An affine Invariant Hyperspectral Texture Descriptor Based Upon Heavy-tailedDistributions and Fourier Analysis ∗

Pattaraporn Khuwuthyakorn1,2,3 and Antonio Robles-Kelly2,3 and Jun Zhou2,3

1Cooperative Research Centre for National Plant Biosecurity, Bruce ACT 2617, Australia2RSISE, Bldg. 115, Australian National University, Canberra ACT 0200, Australia

3National ICT Australia (NICTA†), Locked Bag 8001, Canberra ACT 2601, Australiau4420081, antonio.robles-kelly, jun.zhou@anu.edu.au

Abstract

In this paper, we address the problem of recovering ahyperspectral texture descriptor. We do this by viewing thewavelength-indexed bands corresponding to the texture inthe image as those arising from a stochastic process whosestatistics can be captured making use of the relationshipsbetween moment generating functions and Fourier kernels.In this manner, we can interpret the probability distributionof the hyper-spectral texture as a heavy-tailed one whichcan be rendered invariant to affine geometric transforma-tions on the texture plane making use of the spectral powerof its Fourier cosine transform. We do this by recoveringthe affine geometric distortion matrices corresponding tothe probability density function for the texture under study.This treatment permits the development of a robust descrip-tor which has a high information compaction property andcan capture the space and wavelength correlation for thespectra in the hyperspectral images. We illustrate the utilityof our descriptor for purposes of recognition and provideresults on real-world datasets. We also compare our resultsto those yielded by a number of alternatives.

1. Introduction

The development of image sensor technology has madeit possible to capture image data in hundreds of wavelength-resolved images covering a broad spectral range in the visi-ble and near-infrared range. The information-rich represen-tation of the object under study provided by hyperspectralimagery poses significant opportunities and challenges for

∗This research is supported by the Cooperative Research Centre forNational Plant Biosecurity (CRCNPB) grant: CRC60075 and NICTA.

†NICTA is funded by the Australian Government as represented by theDepartment of Broadband, Communications and the Digital Economy andthe Australian Research Council through the ICT Centre of Excellenceprogram.

localisation and classification tasks.

Hyperspectral imaging can be intuitively related to ma-terial chemistry so as to employ spectral signatures for pur-poses of classification and localisation. This hinges on thenotion that different materials have a characteristic responseas a function of wavelength which can be used so as to pro-vide a description of the object under study. So far, algo-rithms have been proposed for purposes of recognition andclassification based upon spectral imaging [6, 4]. Theseare often based upon subspace projection methods such asPrincipal Component Analysis, Linear Discriminant Anal-ysis, Decision Boundary , Projection Pursuit, and kernelmethods[10]. All these algorithms treat the raw pixel spec-tra as input vectors in high dimensional spaces and lookfor linear or nonlinear mappings to the feature space, of-ten with reduced dimensionality, by optimizing certain cri-terion, leading to statistically optimal solutions to classifi-cation.

Despite effective, the use of higher-level features, suchas those based upon texture or shape for purposes of recog-nition, classification and localisation in hyperspectral imag-ing are less common in the literature. Moreover, to ourknowledge, there are no hyperspectral texture descriptorselsewhere in the literature. In trichromatic imaging, texturehas found applications not only as a shape queue [7, 18],but has also attracted broad attention for recognition andclassification tasks [13]. Moreover, from the shape recov-ery perspective, static texture planes can be recovered mak-ing use of the structural analysis of predetermined textureprimitives [9, 1]. This treatment provides an intuitive geo-metrical meaning to the task of recovering the parametersgoverning the pose by making use of methods akin to 3Dview geometry.

In this paper, we focus in the development of a hyper-spectal texture descriptor. To do this, we use an approachbased upon the higher-order statistics of the texture understudy. This is somewhat akin to the treatment given to dy-

namic textures in computer vision, where image sequencesare modelled via probability density functions which ex-hibit first and second order time-shift invariant moments [5].As their static counterparts, dynamic textures have attractedconsiderable interest for purposes of recognition. [17, 19].

