SIMPLE GABOR FEATURE SPACE FOR INVARIANT OBJECT RECOGNITION
|
|
- Amie Adams
- 6 years ago
- Views:
Transcription
1 SIMPLE GABOR FEATURE SPACE FOR INVARIANT OBJECT RECOGNITION Ville Kyrki Joni-Kristian Kamarainen Laboratory of Information Processing, Department of Information Technology, Lappeenranta University of Technology, P.O. Box 0, FIN Lappeenranta, Finland Abstract Invariant object recognition is one of the most challenging problems in computer vision. Authors propose a simple Gabor feature space, which has been successfully applied to facial feature extraction in invariant face detection in demanding environments (Kamarainen et al., 00c; Hamouz et al., 003). In the proposed feature space, illumination, rotation, scale, and translation invariant recognition of objects can be established within a reasonable amount of computation. In this study, the fundamental properties of Gabor features, construction of the simple feature space, and invariant search operations in the feature space are discussed in more detail. Key words: Gabor filtering, invariant object recognition 1 Introduction Invariant object recognition has been one of the most important topics in the computer vision research for decades. Often the research has focused on invariant features. They make invariant recognition possible in a straightforward manner but also present some pitfalls as an invariant feature is always a generalization and may lack some useful information. In addition, a single feature often does not allow accurate recognition and the relationships between several features need to be examined. It seems that the requirements of a truly invariant system cannot be exclusively met with a single invariant feature, Corresponding author. Tel.: ; fax: addresses: Ville.Kyrki@lut.fi (Ville Kyrki), Joni.Kamarainen@lut.fi (Joni-Kristian Kamarainen).
2 such as the moment invariants; at least not within a reasonable computing complexity. Since the study of a general image processing operator by Granlund (1978), Gabor filters have been used in the feature extraction from images. With properly chosen filter parameters, the Gabor filters have similar characteristics to multiresolution techniques, such as the Gabor expansion and wavelets (e.g. Bastiaans, 1980; Daubechies, 1990; Lee, 1996). Thus, it is reasonable to assume that similar properties can be established also for the Gabor filters. The authors have recently proposed a novel framework utilizing Gabor filter based features in rotation, scale, and translation invariant recognition of objects (Kamarainen et al., 00c; Hamouz et al., 003). The framework is based on a feature space constructed from the Gabor filter responses and invariant search operations established in the feature space. In this study the research results are concluded by showing the most important properties of the Gabor filters for invariance and introducing the feature space in more detail. The feature space assumes that the objects can be distinguished by features at only one spatial location. Despite this restriction, the method performs outstandingly well in the applications, and on the present knowledge, there is no evidence that the method could not be extended to handle several simultaneous locations. Feature Extraction with Gabor Filters There are two basic approaches to use Gabor functions in the feature extraction: 1) Gabor expansion and ) Gabor filtering. In the expansion, the Gabor functions represent distinct, not necessarily orthogonal, pieces of image information. A distinct advantage of the Gabor functions is their optimality in the time frequency, or space spatial-frequency in two dimensions, plane, providing the smallest possible pieces of information about time-frequency events (Gabor, 196). Any well-behaving function can be represented as a linear combination of the Gabor functions (Bastiaans, 1980; Gabor, 196). However, the Gabor expansion requires a computation of biorthogonal analysis functions, which may turn to be a very time consuming task. Thus, it has been more common to use Gabor functions as analysis filters in the image processing. Lades et al. (1993) use Gabor filter responses at single spatial location but do not take advantage of the invariance properties presented here. The use of Gabor filters is also motivated by the physiology of the mammalian visual system (Daugman, 1985). Nevertheless, image analysis with the biorthogonals can be thought as a dual to the analysis with the Gabor filters, and thus, also the Gabor filters extract space spatial-frequency events from images. The Gabor s original 1-D function of time and frequency has been generalized
3 to a -D function of space and spatial-frequency and several forms have been proposed (e.g. Granlund, 1978; Daugman, 1985). The authors have defined the following form of a normalized -D Gabor filter function in the continuous spatial domain (Kyrki, 00) ψ(x, y; f, θ) = f ( f γ x + f πγη e x = x cos θ + y sin θ, y = x sin θ + y cos θ, η y )e jπfx, where f is the frequency of a sinusoidal plane wave, θ is the anti-clockwise rotation of the Gaussian envelope and the sinusoid, γ is the spatial width of the filter along the plane wave, and η the spatial width perpendicular to the wave. The form in Eq. (1) is centered to the origin and a filter response for an image function ξ(x, y) can be calculated at any location (x, y) with the convolution (1) r ξ (x, y; f, θ) = ψ(x, y; f, θ) ξ(x, y) = ψ(x x τ, y y τ ; f, θ)ξ(x τ, y τ )dx τ dy τ. () The response of the Gabor filter in Eq. () is the low-level Gabor feature used in this study. Next it is shown that the Gabor features can be used to extract exact information of pose and scale changes when the geometric manipulation of an input image ξ(x, y) is a function of the Gabor filter parameters; (x, y) for translation, θ for rotation, and f for scale. In a feature space constructed from the Gabor filter responses, invariant search operations can be established based on the rotation, scale, and translation properties of the Gabor features, which are considered next. 3 Invariance properties of Gabor features The invariance theorems in this section are not complicated or completely novel, but will be briefly revisited as the literature of Gabor filtering research seems to lack them. Similar theorems generally hold for any function, which maintains the same shape in all translations, scales, and orientations; probably the most popular ones being the wavelets over scales and translations (e.g. Daubechies, 1990). The Gabor features provide properties similar to the multiresolution analysis, but which behave smoothly (Kamarainen et al., 00b), which is difficult to achieve with wavelets. In this sense the Gabor filters approximate the joint shiftability conditions better than the orthogonal wavelets (Simoncelli et al., 199). The shiftability concept becomes important since the 3
