Positive Signal Spaces and the Mehler-Fock Transform

07/11/2017
Auteurs : Reiner Lenz
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22359
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit
 

Résumé

Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.

Positive Signal Spaces and the Mehler-Fock Transform

Collection

application/pdf Positive Signal Spaces and the Mehler-Fock Transform Reiner Lenz
Détails de l'article
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.
Positive Signal Spaces and the Mehler-Fock Transform

Média

Voir la vidéo

Métriques

0
0
1.24 Mo
 application/pdf
bitcache://5a8f23c9bede9cd024904e1d4e2771dae120f4a9

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors Platine

alanturinginstitutelogo.png
logothales.jpg

Sponsors Bronze

logo_enac-bleuok.jpg
imag150x185_couleur_rvb.jpg

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors

smai.png
logo_gdr-mia.png
gdr_geosto_logo.png
gdr-isis.png
logo-minesparistech.jpg
logo_x.jpeg
springer-logo.png
logo-psl.png

Organisateurs

logo_see.gif
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/17410/22359</identifier><creators><creator><creatorName>Reiner Lenz</creatorName></creator></creators><titles>
            <title>Positive Signal Spaces and the Mehler-Fock Transform</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2018</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Sun 18 Feb 2018</date>
	    <date dateType="Updated">Sun 18 Feb 2018</date>
            <date dateType="Submitted">Mon 22 Oct 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">5a8f23c9bede9cd024904e1d4e2771dae120f4a9</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>37014</version>
        <descriptions>
            <description descriptionType="Abstract">Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.
</description>
        </descriptions>
    </resource>
.