We propose a descriptor based upon the use of statisticsand Fourier kernels for purposes of capturing a discrimi-native and descriptive representation of the hyperspectraltextures in the scene. This provides a principled link be-tween statistical approaches, signal processing methods fortexture recognition and shape modeling approaches basedupon measures of spectral distortion [12, 15]. Hence, this isa means to both, a compact representation of the hyperspec-tral texture based upon its statistical characterisation andaffine invariance via the estimation of the transformationparameters on a locally planar texture plane. Here, we viewthe hyperspectral texture as arising from a heavy-tailed dis-tribution which can be related to a Fourier transform whosekernel, for purposes of efficiency has been set to a cosinefunction. This derives into a cosine transform which, wheneffected with respect to the pixel-coordinates can be relatedto the affine distortion of the texture. As a result, our de-scriptor is affine-invariant at the image-level and, at a wave-length level, captures the correlation of the moment gener-ating functions for each of the bands under study.

The paper is organised as follows. In Section 2, we in-troduce heavy tailed distributions and relate them to Fourierkernels. We also elaborate on the compactness of represen-tation making use of a cosine transform and provide a for-mulation for the local geometric distortion matrix employedhere for purposes of affine invariance. In Section 3, we turnour attention to the computation of the descriptor. In Sec-tion 4, we provide further discussion on our descriptor andits affine invariance. In Section 5 we show experimentalresults making use of our descriptor for hyperspectral tex-ture recognition on two real-world datasets. We also pro-vide comparison to alternatives elsewhere in the literatureand provide discussion on the application of the descriptorto other settings. Finally, we provide conclusion in Section6.

2. Heavy-tailed DistributionsHere, we view hyperspectral textures as stochastic pro-

cesses whose moment generating functions are invariantwith respect to shifts in the image-coordinates. That is, themean, covariance, kurtosis, etc. for the corresponding jointprobability distribution are required to be invariant with re-spect to changes of location on the image. Due to their den-sities have high dispersion, their probability density func-tions are governed by further-order moments. These intro-duces a number of statistical “skewness” variables that al-low high variability in terms of wavelength-dependent be-haviour. This implies that the spectra in the hyperspectral

texture results in spectral values that can be rather high interms of their deviation from the texture-spectra mean andvariance. This variation, which cannot be ignored in hy-perspectral data, is characteristic to the stochastic processgoverning the texture under study.

Thus, we formulate our descriptor so as to model “rare”stationary wavelength-dependent events on the image plane.This is reminiscent of simulation approaches where im-portance sampling cannot be effected via an exponentialchanges of measure due to the fact that the moments arenot exponential in nature. This applies to distributions suchas the log-normal, Weibull with increasing skewness andregularly varying distributions such as Pareto, stable andlog-gamma distributions [2]. We formulate the density ofthe pixel-values for the wavelength λ at the pixel u in theimage-band Iλ of the texture as random variables Yu whoseinherent basis Xu = χu(1), χu(2), . . . , χu(|Xu|) is suchthat

P (Yu) =|Xu|∑k=1

P (χu(k)) (1)

where, χu(k) are identically distributed variables and, asusual for probability distributions of real-valued variables,we have written P (Yu) = Pr[y ≤ Yu] for all y ∈ R.

In other words, we view the pixel values for each band inthe image for the texture under study as arising from a fam-ily of heavy-tailed distributions whose variance is not nec-essarily finite. It is worth noting in passing that, for finitevariance, the formalism above implies that P (Yu) is nor-mally distributed. Nonetheless, this treatment generalisesthe stochastic process to a number of independent influ-ences, each of which is captured by the corresponding vari-able χu(k).

In practice, the Probability Density Function (PDF)f(Yu) is not available in close form. As a result, we canre-parameterise the PDF recasting it as a function of thevariable ς making use of the characteristic function

ψ(ς) =∫ ∞

−∞exp(iςYu)f(Yu)dYu (2)

= exp(iuς − γ |ς|α (1 + iβ sign(ς)ϕ(ς, α))

)where i =

√−1, u is, as before, the pixel-index on the

image plane and γ ∈ R+ are function parameters, β ∈[−1, 1] and α ∈ (0, 2] are the skewness and characteristicexponent, respectively, and ϕ(.) is defined as follows

φ(ς, α) =

tan(απ

2 ) if α 6= 1−π

2 log|ς| if α = 1(3)

Note that, for the characteristic function above, α = 2 im-plies a normal distribution, β = 0 and α = 1 corresponds toa Cauchy distribution and, for the Levy distribution we have

2

α = 12 and β = 1. Thus, nonetheless the formalism above

can capture a number of cases in exponential families, it isstill quite general in nature so as to allow the modelling of alarge number of distributions that may apply to hyperspec-tral data.