4 following theorems are given in the continuous domain, where for example the translation invariance is trivial, but in the discrete case the sampling of the parameters destroys the invariance of non-shiftable functions and transforms. 3.1 Rotation For an image ξ 1 (x, y), the Gabor feature, the response of the normalized Gabor filter in Eq. (), at location (x 0, y 0 ) is r ξ1 (x 0, y 0 ; f, θ) = ψ(x 0 x τ, y 0 y τ ; f, θ)ξ 1 (x τ, y τ )dx τ dy τ. (3) Next, a rotated version ξ (x, y) of an image ξ 1 (x, y) is considered, where the image ξ 1 (x, y) is rotated anti-clockwise around the spatial location (x 0, y 0 ) by the angle φ as ξ (x, y) = ξ 1 (ˆx, ŷ) ˆx = (x x 0 ) cos φ + (y y 0 ) sin φ + x 0 ŷ = (x x 0 ) sin φ + (y y 0 ) cos φ + y 0. () By substituting the rotated image in Eq. () into the convolution formula in Eq. (3) and by integrating over integration axes (x τ, y τ) rotated around the same point (x 0, y 0 ) by an angle φ it is straightforward to show that r ξ (x 0, y 0 ; f, θ) = ψ(x 0 x τ, y 0 y τ; f, θ φ)ξ 1 (x τ, y τ)dx τdy τ = r ξ1 (x 0, y 0 ; f, θ φ), (5) which is due to the rotation property of the Gabor features. The rotation property appearing as the equivalence in Eq. (5) proves that the response of the Gabor filter for a rotated image is equal to the response of the correspondingly rotated filter for the original image. 3. Scaling Next, an image which is homogeneously scaled version of the image ξ 1 (x, y) is considered. The new image is scaled by a factor a as ξ 3 = ξ 1 (ax, ay). A response for the scaled image is r ξ3 (x, y; f, θ) = ψ(x x τ, y y τ ; f, θ)ξ 3 (x τ, y τ )dx τ dy τ = ξ 3 (x x τ, y y τ )ψ(x τ, y τ ; f, θ)dx τ dy τ (6) = ξ 1 (ax ax τ, ay ay τ )ψ(x τ, y τ ; f, θ)dx τ dy τ.
5 By making substitutions ˆx τ = ax τ dx τ = dˆx τ a, ŷ τ = ay τ dy τ = dŷ τ a, (7) and by respectively re-ordering the variables in Eq. (1), Eq. (6) can be rewritten as r ξ3 (x, y; f, θ) = ξ 1 (ax ˆx τ, ay ŷ τ )ψ(ˆx τ, ŷ τ ; f a, θ)dˆx τdŷ τ = r ξ1 (ax, ay; f (8) a, θ). The scale property of the Gabor features, shown by the equality in Eq. (8), proves the response of the Gabor filter for a scaled image to be equal to the response of the correspondingly scaled Gabor filter at the same relative location of the original image. The scale property is fundamental for the wavelets (Daubechies, 1990) and holds also for the normalized Gabor filters in Eq. (1) (Porat and Zeevi, 1988). 3.3 Translation In the case of the centered filters, Eq. (1), the translation invariant search is trivial as the features in Eq. () can be calculated at any location (x, y). For a translated image ξ = ξ 1 (x x 1, x y 1 ) (9) the response at (x, y) is r ξ (x, y; f, θ) = ξ (x x τ, y y τ )ψ(x τ, y τ ; f, θ)dx τ dy τ = ξ 1 (x x 1 x τ, y y 1 y τ )ψ(x τ, y τ ; f, θ)dx τ dy τ = r ξ1 (x x 1, y y 1 ; f, θ), (10) which implies the translation property. 3. Illumination Uniform illumination change can be modeled as a multiplication by a constant. Let ξ 5 (x, y) = c ξ 1 (x, y). By the linearity of the convolution, it can be written r ξ5 (x, y; f, θ) = c r ξ1 (x, y; f, θ). (11) 5
6 Simple Gabor feature space According to the properties defined in the previous section it can be stated for an image ξ 6, which equals ξ 1 rotated by φ, scaled by a and intensity multiplied by c, that r ξ6 (x 0, y 0 ; f, θ) = c r ξ1 (ax 0, ay 0 ; f, θ φ). (1) a In general, an equality similar to Eq. (1) can be obtained for all functions, which retain the same shape regardless of the translation, scale or orientation parameters. Yet the result is not easily obtained by e.g. orthogonal wavelets but it can be easier to inherit the wavelet properties to the Gabor functions (Lee, 1996). So far only features at a single location (x, y) have been considered, but even then the Gabor filters capture information from a remarkable large area, and thus, by combining the responses of several filters in different orientations and frequencies, surprisingly complex objects can be represented (e.g. Krüger and Sommer, 00). For more complicated object structures and objects where the discriminative information at one location is not sufficient, the features must be combined from several spatial locations (Lades et al., 1993; Krüger and Sommer, 00). In that case the invariant properties are more difficult to use, yet still possible. Since the presented method is restricted to a single spatial location, it is referred as simple, but still being surprisingly efficient (Kamarainen et al., 00c; Hamouz et al., 003)..1 Sampling filter parameters It is clear that a filter bank, consisting of several filters, needs to be used, as the relationships between responses of filters provide the basis for distinguishing objects. Next, it is considered how the filter parameters should be chosen for the bank of filters. The selection of discrete rotation angles θ k has already been demonstrated by Kyrki et al. (001) and Park and Yang (001), where it was shown how the orientations must be spaced uniformly, that is, θ k = kπ n k = {0,..., n 1}, (13) where θ k is the kth orientation and n is the number of orientations to be used. However, the computation can be reduced to half since responses on angles [π, π[ are 90 phase shifted from responses on [0, π[ in a case of a real valued input. 6