Positive Signal Spaces and the Mehler-Fock Transform Reiner Lenz Linköping University, SE-58183 Linköping, Sweden reiner.lenz@liu.se Abstract. Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a ra- dial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as sym- metry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The con- structed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual proper- ties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribu- tion. 1 Introduction and Overview Many applications are based on measurements which have only positive values. Length, weight and age/duration are some common examples. In the following we will consider a theoretical framework in which a signal s is an element in a Hilbert space H . We also have a number of subspaces Hk, (k = 1, . . . K) and ck is the length of the projection of s in Hk. Being a length measurement (and assum- ing that the signal is never perfect) we assume that for all k the ck are positive. In the following the space H is the nite-dimensional vector space of gray value distributions on a collections of pixels. The subspaces Hk are given by the ir- reducible representations of the underlying symmetry group of the pixel grid. Descriptions of group representations can be found in [3,4,13,8,9,2]. Applying principal component analysis (PCA) and using the Perron-Frobenius theorem one can introduce a conical coordinate system in which the rst coordinate cor- responds to the rst eigenvector coecient. Using perspective projection along 2 this axis shows that the remaining coordinates describe a solid unit ball. The angular part of the new system can be analyzed with the help of spherical tech- niques like expansion in surface harmonics whereas the radial part can be studied with the help of the Mehler-Fock-Transform (MFT) which corresponds to the Fourier transform in the case where the underlying transformation group is the group of hyperbolic rotations. Three-dimensional color spaces (such as the com- mon RGB-system) have the same structure. Here the main axis of the cone describes the gray colors and the projection the intensity. The solid ball is the unit disk where the radius measures saturation and the angle corresponds to hue. In the second contribution of this paper the conical framework is used to investigate statistical properties of image patches from calibrated images taken in a habitat similar to the one in which the human eye developed. The con- struction leads to texture descriptors which are orientation invariant (due to the usage of the edge-magnitudes) and independent of overall intensity changes (be- cause the perspective projection). They are also independent of the distributions between the higher order eigenvectors since only the radial variation in the unit ball is analyzed. The results show that the family of generalized-extreme-value distributions (GEV's) provide a class of probability distributions that give good tting results for these measurements. In the third contribution the group of hyperbolic rotations is used as natural class of transformations acting on the radial variable. This is a one-parameter subgroup of the Lorentz group and the corresponding Fourier transform in this variable is the Mehler-Fock transform. The study of the relation between the visual properties of a patch and the properties of the MFT are the topic of the nal experiments. 2 Conical structure of positive signals Fourier related techniques (like the continuous Fourier transform (CFT), the dis- crete (DFT) and the fast Fourier transform(FFT)) are some of the most powerful signal processing tools. For images whose pixels are located on a square grid the corresponding symmetry group is the dihedral group D(4) and the Fourier trans- form are the dihedral lters. A special class of dihedral lters corresponds to the traditional edge detectors in low-level image processing. These edge lters come in pairs and correspond to the two-dimensional irreducible representations of the dihedral group. The magnitude of a pair of such lter results corresponds to the projection length mentioned above. In the following the space of all gray value distributions on a 5x5 window is used. Its dimension is 25, the six edge detector pairs span a 12-dimensional subspace and dene six edge magnitudes. Applica- tion of the dihedral ltering results thus for every pixel x in a six-dimensional vector f (x) with non-negative elements (the value zero requires all underlying pixel values to be zero). Next the 6-D (local) mean vector m (x) and the 6×6 matrix C (x) of (local) second order moments of f (x) are computed. The matrix C (x) is symmetric, 3 positive-denite and positive and the Perron-Frobenius theorem shows that the eigenvector b0(x) belonging to the highest eigenvalue has only positive entries. At pixel x a new coordinate system b0(x), b1(x), . . . , b5(x) is dened spanned