Note that, so far, we have limited ourselves to the image-plane for a fixed wavelength λ. That is, we have concen-trated on the distribution of spectral values accross everywavelength-resolved band in the image. We can extendEquation 2 to the wavelength domain, i.e. the spectra ofthe texture across a segment of bands, by noting that theequation above is essentially a cross-correlation. Hence, wecan write the characteristic function for the texture parame-terised with respect to the wavelength λ as follows

ϑ(t) =∫ ∞

−∞

∫ ∞

−∞exp(iλς)exp(iςYu)f(Yu)dYudς

=∫ ∞

−∞exp(iλς)ψ(ς)dς (4)

which captures the spectral cross-correlation for the charac-teristic functions for each band.

In this manner, we view the characteristic function forthe hyperspectral texture as a heavy-tailed distribution ofanother set of heavy-tailed PDFs, which correspond to eachof the band in the image. This can also be interpreted as acomposition of two heavy-tailed distributions, where Equa-tion 2 corresponds to the image-band domain ς of the tex-ture. Equation 4 is then the wavelength-dependent domainλ. This composition operation also opens-up the possibil-ity of following a two-step process. Firstly, at band-level,the information is represented in a compact fashion and ren-dered invariant to geometric distortions on the texture plane.In the second step, wavelength-dependent correlation be-tween bands is captured making use of the operation in 4.

2.1. Compactness of Representation

To achieve a compact representation making use of theequations above, we refer to the fundamentals of integraltransforms [16]. We can view Equations 2 and 4 as char-acteristic functions obtained via the integral of the prod-uct of the function g(η), i.e. f(Yu) and ψ(ς), multipliedby a kernel K(ω, η), which above becomes exp(iλς) andexp(iςYu), respectively. As a result, we have

F (ω) =∫ ∞

−∞g(η)K(ω, η)dη (5)

where K(ω, η) is a Fourier kernel.To see the relation between Fourier transforms and the

equations in previous sections, we can examine ψ(ϕ) inmore detail and write

log[ψ(ς)] = iuς − γ | ς |α(1 + iβ sign(ς)ϕ(ς, α)

)= iuς− | ς |α γ∗α exp

(− iβ∗

π

2ϑsign(ς)

)

where ϑ = 1− | 1 − α | and the parameters γ∗ and β∗ aregiven by

β∗ =2πϑ

arccos(

cos(απ

2

)√

Ω

)γ∗ =

(γ√

Ωcos

(απ

2

)) 1α

(6)

and Ω = cos2(απ

2

)+ β2 sin2

(απ

2

).

To obtain the kernel for Equation 2, we can use Fourierinversion on the characteristic function and, making use ofthe shorthands defined above, the PDF may be computedvia the equation

f(Yu; v, β∗, γ∗, α) =1πγ∗

∫ ∞

0

cos(

(u− Yu)sγ∗

+ sα sin(φ))

exp(− sα sin(φ)

)ds

(7)

where φ = β∗πη2 .

This treatment opens-up not only the possibility of ef-fecting functional analysis on the characteristic functionmaking use of the techniques in the Fourier domain, butalso allows the use of other Fourier kernels for purposesof compactness and ease of computation. Actually, the ex-pression above can be greatly simplified making use of theshorthandsA = (u−Yu)

γ∗ , η = sα and ωη = As+sαsin (φ),which yields

sαsin (φ) = ωη −Aη1α (8)

Substituting Equation 8 into equation 7, the PDF can beexpressed as follows

f (Yu; υ, β∗, γ∗, α) =

√2π

∫ ∞

0

exp(−ωη +Aη

1α

)√

2πγ∗αη(α−1

α )

cos (ωη) dη (9)

where the Fourier kernel becomes

K(ω, η) = cos(ωη) (10)

which can be related, in a straightforward manner, to theFourier Cosine Transform (FCT) of the form

F (ω) =

√2π

∫ ∞

0

exp(−ωη + (u−Yu)

γ∗ η1α

)√

2πγ∗αη(α−1

α )

cos (ωη) dη (11)

which is analogous to the expression in Equation 7.Nonetheless, the transform above does not have imaginary

3

coefficients. This can be viewed as a representation in thepower rather than in the phase spectrum. Moreover, it hasthe advantage of compacting the texture information in thelower-order Fourier terms, i.e. those for which w is closeto the origin. This follows the strong “information com-paction” property of FCTs introduced in [14] and assures agood trade-off between discriminability and complexity.