7 In the selection of discrete frequencies f k, exponential sampling must be used (e.g. Kamarainen et al., 00c; Daugman, 1988), that is, f k = a k f max k = {0,..., m 1}, (1) where f k is the kth frequency, f 0 = f max is the highest frequency desired, and a is the frequency scaling factor (a > 1). Useful values for a include a = for octave spacing and a = for half-octave spacing.. Feature matrix Now, using the features in Eq. () and the parameter selection schemes in Eq. (13) and Eq. (1) to cover frequencies of interest f 0,..., f m 1 and the orientations for desired angular discrimination, one can construct a feature matrix G at an image location (x 0, y 0 ) G = r(x 0, y 0 ; f 0, θ 0 ) r(x 0, y 0 ; f 0, θ 1 ) r(x 0, y 0 ; f 0, θ n 1 ) r(x 0, y 0 ; f 1, θ 0 ) r(x 0, y 0 ; f 1, θ 1 ) r(x 0, y 0 ; f 1, θ n 1 ) r(x 0, y 0 ; f m 1, θ 0 ) r(x 0, y 0 ; f m 1, θ 1 ) r(x 0, y 0 ; f m 1, θ n 1 ). (15) For convenience, typical matrix indexing is used in G, i.e., G(1, 1) = g 1,1 = r(x 0, y 0 ; f 0, θ 0 ) and G(m, n) = g m,n = r(x 0, y 0 ; f m 1, θ n 1 ). For illumination invariance, the response matrix can be normalized as G = G i,j g i,j. (16) The feature matrix in Eq. (15) or Eq. (16) can be used as an input feature for any classifier, e.g., as used in face evidence detection in Kamarainen et al. (00c). If features are extracted using the Gabor feature matrices from objects in a standard pose, it is possible to introduce matrix manipulations that allow invariant search. First, a column-wise circular shift of the feature matrix can be defined as ( ) G (θ+k) = G(1 : m, k : n) G(1 : m, 1 : k 1), (17) where G(i : j, u : v) represents the sub-matrix of G containing rows i... j and columns u... v. It should be noted that if the responses are calculated only for half orientation space, e.g., [0, π[, the phase wrapping must be taken 7
8 into consideration in the column-wise shift. Similarly a row-wise shift for scale manipulation can be defined G (f+k) = ( ) G(k + 1 : m, 1 : n) G(m + 1 : m + k, 1 : n). (18) The column-wise circular shift in Eq. (17) corresponds to search over all rotation angles, and the row-wise shift in Eq. (18) corresponds to search over all down-scales. Note that the row-wise shift is not circular but the highest frequencies (f 0,..., f k 1 ) vanish and new lower frequencies (f m+1,..., f m+k ) are mapped to the Gabor feature matrix as replacements. The shifts for the feature matrix manipulation are results from the rotation and translation properties of the Gabor filter. To provide a smooth behavior of the features, the sharpness parameters γ and η must be adjusted to have a sufficient overlap of the Gabor filters in the feature space (Daugman, 1988; Lee, 1996). As a conclusion it can be said that a very versatile recognition is possible, as rotation, scaling and illumination invariance can be achieved. 5 Experimental results The simple Gabor feature space introduced in this study has already been used in several successful applications, e.g., in rotation invariant and noise robust recognition of electric components by Kamarainen et al. (00a), and illumination, scale, translation, and orientation invariant detection of facial evidences by Kamarainen et al. (00c) and Hamouz et al. (003). In this study, the main contributions are the formulation of the response matrix, representing the simple Gabor feature space, and the simple row-wise and column-wise shifts of the matrix representing the orientation and scale manipulation of an image. The response matrix and the shift operations are demonstrated in Fig. 1, which shows the application in the invariant detection of facial evidences. Rotation of an image induces a circular column-wise shift of the response matrix, Fig. 1(b), and scaling induces a row-wise shift, Fig. 1(c). 6 Discussion Motivated by the excellent results in applications, the authors have introduced the simple Gabor feature space and its theoretical framework in this study. The proposed feature space is especially useful in the scale, rotation, 8
9 scale orientation (a) original image scale () (3) 3 () (1) orientation (b) rotated anti-clockwise by () (3) scale 3 () (5) () (3) 3 () (1) orientation 0 (c) rotated and scaled by 0.5 Fig. 1. Original image and its rotated and scaled versions, and corresponding feature matrices computed at the centroid of the right eye (f k = 1/1, 1/8, 1/56, 1/11, γ = 1, η = 1, θ k = 0, 5, 90, 135 ). For rotation note that the image origin is in the upper left corner. and translation invariant recognition of objects and furthermore it provides robustness to noise and illumination changes. The feature space is considered simple since it occupies only one spatial location and thus is aimed to be 9
10 used with sufficiently distinguishable objects. Sometimes the presence of the objects is too general to be distinguished in the simple Gabor feature space, but even then it may be possible to distinguish salient sub-parts, which can be used as input to a more general classification systems. This approach has already been successfully utilized in the face detection presented by Hamouz et al. (003). In this study, the motivation was to inspect the feature space itself and to present the theory behind the methods, including the invariant properties in Eqs. (5), (6) and (10), the construction of the feature matrix at one location in Eq. (15), and the rotation and scale invariant search operations by formulas in Eqs. (17) and (18). The proposed Gabor features lie somewhere between the shiftable operators (Simoncelli et al., 199) and the wavelets (Daubechies, 1990). In the future it would be intriguing to study what kind of advantages can be achieved with the features, which approximate the shiftability of the steerable functions and the orthogonality of the separable wavelets. References Bastiaans, M. J., Gabor s expansion of a signal into Gaussian elementary signals. Proceedings of the IEEE 68 (). Daubechies, I., The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory 36 (5), Daugman, J. G., July Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America A (7), Daugman, J. G., Complete discrete -d Gabor transforms by neural networks for image analysis and compression. IEEE Transactions on Acoustics, Speech, and Signal Processing 36 (7), Gabor, D., 196. Theory of communications. Journal of Institution of Electrical Engineers 93, Granlund, G. H., In search of a general picture processing operator. Computer Graphics and Image Processing 8, Hamouz, M., Kittler, J., Kamarainen, J.-K., Kälviäinen, H., 003. Hypothesesdriven affine invariant localization of faces in verification systems. In: Proceedings of the International Conference on Audio- and Video-Based Biometric Person Authentication (to appear). Kamarainen, J.-K., Kyrki, V., Kälviäinen, H., 00a. Fundamental frequency Gabor filters for object recognition. In: Proceedings of the 16th International Conference on Pattern Recognition. Vol. 1. Quebec, Canada, pp Kamarainen, J.-K., Kyrki, V., Kälviäinen, H., 00b. Robustness of Gabor 10
