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
- Lindenstrauss, Preiss, et al.
- 2012
(Show Context)
Citation Context ...uch stronger. If Φ is Lipschitz continuous to deformations τ then the Radon-Nyko´ym property proves that the map which transforms τ into Φxτ is almost everywhere differentiable in the sense of Gateau =-=[22]-=-. It means that for small deformations, Φx − Φxτ is closely approximated by a bounded linear operator of τ, which is the Gateau derivative. Deformations are thus linearized by Φ, which enables linear ... |

21 |
Covariance regularization by thresholding. The Annals of Statistics 36:2577–604.
- PJ, Levina
- 2008
(Show Context)
Citation Context ...number of training samples per class. Indeed, if there are few training samples per class then variance terms dominate bias errors when estimating off-diagonal covariance coefficients between classes =-=[4]-=-. An affine space approximation classifier can also be interpreted as a robust quadratic discriminant classifier obtained by coarsely quantizing the eigenvalues of the inverse covariance matrix. For e... |

20 |
Statistical estimation of histogram variation for texture classification.
- Broadhurst
- 2005
(Show Context)
Citation Context ...assification errors obtained on this dataset are 2% for Markov Random Fields [36], 1.53% for a dictionary of textons [15], 1.4% for Basic Image Features [11] and 1% for histograms of image variations =-=[6]-=-. A PCA classifier applied to a Fourier power spectrum estimator also reaches 1% error. The power spectrum is estimated with windowed Fourier transforms calculated over half-overlapping windows, whose... |

15 | Combined Scattering for Rotation Invariant Texture Analysis
- Sifre
- 2012
(Show Context)
Citation Context ...oss of discriminability. However, a more efficient rotation invariant texture classification is obtained by cascading this translation invariant scattering with a second rotation invariant scattering =-=[24]-=-. It transforms each layer of the translation invariant scattering network with new wavelet convolutions along rotation parameters, followed by modulus and average pooling operators, which are cascade... |

15 | Actionable information in vision
- Soatto
- 2013
(Show Context)
Citation Context ...lightly deformed into x0, then kx x0k must be bounded by the size of the deformation, as defined in Section 2. Translation invariant representations can be constructed with registration algorithms =-=[33]-=-, autocorrelations, or with the Fourier transform modulus. However, Section 2.1 explains that these invariants are not stable to deformations and hence not adapted to image classification. Trying to a... |

11 |
spectrum analysis and 1/f processes. Wavelets and statistics
- Abry, Gonçalves, et al.
- 1995
(Show Context)
Citation Context ...ures have a spectrum which typically decays like |ω| −α . For such spectrum, an estimation over dyadic frequency bands provide a better bias versus variance trade-off than a windowed Fourier spectrum =-=[1]-=-. For mmax = 2, the error further drops to 0.2%. Indeed, scattering coefficients of order m = 2 depend upon moments of order 4, which are necessary to differentiate textures having same second order m... |

10 | Role of homeostasis in learning sparse representations. Neural computation 22
- Perrinet
- 2010
(Show Context)
Citation Context ...tation and inverse scaling on the path variable: SX2lrðpÞ SXð2lrpÞ where 2lrp 2lr1; . . . ; 2 lrm ; and 2lrk 2ljk rrk. If natural images can be considered as randomly rotated and scaled =-=[29]-=-, then the path p is randomly rotated and scaled. In this case, the scattering transform has stationary variations along the scale and rotation variables. This suggests approximating the Karhunen-Loè... |

9 | M A Ranzato, and Y LeCun. What is the best multi-stage architecture for object recognition - Jarrett, Kavukcuoglu - 2009 |

5 | Recursive Interferometric Representation
- Mallat
- 2010
(Show Context)
Citation Context ...e. We address these questions from mathematical and algorithmic point of views, by concentrating on a particular class of deep convolution networks, defined by the scattering transforms introduced in =-=[24]-=-, [25]. A scattering transform computes a translation invariant representation by cascading wavelet transforms and modulus pooling operators, which average the amplitude of iterated wavelet coefficien... |

5 |
LeCun: “Unsupervised Learning of Invariant Feature Hierarchies with Applications to Object Recognition”, CVPR
- Ranzato, Huang, et al.
- 2007
(Show Context)
Citation Context ... comprises at most 60000 training samples and 10000 test samples. If the training dataset is not augmented with deformations, the state of the art was achieved by deep-learning convolutional networks =-=[29]-=-, deformation models [15], and dictionary learning [25]. These results are improved by a scattering classifier. All computations are performed on the reduced cosine scattering representation described... |

5 | Gabor Feature Space Diffusion via the Minimal Weighted Area Method
- Sagiv, Sochen, et al.
- 2001
(Show Context)
Citation Context ...l adapted to detect edges or sharp transitions but do not have enough frequency and directional resolution to discriminate complex directional structures. For texture analysis, many researchers [19], =-=[30]-=-, [28] have been using averaged wavelet coefficient amplitudes |x ⋆ ψλ| ⋆ φJ(u), but calculated with a complex wavelet ψ having a better frequency and directional resolution. A scattering transform co... |