2.2. Invariance to Affine Distortions

Having introduced the notion of the FCT for purposes ofrepresenting the PDF of the dynamic texture under study,we now focus on relation between distortions on the tex-ture plane and the Fourier domain. To this end, we fol-low [3] and relate the Fourier domain distortions to affinetransformations on the texture shape. As mentioned ear-lier, the function f(Yu) corresponds to the band-dependentcomponent of the texture and, as a result, its prone to affinedistortion. This hinges in the notion that a distortion onthe texture plane will affect the geometric factor in the sur-face reflectance, but not its photometric properties. In otherwords, the material index of refraction, roughness, etc. re-mains unchanged, whereas the geometry of the reflectiveprocess does vary with respect to affine distortions on theimage plane. The corresponding 2D FCT of the functionf(Yu) which, as introduced in the previous sections, cor-responds to the pixel values for the image-band Iλ in thetexture under study is given by

F (ξ) =∫

Γ

f (Yu) cos(2π

(ξTu

))du (12)

where u = [x, y]T is the vector of two-dimensional coordi-nates for the compact domain Γ ∈ <2. It is worth noting inpassing that, in practice, the coordinate-vectors u = [x, y]T

will be given by discrete quantities on the image lattice.For purposes of analysis, we consider the continuous caseand note that the affine coordinate transformation can beexpressed in matrix notation as follows

u′ =[x′

y′

]=

[a bd e

] [xy

]+

[ch

](13)

This observation is important because we can relate thekernel for the FCT in Equation (12) to the transformed co-ordinate u′ = [x′, y′]T . Also, note that, for patches centeredat keypoints in the image, the texture can be considered de-void of translation. Thus, we can set f = c = 0 and write

ξTu = ξT

[xy

](14)

=[ξx ξy

] [a bd e

]−1 [x′

y′

]=

1ae− bd

[eξx − dξy −bξx + aξy

] [x′

y′

]

Figure 1. Examples of reference, input and distortion correctedsingle-band textures. In the panels, the left-hand image shows thesingle-band textures whereas the right-hand panel shows the powerspectrum of the corresponding FCT.

where ξ = [ξx, ξy]T is the vector of spectral indexes for the2D FCT.

Hence, after some algebra, and using the shorthand ∆ =ae− bd, we can show that the FCT for the transformed co-ordinates u′ is

F (ξ) =1|∆|

∫ ∞

−∞

∫ ∞

−∞f (Yu′) cos

(2π∆

(eξx − dξy)x′

+(bξx − aξy)y′)dx′dy′ (15)

This implies that

F (ξ) =1|∆|

F (ξ′). (16)

where ξ′ is the “distorted” analogue of ξ. The distortionmatrix T is such that

ξ =[ξxξy

]=

[a db e

] [ξ′xξ′y

]= Tξ′ (17)

As a result, let the FCT for the affinely transformed pixelYu′ be given by U′. Analogously, suppose the FCT for thepixel Yu be given by U. From Equation 15, we can con-clude that the effect of the affine coordinate transformationmatrix T is to produce a distortion equivalent to (TT )−1

in the Fourier domain for the corresponding FCT. This ob-servation is an important one since it permits achieving in-variance to affine transformations on the texture plane via aFourier-domain distortion correction operation of the form

F (ξ) = (TT )−1F (ξ′) (18)

3. Descriptor ComputationWith the formalism presented in the previous sections,

we now proceed to elaborate further on the descriptor com-putation. Succinctly, this is a two-step process. Firstly,

4

we compute the affine-invariant 2D-FCT for every band inthe hyperspectral texture under study. This is equivalent tocomputing the band-dependent component of the character-istic function ψ(ς). Secondly, we capture the wavelength-dependent behaviour of the dynamic texture by computingthe FCT with respect to the spectral domain for the “train”of distortion-invariant FCTs. Thus, the descriptor becomesan FCT with respect to the band index for the cosine trans-forms corresponding to wavelength-resolved image in thesequence.