11 feature parameter selection. In: Proceedings of the IAPR Workshop on Machine Vision Applications. pp Kamarainen, J.-K., Kyrki, V., Kälviäinen, H., Hamouz, M., Kittler, J., 00c. Invariant Gabor features for evidence extraction. In: Proceedings of the IAPR Workshop on Machine Vision Applications. pp Krüger, V., Sommer, G., 00. Gabor wavelet networks for efficient head pose estimation. Image and Vision Computing 0 (9-10), Kyrki, V., 00. Local and global feature extraction for invariant object recognition. Ph.D. thesis, Lappeenranta University of Technology. Kyrki, V., Kamarainen, J.-K., Kälviäinen, H., 001. Content-based image matching using Gabor filtering. In: Proceedings of the International Conference on Advanced Concepts for Intelligent Vision Systems Theory and Applications. Baden-Baden, Germany, pp Lades, M., Vorbrüggen, J., Buhmann, J., Lange, J., von der Malsburg, C., Würtz, R., Konen, W., Distortion invariant object recognition in the dynamic link architecture. IEEE Transactions on Computers (3), Lee, T. S., Image representation using D Gabor wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence 18 (10), Park, H. J., Yang, H. S., 001. Invariant object detection based on evidence accumulation and gabor features. Pattern Recognition Letters, Porat, M., Zeevi, Y. Y., The generalized Gabor scheme of image representation in biological and machine vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 10 (), Simoncelli, E. P., Freeman, W. T., Adelson, E. H., Heeger, D. J., Mar Shiftable multiscale transforms. IEEE Transactions on Information Theory 38 (),
INVARIANCE of features has been an active area in object
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. Y, MONTH Invariance Properties of Gabor Filter Based Features Overview and Applications Joni-Kristian Kamarainen, Ville Kyrki, Member, IEEE, Heikki Kälviäinen,
More informationInvariant Pattern Recognition using Dual-tree Complex Wavelets and Fourier Features
Invariant Pattern Recognition using Dual-tree Complex Wavelets and Fourier Features G. Y. Chen and B. Kégl Department of Computer Science and Operations Research, University of Montreal, CP 6128 succ.
More informationarxiv: v1 [cs.cv] 10 Feb 2016
GABOR WAVELETS IN IMAGE PROCESSING David Bařina Doctoral Degree Programme (2), FIT BUT E-mail: xbarin2@stud.fit.vutbr.cz Supervised by: Pavel Zemčík E-mail: zemcik@fit.vutbr.cz arxiv:162.338v1 [cs.cv]
More information6.869 Advances in Computer Vision. Bill Freeman, Antonio Torralba and Phillip Isola MIT Oct. 3, 2018
6.869 Advances in Computer Vision Bill Freeman, Antonio Torralba and Phillip Isola MIT Oct. 3, 2018 1 Sampling Sampling Pixels Continuous world 3 Sampling 4 Sampling 5 Continuous image f (x, y) Sampling
More informationEnhanced Fourier Shape Descriptor Using Zero-Padding
Enhanced ourier Shape Descriptor Using Zero-Padding Iivari Kunttu, Leena Lepistö, and Ari Visa Tampere University of Technology, Institute of Signal Processing, P.O. Box 553, I-330 Tampere inland {Iivari.Kunttu,
More informationJoni-Kristian Kämäräinen
Lappeenrannan teknillinen yliopisto Lappeenranta University of Technology Joni-Kristian Kämäräinen FEATURE EXTRACTION USING GABOR FILTERS Thesis for the degree of Doctor of Science (Technology) to be presented
More informationImage Representation by Complex Cells. Responses
Neural Computation MS number 2831 Image Representation by Complex Cell Responses Ingo J. Wundrich, Christoph von der Malsburg, and Rolf P. Würtz Institut für Neuroinformatik Ruhr-Universität Bochum D 44780
More informationScale-Invariance of Support Vector Machines based on the Triangular Kernel. Abstract
Scale-Invariance of Support Vector Machines based on the Triangular Kernel François Fleuret Hichem Sahbi IMEDIA Research Group INRIA Domaine de Voluceau 78150 Le Chesnay, France Abstract This paper focuses
More informationTwo-Dimensional Orthogonal Filter Banks with Directional Vanishing Moments
Two-imensional Orthogonal Filter Banks with irectional Vanishing Moments Jianping Zhou and Minh N. o epartment of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana,
More informationA new class of shift-invariant operators
1 A new class of shift-invariant operators Janne Heiilä Machine Vision Group Department of Electrical and Information Engineering P.O. Box 4500, 90014 University of Oulu, Finland Tel.: +358 8 553 2786,
More informationLogarithmic quantisation of wavelet coefficients for improved texture classification performance
Logarithmic quantisation of wavelet coefficients for improved texture classification performance Author Busch, Andrew, W. Boles, Wageeh, Sridharan, Sridha Published 2004 Conference Title 2004 IEEE International
More informationImage Representation by Complex Cell Responses
LETTER Communicated by Eero Simoncelli Image Representation by Complex Cell Responses Ingo J. Wundrich Ingo.Wundrich@neuroinformatik.ruhr-uni-bochum.de Institut für Neuroinformatik, Ruhr-Universität Bochum,
More informationMultiscale Autoconvolution Histograms for Affine Invariant Pattern Recognition
Multiscale Autoconvolution Histograms for Affine Invariant Pattern Recognition Esa Rahtu Mikko Salo Janne Heikkilä Department of Electrical and Information Engineering P.O. Box 4500, 90014 University of
More informationA New Efficient Method for Producing Global Affine Invariants
A New Efficient Method for Producing Global Affine Invariants Esa Rahtu, Mikko Salo 2, and Janne Heikkilä Machine Vision Group, Department of Electrical and Information Engineering, P.O. Box 45, 94 University
More informationImage Reconstruction from Gabor Magnitudes
From: Heinrich H. Bülthoff and Seong-Whan Lee and Tomaso A. Poggio and Christian Wallraven (eds.), Biologically Motivated Computer Vision, pp. 117 126, Springer, 2002. Image Reconstruction from Gabor Magnitudes
More informationFACE RECOGNITION BY EIGENFACE AND ELASTIC BUNCH GRAPH MATCHING. Master of Philosophy Research Project First-term Report SUPERVISED BY