5 |
Group invariant scattering,” Communications in Pure and
- Mallat
- 2012
(Show Context)
Citation Context ...address these questions from mathematical and algorithmic point of views, by concentrating on a particular class of deep convolution networks, defined by the scattering transforms introduced in [24], =-=[25]-=-. A scattering transform computes a translation invariant representation by cascading wavelet transforms and modulus pooling operators, which average the amplitude of iterated wavelet coefficients. It... |

4 |
Recovering the phase of a complex wavelet transform,” CMAP, Ecole Polytechnique
- Waldspurger, Mallat
- 2012
(Show Context)
Citation Context ... norms {‖x ⋆ ψλ‖1}λ form a crude signal SJ[p]x(u) = | ||x⋆ψλ1|⋆ψλ2|...|⋆ψλm|⋆φ 2J(u) , representation, which measures the sparsity of the wavelet coefficients. For appropriate wavelets, one can prove =-=[36]-=- thatxcan be reconstructed from{|x⋆ψλ(u)|}λ, up to a multiplicative constant. The information loss thus comes from the integration of |x ⋆ ψλ(u)|, which removes all non-zero frequency components. Thes... |

4 | C.Schmid, J.Ponce Beyond Bags of Features: Spatial Pyramid Matching for Recognizing Natural Scene Categories CVPR - Lazebnik |

3 |
Group Invariant Scattering”, to appear
- Mallat
(Show Context)
Citation Context ...numerical experimentations that require significant expertise. Deformation stability is obtained with localized wavelet filters which separate the image variations at multiple scales and orientations =-=[22]-=-. Computing a nonzero translation invariant representation from wavelet coefficients requires introducing a non-linearity, which is chosen to be a modulus to optimize stability [6]. Wavelet scattering... |

2 | F.Lv, T.Huang, Y.Gong, “Localityconstrained Linear Coding for Image Classification”, CVPR - Wang, Yu - 2010 |

1 |
Operators commuting with diffeomorphisms
- Bruna
- 2012
(Show Context)
Citation Context ...s and orientations [22]. Computing a nonzero translation invariant representation from wavelet coefficients requires introducing a non-linearity, which is chosen to be a modulus to optimize stability =-=[6]-=-. Wavelet scattering networks, introduced in [23], [22], build translation invariant representations with average poolings of wavelet modulus coefficients. The output of the first network layer is sim... |

1 |
Role of Homeostasis
- Perrinet
- 2010
(Show Context)
Citation Context ...) = SX(2 l rp) where 2 l rp = (2 l+j1 rr1,...,2 l+jm rrm). cients by iterating on this propagator. If images are randomly rotated and scaled by 2 l r −1 then the path p is randomly rotated and scaled =-=[27]-=-. In this case, the scattering transform has stationary variations along the scale and rotation variables. This suggests approximating the Karhunen-Loève basis by a cosine basis along these variables.... |

1 |
Scattering audio representations”, subm. to
- Anden, Mallat
(Show Context)
Citation Context ...fficients are zero produces an error. This error is further amplified as the inversion of ˜ W progresses across layers from m to 0. Numerical experiments conducted over one-dimensional audio signals, =-=[2]-=-, [7] indicate that reconstructed sig-8 nals have a good audio quality with m = 2, as long as the number of scattering coefficients is comparable to the number of signal samples. Audio examples in ww... |

1 |
Scattering representations for pattern and texture recognition
- Bruna
- 2012
(Show Context)
Citation Context ...ism. To preserve stability to additive noise we also want M to be nonexpansive: ‖My − Mz‖ ≤ ‖y−z‖. If M is a nonexpansive operator which commutes with the action of diffeomorphisms then one can prove =-=[7]-=- that M is necessarily a pointwise operator. It means that My(u) is a function of the value y(u) only. To build invariants which also preserve the signal energy requires to choose a modulus operator o... |

1 |
Robust Higher Order Statistics,” AISTATS
- Welling
- 2005
(Show Context)
Citation Context ...nary processes depends upon second order and higher-order moments, and can thus discriminate such textures. Moreover, it does not suffer from the large variance curse of high order moments estimators =-=[37]-=-, because it is computed with a nonexpansive operator. If X(u) is stationary then U[p]X(u) remains stationary because it is computed with a cascade of convolutions and modulus which preserve stationar... |

1 |
Scattering Audio Representations
- Anden, Mallat
(Show Context)
Citation Context ...oefficients are zero produces an error. This error is further amplified as the inversion of eW progresses across layers from m to 0. Numerical experiments conducted over one-dimensional audio signals =-=[2]-=-, [7] indicate that reconstructed signals have good audio quality withm 2 as long as the number of scattering coefficients is comparable to the number of signal samples. Audio examples in www.di.ens... |

1 |
Scattering Representations for Pattern and Texture
- Bruna
- 2012
(Show Context)
Citation Context ...ally, to preserve stability to additive noise we also want M to be nonexpansive: kMyMzk ky zk. If M is a nonexpansive operator that commutes with the action of diffeomorphisms, then one can prove =-=[7]-=- that M is necessarily a pointwise operator. It means that MyðuÞ is a function of the value yðuÞ only. If, moreover, we want invariants which also preserve the signal energy, we shall choose a modulus... |