Following the rationale above, we commence by com-puting the distortion invariant FCT for each band in the im-age. To do this, we make use of the Fourier-domain prop-erty in Equation 18 to estimate the distortion matrix withrespect to a predefined reference. Here, we make use ofthe peaks of the power spectrum and express the relation ofthe FCTs for two texture planes. We have done this follow-ing the notion that a blob-like texture composed of a singletranscendental function on the texture plane would producetwo peaks in the Fourier domain. That is, we have set, asour reference, a moment generating function arising from atexture modelled as a cosine on a texture plane parallel tothe camera plane.

Let the peaks of the power spectrum for two locally pla-nar texture patches be given by UA and UB . Those for thereference are UR. As a result, the matrices UA, UB andUR are such each of their columns correspond to the x-ycoordinates for one of the two peaks in the power spectrum.These relations are given by

UA = (TAT )−1 UR (19)

UB = (TBT )−1 UR (20)

Where TA : UA ⇒ UR and TB : UB ⇒ UR are theaffine coordinate transformation matrices of the planar sur-face patches under consideration.

Note that, this is reminiscent of the shape-from-textureapproaches hinging in the use of the Fourier transform forthe recovery of the local distortion matrix [15]. Nonethe-less, in [15], the normal is recovered explicitly making useof the Fourier transform, whereas here, we employ the co-sine transform and aim at relating the FCTs for the two lo-cally planar patches with that of the reference. We can dothis making use of the composition operation given by

UB =(TAT−1

B

)TUA (21)

= ΦUA (22)

where Φ =(TAT−1

B

)Tis the distortion matrix. This matrix

represents the distortion of the power spectrum of UA withrespect to UB .

In practice, note that, if UR is known and fixed for ev-ery locally planar patch, we can use the shorthands TT

A =

URUA−1 and TT

B−1 = UBU−1

R to write

Φ =(URUA

−1) (

UBU−1R

)(23)

Which contrasts with other methods in the fact that, forour descriptor computation, we do not recover the principalcomponents of the local distortion matrix, but rather com-pute the matrix Φ directly through the expression above.Thus, we can construct a band-level descriptor of the form

V = [F (I1)∗ | F (I2)∗ | . . . | F (I|I|)∗] (24)

which is the concatenation of the affine invariant FCTsF (·)∗ for the frames in the texture. Moreover, we renderthe band-level FCT invariant to affine transformations mak-ing use of the reference peak matrix UR such that the FCTfor the frame indexed t is given by

F (IR) = F (It)∗Φ−1t (25)

where Φ−1t is the matrix which maps the FCT for the band

corresponding to the wavelength λ to the cosine transformF (IR) for the reference FCT. Here, as mentioned earlier,we have used as reference the power spectrum given by twojuxtaposed peaks rotated 45o degrees about the upper leftcorner of the 2D FCT. The reference FCT is shown in Figure1.

With the band-level representation V at hand, we canperform the second FCT computation. This is done by us-ing the discrete analogue of Equation 4. Thus, the kth coef-ficient for the texture descriptor G becomes

Gk = F (V) =|I|−1∑n=0

F (In)∗ cos(

π

| I |(n+

12)(k +

12))

(26)

where | G |=| I |.

4. DiscussionIn this section, we provide a discussion on the descrip-

tor presented previously. We commence by illustrating thedistortion correction operation at the band level in Figure 1.In the panels, we show the reference, corrected and inputtextures in their spatial and frequency domains. Note that,at input, the texture shows an affine distortion which affectsthe distribution of the peaks in its power spectrum. The dis-tortion corrected texture patch is in good accordance withthe reference.