FACE RECOGNITION BY EIGENFACE AND ELASTIC BUNCH GRAPH MATCHING Master of Philosophy Research Proect First-term Report SUPERVISED BY Professor LYU, Rung Tsong Michael PREPARED BY JANG KIM FUNG (01036550)
More informationAffine Normalization of Symmetric Objects
Affine Normalization of Symmetric Objects Tomáš Suk and Jan Flusser Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Prague 8, Czech
More informationLearning features by contrasting natural images with noise
Learning features by contrasting natural images with noise Michael Gutmann 1 and Aapo Hyvärinen 12 1 Dept. of Computer Science and HIIT, University of Helsinki, P.O. Box 68, FIN-00014 University of Helsinki,
More informationWavelet Footprints: Theory, Algorithms, and Applications
1306 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 5, MAY 2003 Wavelet Footprints: Theory, Algorithms, and Applications Pier Luigi Dragotti, Member, IEEE, and Martin Vetterli, Fellow, IEEE Abstract
More informationUnsupervised Learning with Permuted Data
Unsupervised Learning with Permuted Data Sergey Kirshner skirshne@ics.uci.edu Sridevi Parise sparise@ics.uci.edu Padhraic Smyth smyth@ics.uci.edu School of Information and Computer Science, University
More informationMultiscale Image Transforms
Multiscale Image Transforms Goal: Develop filter-based representations to decompose images into component parts, to extract features/structures of interest, and to attenuate noise. Motivation: extract
More informationEdges and Scale. Image Features. Detecting edges. Origin of Edges. Solution: smooth first. Effects of noise
Edges and Scale Image Features From Sandlot Science Slides revised from S. Seitz, R. Szeliski, S. Lazebnik, etc. Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity
More informationSparse linear models
Sparse linear models Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda 2/22/2016 Introduction Linear transforms Frequency representation Short-time
More informationLecture Notes 5: Multiresolution Analysis
Optimization-based data analysis Fall 2017 Lecture Notes 5: Multiresolution Analysis 1 Frames A frame is a generalization of an orthonormal basis. The inner products between the vectors in a frame and
More informationDiscriminative Direction for Kernel Classifiers
Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering
More informationModeling Multiscale Differential Pixel Statistics
Modeling Multiscale Differential Pixel Statistics David Odom a and Peyman Milanfar a a Electrical Engineering Department, University of California, Santa Cruz CA. 95064 USA ABSTRACT The statistics of natural
More informationOrientation Map Based Palmprint Recognition
Orientation Map Based Palmprint Recognition (BM) 45 Orientation Map Based Palmprint Recognition B. H. Shekar, N. Harivinod bhshekar@gmail.com, harivinodn@gmail.com India, Mangalore University, Department
More informationDetectors part II Descriptors
EECS 442 Computer vision Detectors part II Descriptors Blob detectors Invariance Descriptors Some slides of this lectures are courtesy of prof F. Li, prof S. Lazebnik, and various other lecturers Goal:
More informationThe New Graphic Description of the Haar Wavelet Transform
he New Graphic Description of the Haar Wavelet ransform Piotr Porwik and Agnieszka Lisowska Institute of Informatics, Silesian University, ul.b dzi ska 39, 4-00 Sosnowiec, Poland porwik@us.edu.pl Institute
More informationInvariant Scattering Convolution Networks
Invariant Scattering Convolution Networks Joan Bruna and Stephane Mallat Submitted to PAMI, Feb. 2012 Presented by Bo Chen Other important related papers: [1] S. Mallat, A Theory for Multiresolution Signal
More informationSteerable Wedge Filters for Local Orientation Analysis
IEEE Transactions on Image Processing, 5(9): 377-382, 996 Steerable Wedge Filters for Local Orientation Analysis Eero P Simoncelli and Hany Farid GRASP Laboratory, Dept of Computer & Information Science
More informationNew signal representation based on the fractional Fourier transform: definitions
2424 J. Opt. Soc. Am. A/Vol. 2, No. /November 995 Mendlovic et al. New signal representation based on the fractional Fourier transform: definitions David Mendlovic, Zeev Zalevsky, Rainer G. Dorsch, and
More informationPROBABILISTIC CONTINUOUS EDGE DETECTION USING LOCAL SYMMETRY
PROBABILISTIC CONTINUOUS EDGE DETECTION USING LOCAL SYMMETRY Gerald Mwangi, Paul Fieguth, Christoph S. Garbe : University of Heidelberg, Germany, : University of Waterloo, Canada ABSTRACT We describe a
More informationA Flexible Framework for Local Phase Coherence Computation
A Flexible Framework for Local Phase Coherence Computation Rania Hassen, Zhou Wang, and Magdy Salama Department of Electrical and Computer Engineering, University of Waterloo, 200 University Avenue West,
More informationMulti-Robotic Systems
CHAPTER 9 Multi-Robotic Systems The topic of multi-robotic systems is quite popular now. It is believed that such systems can have the following benefits: Improved performance ( winning by numbers ) Distributed
More informationStudy of Wavelet Functions of Discrete Wavelet Transformation in Image Watermarking
Study of Wavelet Functions of Discrete Wavelet Transformation in Image Watermarking Navdeep Goel 1,a, Gurwinder Singh 2,b 1ECE Section, Yadavindra College of Engineering, Talwandi Sabo 2Research Scholar,
More informationA NO-REFERENCE SHARPNESS METRIC SENSITIVE TO BLUR AND NOISE. Xiang Zhu and Peyman Milanfar
A NO-REFERENCE SARPNESS METRIC SENSITIVE TO BLUR AND NOISE Xiang Zhu and Peyman Milanfar Electrical Engineering Department University of California at Santa Cruz, CA, 9564 xzhu@soeucscedu ABSTRACT A no-reference
More informationWavelet Packet Based Digital Image Watermarking
Wavelet Packet Based Digital Image ing A.Adhipathi Reddy, B.N.Chatterji Department of Electronics and Electrical Communication Engg. Indian Institute of Technology, Kharagpur 72 32 {aar, bnc}@ece.iitkgp.ernet.in
More informationA Robust Modular Wavelet Network Based Symbol Classifier
A Robust Modular Wavelet Network Based Symbol Classifier A.K. Mishra, P.W. Fieguth, D.A. Clausi University of Waterloo, Vision and Image Processing Group akmishra, pfieguth, dclausi@uwaterloo.ca Abstract.