by the eigenvectors of C (x) (ordered by the value of the eigenvalues). This gives the expansion of the vector f (x) = P5 k=0 hf (x), bk(x)i bk(x) = P5 k=0 ck(x) bk(x), where h., .i denotes the scalar product. The vectors f (x) and b0(x) have both positive entries and their scalar product is therefore positive and the ratios qk(x) = ck(x) c0(x) , k = 1, 2 are well- dened. Eigenvectors are orthogonal and the eigenvectors bk(x), k > 0 must have negative elements. In general one can expect that c0(x) > 0 will be big and |ck(x)|, k > 0 will be small. Furthermore, the values qk(x) are invariant under scaling of the original pixel values. If q(x) denotes the vector containing these ratios then it is assumed that its norm is bounded by some value R in- dependent of x. This is the case for all examples used later. Therefore polar coordinates (ρ(x) , θ(x)) with (q1(x) , q2(x)) = (Rρ(x) cos θ(x) , Rρ(x) sin θ(x)) can be used. The result describes f (x) in the coordinate system (c0(x) , ρ(x) , θ(x)) where c0(x) measures the projection along the mean-direction, ρ(x) is the distance from the mean-vector in the space spanned by the second and third eigenvector and θ(x) depends on the relation between the second and third eigenvector. In human color perception this corresponds to a characterization in terms of intensity c0(x), saturation ρ(x) and hue θ(x) (more information can be found in [11] and [10]). This construction is not limited to the analysis of the rst three eigenvector coecients. Using more then three components only leads to the replacement of the angular variable by spherical coordinates. 3 The action of the group SU(1, 1) In a group theoretical context the groups used so far are: the group D(4) related to the sensor array. The scaling group R+ acting by multiplication on c0(x) and the group SU(1, 1) acting on the points on the unit disk. The details of this constructions are as follows: points on the unit disk are complex variables z and the group SU(1, 1) is dened as the 2 × 2 matrices with complex elements: SU(1, 1) =  M =  a b b a  , a, b ∈ C, |a| 2 − |b| 2 = 1 (1) The group operation is the usual matrix multiplication and the group acts as a transformation group on the open unit disk D (consisting of all points z ∈ C with |z| < 1) as the Möbius transforms: (M, z) =  a b b a  , z  7→ M hzi = az + b bz + a , z ∈ D (2) with (M1M2) hzi = M1 hM2 hzii for all matrices and all points. The notation M will be used when the group elements are represented as matrices. otherwise the 4 symbol g is used. An ordinary three-dimensional rotation can be written as a product of three rotations around the coordinate axes. A similar decomposition holds also for the group SU(1, 1). Denoting the three parameters by ψ, τ, ϕ the decomposition is given as: g(ϕ, τ, ψ) = g(ϕ, 0, 0)g(0, τ, 0)g(0, 0, ψ) =  cosh τ 2 ei(ϕ+ψ)/2 sinh τ 2 ei(ϕ−ψ)/2 sinh τ 2 e−i(ϕ−ψ)/2 cosh τ 2 e−i(ϕ+ψ)/2  (3) Introducing the two one-parameter subgroups K =  g(ϕ, 0, 0) =  eiϕ/2 0 0 e−iϕ/2  : −2π ≤ ϕ < 2π (4) A =  g(0, τ, 0) =  cosh τ 2 sinh τ 2 sinh τ 2 cosh τ 2  : τ ∈ R . (5) shows that g(ϕ, 0, 0), g(0, 0, ψ) ∈ K and g(0, τ, 0) ∈ A. This is known as the Cartan or the polar decomposition of the group and the ψ, τ, ϕ are the Cartan coordinates. The elements in K are rotations and leave the origin xed. For a general element in SU(1, 1) the Cartan decomposition gives: g(ϕ, τ, ψ) h0i = g(ϕ, 0, 0)g(0, τ, 0)g(0, 0, ψ) h0i = tanh τ 2 eiϕ . This shows that D = SU(1, 1)/K and functions on the unit disk are functions on the group that are independent of the last argument of the Cartan decompo- sition. Next consider two points on the unit disk, corresponding to group elements g = g(ϕ0, τ0, 0) and h = g(ϕ1, τ1, 0). The dierence between these two elements is given by h−1 g with its own decomposition h−1 g = g(ϕ, τ, 0) with g(ϕ, τ, 0) = g(ϕ1, τ1, 0)−1 g(ϕ0, τ0, 0) = g(0, −τ1, 0)g(ϕ1 −ϕ0, τ0, 0). In [17], (Vol 1, page 271) the following relation between the parameters of the three group elements is derived: cosh τ = cosh τ1 cosh τ0 + sinh τ1 sinh τ0 cos(ϕ1 − ϕ0). (6) For SU(1, 1) the role of the exponential function is played by the associated Legendre functions (zonal or Mehler functions, [17], page 324) of order m and degree α = −1/2 + iκ. They are dened as (see Eq.