To further illustrate the effects of affine distortions, inFigure 2, we show a sample texture which has been affinelydistorted. In the figure, we have divided the distorted inputtexture into patches that are assumed to be locally planar.We then apply the FCT to each of these patches, representedin the form of a lattice on the input image in the left-handpanel. The corresponding power spectrums are shown in

5

Figure 2. From left-to-right: Affine distortion of a sample single-band texture, FCT of the texture in the left-hand panel, distortion-correctedpower spectrums for the FCTs in the second panel, inverse FCTs for the power spectrum in the third panel.

the second column of the figure. Note that, as expected,the affine distortions produce a displacement on the powerspectrum peaks. In the third panel, we show the power spec-trums after the matrix Φ has been recovered and multipliedso as to obtain the corrected FCTs given by F (·)∗. The dis-tortion corrected textures in the spatial domain are shown inthe right-most panel in the figure. These have been obtainedby applying the inverse cosine transform to the power spec-trums in the third column. Note that, from both, the cor-rected power spectrums and the inverse cosine transforms,we can conclude that the correction operation can cope withlarge degrees of shear in the input texture-plane patches.

Now, we turn our attention to the wavelength depen-dency in hyperspectral textures. To this end, we illustrate,in Figure 3, the step-sequence of the descriptor computationprocedure. We depart from a series of bands in the imageand compute the band-by-band FCT. With the band FCTs athand, we apply the distortion correction approach presentedin the previous sections so as to obtain a “power-aligned”series of cosine transforms that can be concatenated into V.The descriptor is then given by the cosine transform of Vover the wavelength-index. Note that the descriptor will bethree-dimensional in nature, with sizeNx×Ny×Nλ, whereNx andNy are the texture sizes on the image lattice andNλ

is equivalent to the wavelength range for the dynamic tex-ture sequence. In the figure, for purposes of visualisation,we have raster-scanned the descriptor so as to display a 2Dmatrix whose rows correspond to the time-indexes of thedynamic textures under study.

5. Experiments

For purposes of illustrating the utility of our descriptorfor purposes of texture recognition, we make use of threedatasets. The first two of these are hyperspectral imagingones. The third is the DynTex dataset. The first of the hy-perspectral datasets comprises 100 hyperspectral images ofOriental Fruit Moths (OFM). In each view, together withthe insects, there is an amount of debris present, which actsas a confounding factor. The second hyperspectral data setis comprised by 50 images of urban scenes in high-obliqueviews. For the OFM dataset, the images have been acquired

in 10nm-step bands spanning from 380 to 1050nm. In thecase of our urban views, these are comprised of 10 bands inregular wavelength intervals of 25nm from 420 to 650nm.Also, note that the third of our datasets is not hyperspectralin nature, but rather a dynamic texture one. This is so as toillustrate how the descriptor presented here may be appliedto time-dependent as well as wavelength-resolved textures.

Here, we have used each of these descriptors for pur-poses of recognition as follows. After selecting a subset ofthe textures in each dataset for testing, the rest of the im-age are used as a data-base for purposes of recognition andlocalisation making use of a k-nearest neighbour classifier.We have organised this section as follows. We commenceby presenting our results on hyperspectral data. We thenturn our attention to the application of the descriptor pre-sented here to a dynamic texture setting. Along these lines,note that, as mentioned earlier, to our knowledge, there areno hyperspectral texture descriptors available in the liter-ature. As a result, for purposes of comparison, we haveturned our attention to the dynamic texture literature andselected the the algorithm of Zhao and Pietikainen [20] asthat to compare against. The reasons for this are twofold.Firstly, this is a method based upon local binary patterns(LBPs), which can be viewed as a vehicle which combinesthe statistical and structural models of texture analysis. Sec-ondly, from the literature [20], this method provides a mar-gin of advantage over other alternatives.

5.1. Hyperspectral Imagery

In this section, we illustrate the utility of our descriptorfor purposes of hyperspectral object localisation via texturerecognition. To this end, we have used SIFT [11] keypointsrecovered from the average image luminance so as to re-cover patches on the image plane. We have then used thispatches as candidates for purposes of hyperspectral texturerecognition. To do this, we have used half of the images ineach dataset for purposes of training. The other half wasused for testing.

In the case of the OFM dataset, we aim at localising themoths so as to identify them in each view. This is a chal-lenging task since each image portrays areas with specular-

6

Figure 3. From left-to-right: Hyperspectral texture, the band-wise FCT, the distortion invariant cosine transforms for every band in theimage and the final transform, i.e. correlation, with respect to wavelength.