More informationA Contrario Detection of False Matches in Iris Recognition
A Contrario Detection of False Matches in Iris Recognition Marcelo Mottalli, Mariano Tepper, and Marta Mejail Departamento de Computación, Universidad de Buenos Aires, Argentina Abstract. The pattern of
More informationarxiv: v1 [physics.comp-ph] 22 Jul 2010
Gaussian integration with rescaling of abscissas and weights arxiv:007.38v [physics.comp-ph] 22 Jul 200 A. Odrzywolek M. Smoluchowski Institute of Physics, Jagiellonian University, Cracov, Poland Abstract
More informationCamera Models and Affine Multiple Views Geometry
Camera Models and Affine Multiple Views Geometry Subhashis Banerjee Dept. Computer Science and Engineering IIT Delhi email: suban@cse.iitd.ac.in May 29, 2001 1 1 Camera Models A Camera transforms a 3D
More informationAruna Bhat Research Scholar, Department of Electrical Engineering, IIT Delhi, India
International Journal of Scientific Research in Computer Science, Engineering and Information Technology 2017 IJSRCSEIT Volume 2 Issue 6 ISSN : 2456-3307 Robust Face Recognition System using Non Additive
More information1 Kalman Filter Introduction
1 Kalman Filter Introduction You should first read Chapter 1 of Stochastic models, estimation, and control: Volume 1 by Peter S. Maybec (available here). 1.1 Explanation of Equations (1-3) and (1-4) Equation
More informationWavelet Transform And Principal Component Analysis Based Feature Extraction
Wavelet Transform And Principal Component Analysis Based Feature Extraction Keyun Tong June 3, 2010 As the amount of information grows rapidly and widely, feature extraction become an indispensable technique
More information2D Wavelets. Hints on advanced Concepts
2D Wavelets Hints on advanced Concepts 1 Advanced concepts Wavelet packets Laplacian pyramid Overcomplete bases Discrete wavelet frames (DWF) Algorithme à trous Discrete dyadic wavelet frames (DDWF) Overview
More informationMultiscale Principal Components Analysis for Image Local Orientation Estimation
Multiscale Principal Components Analsis for Image Local Orientation Estimation XiaoGuang Feng Peman Milanfar Computer Engineering Electrical Engineering Universit of California, Santa Cruz Universit of
More informationSingle Channel Signal Separation Using MAP-based Subspace Decomposition
Single Channel Signal Separation Using MAP-based Subspace Decomposition Gil-Jin Jang, Te-Won Lee, and Yung-Hwan Oh 1 Spoken Language Laboratory, Department of Computer Science, KAIST 373-1 Gusong-dong,
More informationScience Insights: An International Journal
Available online at http://www.urpjournals.com Science Insights: An International Journal Universal Research Publications. All rights reserved ISSN 2277 3835 Original Article Object Recognition using Zernike
More informationToward a Realization of Marr s Theory of Primal Sketches via Autocorrelation Wavelets: Image Representation using Multiscale Edge Information
Toward a Realization of Marr s Theory of Primal Sketches via Autocorrelation Wavelets: Image Representation using Multiscale Edge Information Naoki Saito 1 Department of Mathematics University of California,
More informationOrientation Estimation in Ambiguous Neighbourhoods Mats T. Andersson & Hans Knutsson Computer Vision Laboratory Linköping University 581 83 Linköping, Sweden Abstract This paper describes a new algorithm
More informationThe Hermite Transform: An Alternative Image Representation Model for Iris Recognition
The Hermite Transform: An Alternative Image Representation Model for Iris Recognition Alfonso Estudillo-Romero and Boris Escalante-Ramirez Universidad Nacional Autonoma de Mexico, Fac. de Ingenieria, Edif.