(7)): Pm α (cosh τ) = 1 2π Γ(α + m + 1) Γ(α + 1) Z 2π 0 (sinh τ cos θ + cosh τ) α eimθ dθ (7) and satisfy the addition formula ([17], page 327) Pα(cosh τl cosh τ0 + sinh τl sinh τ0 cos θ)= X m∈Z P−m α (cosh τl) Pm α (cosh τ0) e−imθ . (8) The transform for SU(1, 1) corresponding to the Fourier transform is the Mehler-Fock transform. The following theorem shows that a large class of func- tions are combinations of the associated Legendre functions: 5 Theorem 1 (Mehler-Fock Transform; MFT) For a function k dened on the interval [1, ∞) dene its transform c as: c(κ) = Z ∞ 0 k (cosh τ)P−1/2+iκ (cosh τ) sinh τ dτ (9) Then k can be recovered by the inverse transform: k (cosh τ) = Z ∞ 0 κ tanh(πκ)P−1/2+iκ (cosh τ)c(κ) dκ (10) Details about the transform, special cases and its applications can be found in [14],Sec.7.6, [17],[15],[7] and [1]. The MFT also preserves the scalar product (Parseval relation): using the parametrization x = cosh τ and dening cn(κ) = R ∞ 1 fn(x)P1/2+iκ(x) dx gives (see [14](7.6.16) and [14](7.7.1)): Z ∞ 0 c1(κ)c2(κ)κ tanh(πκ) dκ = Z ∞ 1 f1(x)f2(x) dx (11) One possible application of the MFT in statistics is the application as a tool to study parametric probability distributions. Using its connection to group convolutions it can also be used to simplify kernel density estimators of densities dened on the positive half-axis. In the following it will be used to investigate generalized-extreme-value distribution based models of natural image statistics. Kernel-density estimators will be discussed elsewhere. 4 Implementation An implementation of the transform has to take into account at least three prob- lems: (1) the computation of the associated Legendre functions, (2) numerical evaluation of innite integrals and (3) sampling schemes in the signal and trans- form domain. The denition of the associated Legendre functions using Eq.(7) is not useful, instead the relations to other special functions can be used (see [6, 12]). In the following the LegendreP function in Mathematica is used. Compu- tation of the innite integrals involved is dicult in general. In the experiments the density functions of the generalized-extreme-value distribution are used. For these functions the quantiles are known and the integration domain selected ex- tends to the maximum value of the 0.99 quantiles for all distributions in the database. Sampling is another factor to be taken into account. Since the main underlying structure is the one-parameter group A a linear sampling scheme in the signal and the transform domain is chosen. Integrals are computed as scalar products of vectors. The results were compared to the application of numerical integration methods (NIntegrate in Mathematica) which conrmed the reliabil- ity of the results for the functions involved in this study. In all the numerical results reported in the following a matrix of size 101 × 301 was used where 301 sampling points were used to sample the pdfs and 101 points sampling the trans- form domain. The range of the transform domain was determined manually. 6 5 The Botswana Dataset Statistical properties of images are of interest in the study of biological vision systems and technical applications such as image restoration and image classi- cation and compression (see [5] for a review of natural image statistics research). The following experiments make use of a large database of natural images1 and described in [16]). Some of the main characteristics of the database are the following: it contains 5677 calibrated natural images, collected in a single environment: a savanna habitat in Botswana which is thought to be similar to the environment where the human eye evolved. The images in the database are organized in 103 folders (albums) containing images of a common theme. There are 61 albums characterized as scenes and 42 albums showing objects. The original images were taken with a Nikon D70 camera which was calibrated to take into account the optical and signal processing properties of the camera. The images in the LMS format were downloaded. They represent the input to the long/medium/short wavelength sensitive cones in the human retina and the M-channel which is most important for lightness perception was used. The raw images are ltered with the six edge-detection lter pairs and the results combined in the six dimensional vector with the edge-magnitude values. Then the outer product is computed at every pixel and the (matrix valued) image is ltered with a Gaussian lter kernel of size 15 × 15. The result is a matrix valued image with the empirical second-order moments. Next the (normalized) projection variables ρ(x) are computed and converted to hyperbolic form. For every image the 128×128 patch at the center of the image is selected. Pixels with zero-valued rst eigenvectors (produced by constant 5x5 patches) are ignored and only patches with more than 500 valid measurements were evaluated. This resulted in 5560 patches used in the following. The quality of the tting was measured with the adjusted-RSquared value and the lowest values found was 0.9623. The mean value over all 5560 patches was 0.998. As illustration an example with an R-Squared value nearest to the mean value of 0.998 is selected. In Fig. 1 the original scene (cd20A, DSC-0055), the selected patch, the image of the group parameters in the projection, the histogram and the GEV-tting and nally the MFT of the distribution are shown. The next three gures illustrate the eect of an operation similar to low-pass