Figure 4. Sample hyper-spectral OFM images used in our experi-ments.

ities, shadowing, debris and optic distortion. Each imagecontains eight moths. Figure 4 shows some sample OFMimages. So as to deal with photometric artifacts, we haveused the illuminant estimation method in [8] for purposesof photometric calibration. For purposes of training, wehave used 400 texture patches depicting the moths in thedataset. Thus, the aim is to localise the 400 moths in the 50remaining testing images.

For the urban-scene dataset, we have labelled the vehi-cles in each scene. We have done this so as to illustratethe utility of the descriptor for purposes of vehicle local-isation in high-oblique hyperspectral imagery of built en-vironments. Sample hyperspectral urban scene images areshown in Figure 5. As in the case of the OFM data, theimages present a number of photometric artifacts whose ef-fects in the localisation task have been mitigated throughthe use of photometric calibration. As before, the illumi-nant spectrum has been recovered using the method in [8].With the calibrated images at hand, we have hand-labelledthe 387 cars in the imagery. Of these, 189 appeared in the25 training images and the other 198 were in the testing im-ages. Again, in our experiments, we aim at localising these198 vehicles within the testing imagery.

In Table 1, we show recognition results for both, our de-scriptor and the alternative, when applied to the two datasets under study. The accuracy reported in the table cor-responds to the percentage of moths or vehicles correctlylocalised in the testing imagery. Note that, for the two datasets, our method provides a margin of improvement over thealternative. In the table, we also provide details regardingdescriptor-lengths. Recall that our descriptor has a length ofNx×Ny×Nλ. Note that a longer descriptor will potentiallyencode more information. Nonetheless, due to the good in-formation compaction properties of our approach, the com-plexity of the descriptor presented here is lower than that ofthe alternative.

Figure 5. Sample hyper-spectral urban views used in our car local-isation experiments.

Data set MethodDescriptor

LengthAccuracy

rateUrbanScenes

FCTLBP [20]

10004176

83.65%69.20%

OFMFCTLBP [20]

30004176

83.09%76.54%

Table 1. Recognition results (%) for the hyper-spectral image datasets with respect to our method (FCT) and the alternative Zhao &Pietikainen [20]

5.2. Dynamic Textures

As mentioned earlier, we also present results on the Dyn-Tex dataset 1. This is a dynamic texture depicting light can-dles, moving plants and trees, smoke plumes, sea waves,etc. Each texture class contains 12 dynamic texture se-quences. Following the experimental setting in [20], wehave pre-segmented each of the dynamic textures in eachclass so as to divide them into a number of non-overlappingsubsets. Here, we have obtained ten non-overlapping im-age sequences of random sizes Nx × Ny × Nt (here, Nt

is equivalent to Nλ in the equations throughout the paper)from each texture sequence and set k = 6 for our k-nearestneighbour classifier. We have used the Euclidean distancebetween the descriptors corresponding to each of the dy-namic texture pairs comprised by the query, i.e. testing, andthe sequences in the data-base. An example of the segmen-tation operation on the dataset is shown in Figure 6. Thispre-segmentation operation yields 1080 items in the finaldataset, half of which we use for testing and the other halfwe reserve for recognition purposes.

From a more quantitative viewpoint, in Table 2, we showthe recognition rates yielded by our descriptor as comparedto the method in [20]. In the table, we have shown resultsfor three different descriptor lengths and compare our re-

1The DynTex dataset can be downloaded fromhttp://www.cwi.nl/projects/dyntex/

7

(a) (b)

Figure 6. Example segmentation of a sample DynTex dynamic tex-ture sequence. (a): Original; (b): Segmented sequences

MethodDescriptor

LengthRecognition

rateFCT 9000 96.24%FCT 1000 95.71%LBP [20] 4176 95.07%

Table 2. Table showing the average recognition results (%) onthe DynTex database for our descriptor (FCT) and the alternative(LBP).

sults with those in [20]. It is somewhat expected that, as thelength increases, more information can be compacted intothe descriptor. This is in accordance with the fact that, fora length of 1000, i.e. Nx = Ny = Nt = 10, the recogni-tion rate is slightly lower (approx 1%) than that yielded bythe descriptor with length 9000, i.e. Nx = Ny = 30 andNt = 10. Nonetheless, our descriptor with length 1000 stillprovides a margin of advantage over that in [20]. This is asignificant observation since the descriptor in [20] has 4176coefficients. That is, our descriptor outperforms the alter-native regardless of being a factor of four less expensive tostore.