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Spring 2014 TTh 14:30-15:45 CBC C313 Lecture 05 Image Processing Basics 13/02/04 http://www.ee.unlv.edu/~b1morris/ecg782/
More informationAn Introduction to Filterbank Frames
An Introduction to Filterbank Frames Brody Dylan Johnson St. Louis University October 19, 2010 Brody Dylan Johnson (St. Louis University) An Introduction to Filterbank Frames October 19, 2010 1 / 34 Overview
More informationGabor wavelet analysis and the fractional Hilbert transform
Gabor wavelet analysis and the fractional Hilbert transform Kunal Narayan Chaudhury and Michael Unser (presented by Dimitri Van De Ville) Biomedical Imaging Group, Ecole Polytechnique Fédérale de Lausanne
More informationKazuhiro Fukui, University of Tsukuba
Subspace Methods Kazuhiro Fukui, University of Tsukuba Synonyms Multiple similarity method Related Concepts Principal component analysis (PCA) Subspace analysis Dimensionality reduction Definition Subspace
More informationMulti-Layer Boosting for Pattern Recognition
Multi-Layer Boosting for Pattern Recognition François Fleuret IDIAP Research Institute, Centre du Parc, P.O. Box 592 1920 Martigny, Switzerland fleuret@idiap.ch Abstract We extend the standard boosting
More informationFARSI CHARACTER RECOGNITION USING NEW HYBRID FEATURE EXTRACTION METHODS
FARSI CHARACTER RECOGNITION USING NEW HYBRID FEATURE EXTRACTION METHODS Fataneh Alavipour 1 and Ali Broumandnia 2 1 Department of Electrical, Computer and Biomedical Engineering, Qazvin branch, Islamic
More informationDevelopment and Applications of Wavelets in Signal Processing
Development and Applications of Wavelets in Signal Processing Mathematics 097: Senior Conference Paper Published May 014 David Nahmias dnahmias1@gmailcom Abstract Wavelets have many powerful applications
More informationDynamic Data Modeling, Recognition, and Synthesis. Rui Zhao Thesis Defense Advisor: Professor Qiang Ji
Dynamic Data Modeling, Recognition, and Synthesis Rui Zhao Thesis Defense Advisor: Professor Qiang Ji Contents Introduction Related Work Dynamic Data Modeling & Analysis Temporal localization Insufficient
More informationFast Nonnegative Matrix Factorization with Rank-one ADMM
Fast Nonnegative Matrix Factorization with Rank-one Dongjin Song, David A. Meyer, Martin Renqiang Min, Department of ECE, UCSD, La Jolla, CA, 9093-0409 dosong@ucsd.edu Department of Mathematics, UCSD,
More informationA UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS
A UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS Kai Krajsek J. W. Goethe University, Visual Sensorics and Information Processing Lab Robert Mayer Str.2-4, D-60054 Frankfurt am Main, Germany krajsek@vsi.cs.uni-frankfurt.de
More informationFFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations
FFTs in Graphics and Vision Homogenous Polynomials and Irreducible Representations 1 Outline The 2π Term in Assignment 1 Homogenous Polynomials Representations of Functions on the Unit-Circle Sub-Representations
More informationImage Similarity Test Using Eigenface Calculation
2017 IJSRST Volume 3 Issue 6 Print ISSN: 2395-6011 Online ISSN: 2395-602X Themed Section: Science and Technology Image Similarity Test Using Eigenface Calculation Nadya Andhika Putri 1, Andysah Putera
More informationDistance Between Ellipses in 2D
Distance Between Ellipses in 2D David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To
More informationOverview. Introduction to local features. Harris interest points + SSD, ZNCC, SIFT. Evaluation and comparison of different detectors
Overview Introduction to local features Harris interest points + SSD, ZNCC, SIFT Scale & affine invariant interest point detectors Evaluation and comparison of different detectors Region descriptors and
More informationShape of Gaussians as Feature Descriptors
Shape of Gaussians as Feature Descriptors Liyu Gong, Tianjiang Wang and Fang Liu Intelligent and Distributed Computing Lab, School of Computer Science and Technology Huazhong University of Science and
More informationIN many image processing applications involving wavelets
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM 2007 1 Phase-Shifting for Non-separable 2D Haar Wavelets (This correspondence extends our earlier paper also under review with IEEE TIP to 2D non-separable
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx
More informationUsing Skew Gabor Filter in Source Signal Separation and Local Spectral Multi-Orientation Analysis
Using Skew Gabor Filter in Source Signal Separation and Local Spectral Multi-Orientation Analysis Weichuan Yu Gerald Sommer Kostas Daniilidis 3 Dept. of Diagnostic Radiology Institut für Informatik 3 GRASP
More informationVlad Estivill-Castro (2016) Robots for People --- A project for intelligent integrated systems
1 Vlad Estivill-Castro (2016) Robots for People --- A project for intelligent integrated systems V. Estivill-Castro 2 Perception Concepts Vision Chapter 4 (textbook) Sections 4.3 to 4.5 What is the course
More informationwhere =0,, 1, () is the sample at time index and is the imaginary number 1. Then, () is a vector of values at frequency index corresponding to the mag
Efficient Discrete Tchebichef on Spectrum Analysis of Speech Recognition Ferda Ernawan and Nur Azman Abu Abstract Speech recognition is still a growing field of importance. The growth in computing power
More informationAn Improved Cumulant Based Method for Independent Component Analysis
An Improved Cumulant Based Method for Independent Component Analysis Tobias Blaschke and Laurenz Wiskott Institute for Theoretical Biology Humboldt University Berlin Invalidenstraße 43 D - 0 5 Berlin Germany
More informationReflections and Rotations in R 3
Reflections and Rotations in R 3 P. J. Ryan May 29, 21 Rotations as Compositions of Reflections Recall that the reflection in the hyperplane H through the origin in R n is given by f(x) = x 2 ξ, x ξ (1)
More informationMATH 320, WEEK 7: Matrices, Matrix Operations
MATH 320, WEEK 7: Matrices, Matrix Operations 1 Matrices We have introduced ourselves to the notion of the grid-like coefficient matrix as a short-hand coefficient place-keeper for performing Gaussian
More informationNotes on anisotropic multichannel representations for early vision
Notes on anisotropic multichannel representations for early vision Silvio P. Sabatini Department of Biophysical and Electronic Engineering, University of Genoa 1 Introduction Although the basic ideas underlying
More informationSimple Cell Receptive Fields in V1.