ltering in traditional signal processing. Here the rst 10 MFT coecients were extracted. Then the scalar product (see Eq. 11) in the transform domain was used to compute the length of the projection of a patch in this MFT-region. The patches were then ordered and in Figure 2 the patches, the values of the projections and three distributions are shown. The left and the middle column show 3x5 patches. The patches on top are the ve patches with lowest projection value in the MFT-bands, the patches in the middle correspond to positions 251 to 255 and the lowest ve are number 4001 to 4005. The pixel values in these images are all scaled such that the 0.9 quantile of the values in all patches has maximum gray value. The left column (Fig. 2(a)) with the original patches 1 available at http://tofu.psych.upenn.edu/~upennidb 7 (a) Scene (b) Patch (c) Projection 0 0.5 1 1.5 Hyperbolic Data 0 0.5 1 1.5 2 2.5 3 Density rhocd02A53 data GEV-Fit (d) Distribution/GEV-Fit 0 50 100 150 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 MFT( ) Mehler Fock Transform (e) MFT Fig. 1. Scene (cd02A; DSC-0055-LMS) show that the projection eliminates global intensity changes: Some parts of the patches are almost black and others have maximum gray value. The patches in the center of the gure (Fig. 2(b)) show that low contributions to the lower MFT channels correspond to low lter values (high probability of low edge magnitude responses) and the patches at the top are therefore more homogeneous than the others. The plot in the right column (Fig. 2(c)) shows the histograms and the tted GEV-distributions for the three left-most patches. This illustrates the strength of the GEV-approach since the shapes of the three distributions vary widely and the locations of the distributions indicate the increasing value of the projection values of the three images. (a) Patches (b) Hyperbolic data 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Hyperbolic Data 0 1 2 3 4 5 6 Density Low (1) FitLow Mid (250) FitMid High (4000) FitHigh (c) Fitting Fig. 2. Bandpass Filtering 6 Summary and Conclusions Symmetry-based methods entered the study of positive signals at several levels: rst the representation theory of the symmetry group of the sensor is used to split 8 the signal space into dierent components. Then the lengths of the projections in the dierent subspaces form another signal space of positive signals which in turn can be represented as a product of a half-line and a unit ball. The half-axis is related to scaling operations modeled by the scaling group and the unit-ball can be analyzed with the help of Lorentz groups. The description here is only an illustration of how group theoretical tools can be used to analyze the properties of signal spaces. Other applications, for example the study of kernel-density estimators and the usage of the full Lorentz group, are under investigation and will be presented elsewhere. References 1. Bateman, H., Erdélyi, A., States, U. (eds.): Tables of Integral Transforms, vol. 2. McGraw-Hill (1954) 2. Chirikjian, G., Kyatkin, A.: Harmonic Analysis for Engineers and Applied Scien- tists. Dover Publications (2016) 3. Diaconis, P.: Group representations in probability and statistics. JSTOR (1988) 4. Fässler, A., Stiefel, E.: Group theoretical methods and their applications. Birkäuser (1992) 5. Gerhard, H.E., Theis, L., Bethge, M.: Modeling natural image statistics. In: Biologically-inspired Computer Vision, Fundamentals and Applications. John Wi- ley & Sons (2015) 6. Gil, A., Segura, J., Temme, N.M.: Computing the conical function pµ −1/2+iτ (x). SIAM Journal on Scientic Computing 31(3), 17161741 (2009) 7. Lebedev, N.N.: Special functions and their applications. Dover, New York (1972) 8. Lenz, R.: Group Theoretical Methods in Image Processing. LNCS, Vol. 413, Springer Verlag (1990) 9. Lenz, R.: Investigation of receptive elds using representations of dihedral groups. J. Vis. Comm. Im. Repr. 6(3), 209227 (1995) 10. Lenz, R.: Lie methods for color robot vision. Robotica 26(4), 453464 (2008) 11. Lenz, R.: Spectral color spaces: Their structure and transformations. In: Advances in imaging and electron physics, vol. 138, pp. 167. Elsevier (2005) 12. Olver, F.W.J.: NIST handbook of mathematical functions. Cambridge University Press :, New York (2010) 13. Serre, J.: Linear representations of nite groups. Springer (2012) 14. Sneddon, I.N.: The use of integral transforms. McGraw-Hill New York (1972) 15. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications I. Springer (1985) 16. Tkacik, G., Garrigan, P., Ratli, C., Milcinski, G., Klein, J., Seyfarth, L., Sterling, P., Brainard, D., Balasubramanian, V.: Natural images from the birthplace of the human eye. PLoS ONE 6(6) (2011) 17. Vilenkin, N., Klimyk, A.: Representation of Lie Groups and Special Functions: Vol- ume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Springer (2012)