6. ConclusionWe have presented a novel approach to dynamic texture

recognition based upon heavy-tailed distributions. Here, weprovide a link to Fourier kernels and, through the use of acosine transform, we achieve a texture representation that iscompact and invariant to spectral distortions. The resultingdescriptor is given by the time-correlation for the distortioninvariant frame-wise cosine transforms. We have illustratedthe utility of the method for purposes of recognition by per-forming experiments on the DynTex and the MIT temporaltexture dataset, where our method outperforms the alterna-tives.

References[1] J. Aloimonos and M. Swain. Shape from texture. Biological

Cybernetics, 58(5):345–360, 1988. 1[2] S. Asmussen, K. Binswanger, and B. Hojgaard. Rare

events simulation for heavy-tailed distributions. Bernoulli,6(2):303–322, 2000. 2

[3] R. Bracewell, K.-Y. Chang, A. Jha, and Y.-H. Wang. Affinetheorem for two-dimensional fourier transform. ElectronicsLetters, 29(3):304, Feb. 1993. 4

[4] R. N. Clark, G. A. Swayze, A. Gallagher, N. Gorelick, andF. Kruse. Mapping with imaging spectrometer data using thecomplete band shape least-squares algorithm simultaneouslyfit to multiple spectral features from multiple materials. InProceedings of the Third Airborne Visible/Infrared ImagingSpectrometer Workshop, pages 91–28, 1991. 1

[5] G. Doretto, A. Chiuso, Y. Wu, and S. Soatto. Dynamic tex-tures. Int. Journal on Computer Vision, 51(2):91–109, 2003.2

[6] Z. Fu, A. Robles-Kelly, R. T. Tan, and T. Caelli. Invari-ant object material identification via discriminant learning onabsorption features. In Object Tracking and Classification inand Beyond the Visible Spectrum, 2006. 1

[7] J. Garding. Direct estimation of shape from texture. In ISRNKTH, 1993. 1

[8] C. P. Huynh and A. Robles-Kelly. Optimal solution of thedichromatic model for multispectral photometric invariance.In S+SSPR 2008, pages 382–391, 2002. 7

[9] K. Ikeuchi. Shape from regular patterns. Artificial Intelli-gence, 22:49–75, 1984. 1

[10] D. Landgrebe. Hyperspectral image data analysis. IEEESignal Process. Mag., 19:17–28, 2002. 1

[11] D. Lowe. Distinctive image features from scale-invariantkeypoints. International Journal of Computer Vision,60(2):91–110, 2004. 7

[12] J. Malik and R. Rosenholtz. Computing local surface ori-entation and shape from texture for curved surfaces. IJCV,23(2):149–168, June 1997. 2

[13] T. Randen and J. H. Husoy. Filtering for texture classifica-tion: a comparative study. Transactions on Pattern Analysisand Machine Intelligence, 21(4):291–310, 1999. 1

[14] K. R. Rao and P. Yip. Discrete cosine transform: algo-rithms, advantages, applications. Academic Press Profes-sional, Inc., San Diego, CA, USA, 1990. 4

[15] E. Ribeiro and E. Hancock. Shape from periodic tex-ture using the eigenvectors of local affine distortion. IEEETransactions on Pattern Analysis and Machine Intelligence,23(12):1459–1465, 2001. 2, 5

[16] I. N. Sneddon. Fourier Transforms. McGraw-Hill, 1951. 3[17] S. Soatto, G. Doretto, and Y. N. Wu. Dynamic textures. In

ICCV, pages 439–446, 2001. 2[18] B. Super and A. Bovik. Shape from texture using local spec-

tral moments. Transactions on Pattern Analysis and MachineIntelligence, 17(4):333–343, April 1995. 1

[19] M. Szummer and R. Picard. Temporal texture modeling. Im-age Processing, 1996. Proceedings., International Confer-ence on, 3:823–826 vol.3, Sep 1996. 2

[20] G. Zhao and Pietikainen. Dynamic texture recognition us-ing local binary patterns with an application to facial expres-sions. IEEE Transactions on Pattern Analysis and MachineIntelligence, 29(6):915–928, 2007. 6, 7, 8

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