Simple Cell Receptive Fields in V1. The receptive field properties of simple cells in V1 were studied by Hubel and Wiesel [65][66] who showed that many cells were tuned to the orientation of edges and
More informationMicro-Doppler based Recognition of Ballistic Targets using 2D Gabor Filters
Micro-Doppler based ecognition of Ballistic Targets using 2D Gabor Filters Adriano osario Persico, armine lemente, hristos V. Ilioudis, Domenico Gaglione, Jianlin ao and John Soraghan University of Strathclyde,
More informationLie Groups for 2D and 3D Transformations
Lie Groups for 2D and 3D Transformations Ethan Eade Updated May 20, 2017 * 1 Introduction This document derives useful formulae for working with the Lie groups that represent transformations in 2D and
More informationImage representation with multi-scale gradients
Image representation with multi-scale gradients Eero P Simoncelli Center for Neural Science, and Courant Institute of Mathematical Sciences New York University http://www.cns.nyu.edu/~eero Visual image
More informationNeural Network Training
Neural Network Training Sargur Srihari Topics in Network Training 0. Neural network parameters Probabilistic problem formulation Specifying the activation and error functions for Regression Binary classification
More informationAlgorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 1 / 25 Algorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations Dmitry A. Lyakhov 1 1 Computational Sciences
More informationStatistics of Natural Image Sequences: Temporal Motion Smoothness by Local Phase Correlations
Statistics of Natural Image Sequences: Temporal Motion Smoothness by Local Phase Correlations Zhou Wang and Qiang Li Dept. of Electrical & Computer Engineering, University of Waterloo, Waterloo, ON, Canada
More informationBasic Concepts of. Feature Selection
Basic Concepts of Pattern Recognition and Feature Selection Xiaojun Qi -- REU Site Program in CVMA (2011 Summer) 1 Outline Pattern Recognition Pattern vs. Features Pattern Classes Classification Feature
More informationCSC 576: Variants of Sparse Learning
CSC 576: Variants of Sparse Learning Ji Liu Department of Computer Science, University of Rochester October 27, 205 Introduction Our previous note basically suggests using l norm to enforce sparsity in
More informationFace Detection and Recognition
Face Detection and Recognition Face Recognition Problem Reading: Chapter 18.10 and, optionally, Face Recognition using Eigenfaces by M. Turk and A. Pentland Queryimage face query database Face Verification
More informationAccurate Orthogonal Circular Moment Invariants of Gray-Level Images
Journal of Computer Science 7 (5): 75-722, 20 ISSN 549-3636 20 Science Publications Accurate Orthogonal Circular Moment Invariants of Gray-Level Images Khalid Mohamed Hosny Department of Computer Science,
More informationNoise Robust Isolated Words Recognition Problem Solving Based on Simultaneous Perturbation Stochastic Approximation Algorithm
EngOpt 2008 - International Conference on Engineering Optimization Rio de Janeiro, Brazil, 0-05 June 2008. Noise Robust Isolated Words Recognition Problem Solving Based on Simultaneous Perturbation Stochastic
More informationTWO METHODS FOR ESTIMATING OVERCOMPLETE INDEPENDENT COMPONENT BASES. Mika Inki and Aapo Hyvärinen
TWO METHODS FOR ESTIMATING OVERCOMPLETE INDEPENDENT COMPONENT BASES Mika Inki and Aapo Hyvärinen Neural Networks Research Centre Helsinki University of Technology P.O. Box 54, FIN-215 HUT, Finland ABSTRACT
More informationA Generative Perspective on MRFs in Low-Level Vision Supplemental Material
A Generative Perspective on MRFs in Low-Level Vision Supplemental Material Uwe Schmidt Qi Gao Stefan Roth Department of Computer Science, TU Darmstadt 1. Derivations 1.1. Sampling the Prior We first rewrite
More informationWiener System Identification with Four-Segment and Analytically Invertible Nonlinearity Model
Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 FrA4.2 Wiener System Identification with Four-Segment and Analytically Invertible
More informationComplements on Simple Linear Regression
Complements on Simple Linear Regression Terry R. McConnell Syracuse University March 16, 2015 Abstract We present a simple-minded approach to a variant of simple linear regression that seeks to minimize
More informationOn the Equivariance of the Orientation and the Tensor Field Representation Klas Nordberg Hans Knutsson Gosta Granlund Computer Vision Laboratory, Depa
On the Invariance of the Orientation and the Tensor Field Representation Klas Nordberg Hans Knutsson Gosta Granlund LiTH-ISY-R-530 993-09-08 On the Equivariance of the Orientation and the Tensor Field
More informationCS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33
Linear Algebra 1/33 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct
More information8. Diagonalization.
8. Diagonalization 8.1. Matrix Representations of Linear Transformations Matrix of A Linear Operator with Respect to A Basis We know that every linear transformation T: R n R m has an associated standard
More informationDESIGN OF MULTI-DIMENSIONAL DERIVATIVE FILTERS. Eero P. Simoncelli
Published in: First IEEE Int l Conf on Image Processing, Austin Texas, vol I, pages 790--793, November 1994. DESIGN OF MULTI-DIMENSIONAL DERIVATIVE FILTERS Eero P. Simoncelli GRASP Laboratory, Room 335C
More informationTopics in Harmonic Analysis, Sparse Representations, and Data Analysis
Topics in Harmonic Analysis, Sparse Representations, and Data Analysis Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu Thesis Defense,
More informationDistinguishing Causes from Effects using Nonlinear Acyclic Causal Models
JMLR Workshop and Conference Proceedings 6:17 164 NIPS 28 workshop on causality Distinguishing Causes from Effects using Nonlinear Acyclic Causal Models Kun Zhang Dept of Computer Science and HIIT University
More informationMachine Learning! in just a few minutes. Jan Peters Gerhard Neumann
Machine Learning! in just a few minutes Jan Peters Gerhard Neumann 1 Purpose of this Lecture Foundations of machine learning tools for robotics We focus on regression methods and general principles Often
More informationGroup Sparse Non-negative Matrix Factorization for Multi-Manifold Learning
LIU, LU, GU: GROUP SPARSE NMF FOR MULTI-MANIFOLD LEARNING 1 Group Sparse Non-negative Matrix Factorization for Multi-Manifold Learning Xiangyang Liu 1,2 liuxy@sjtu.edu.cn Hongtao Lu 1 htlu@sjtu.edu.cn
More information