Probability Distributions for Gradient Directions in Uncertain 3D Scalar Fields

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1 Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion Group, Technische Universität München, Germny E-Mil: {pfffelmoser, mihel.mihi, Abstrct This report introduces mthemticl frmework to compute probbility density functions for the directions of grdients in Gussin distributed uncertin 3D sclr fields, nd illustrtes results by mens of severl exmples. Introduction The geometry of slient fetures - prominent structures indicted by strong grdients, such s bounding surfces, nd isocontours - in uncertin 3D sclr fields is stble only to some extent. In order to nlyze the stbility of the direction of fetures in sclr fields, we need to investigte the vribility of derived vector fields tht revel the uncertinty in direction of these fetures, such s grdient fields. Grdients t sptil points of sclr field indicte t every loction the direction of the gretest rte of increse of the sclr field, the mgnitude giving the rte of increse. A significnt chnge in the vlues of the sclr field results in strong grdient mgnitude, while the direction of the grdient shows the direction of the gretest chnge. The grdient direction cn thus give further insight into the stbility of orienttions of fetures. Nonetheless, in the context of uncertin sclr fields, derived quntities such s grdients re lso ffected by uncertinty, investigting the grdient vribility being thus consequentil in ddressing the stbility of orienttions of possible fetures in uncertin sclr fields. In situtions where the uncertinty of sclr vlues in sclr fields is completely chrcterized by the men nd covrince prmeters, of which the stndrd model is the multivrite Gussin distribution, closed-form derivtion of the probbilty density function of the grdient direction is possible. The grdients t sptil points of Gussin distributed sclr field, being obtined by pplying liner opertor to stencils of neighbors of the points in the sclr field, cn be shown to follow Gussin distribution s well. This lso holds for ny other vector fields tht cn be obtined similrly. Thus, the uncertinty of the grdients nd other such vector fields cn be modeled vi men nd covrince s well. However, such distribution describes the combined uncertinty of both the grdient mgnitude nd direction. As we re interested only in the direction uncertinty, we seek to isolte the ltter from the grdient strength, by switching from the Crtesin to the sphericl coordinte system. This llows us to perform nlyses in terms of directions in spce, i.e., positions on sphere, where, by integrting over the rdil distnce, we re left with positions on the unit sphere, i.e., the direction uncertinty. The rest of the report detils the derivtion of the probbility density functions of grdient directions nd uses severl exmples to illustrte different probbility density functions. Derivtion Throughout the pper, we ssume discrete smpling of 3D domin on Crtesin grid structure with grid points S m,n,p = {x i, j,k : i m, j n, k p}. The dt uncertinty t every point is modeled by multivrite rndom vrible Y with sclr-vlued components Y x i, j,k. We further suppose tht the rndom vribles follow multivrite Gussin distribution, so tht the distribution t point x i, j,k is chrcterized by men vlue µx i, j,k nd stndrd devition σx i, j,k. Moreover, the correltion vlues between ny pir of rndom vribles, ρy x i, j,k,y x u,v,t, cn be computed. The probbility density function for the grdient t the grid points of trivrite Gussin distributed 3D sclr field cn be determined by first pproximting the grdient vi liner opertor, nd then using this opertor to pproximte the uncertinty prmeters from the corresponding uncertinty prmeters of the points in the sclr field.

2 The grdient t point x i, j,k cn be pproximted vi centrl differences one-sided differences t the domin boundries on the rndom vribles s Y x i, j,k = Asx i, j,k. Here, the 6-element stencil s contins the rndom vribles sx i, j,k = [Y x i+, j,k,y x i, j,k,y x i, j+,k,y x i, j,k,y x i, j,k+,y x i, j,k ] nd the 3 6 mtrix A contins the inverse point distnces A, = xi+, j,k x i, j,k,a, = A,,A,3 = xi, j+,k x i, j,k,a,4 = A,3,A 3,5 = xi, j,k+ x i, j,k, A 3,6 = A 3,5,A,3 = A,4 = A,5 = A,6 = A, = A, = A,5 = A,6 = A 3, = A 3, = A 3,3 = A 3,4 =. Since the stencil sx i, j,k forms 6-component subset of the multivrite rndom vrible Y, its probbility distribution is multivrite Gussin s well. Furthermore, due to the liner reltion between Y nd s, the grdient lso follows trivrite Gussin distribution. In order to fully describe this distribution, we first need to compute the men grdient µ nd the covrince mtrix Σ of Y, which relte to the mens nd covrinces of the rndom vribles vi µ x i, j,k = Aµ s x i, j,k, Σ = AΣ s A, where the q-th component of µ s x i, j,k contins the men of the q-th component of sx i, j,k, µ s x i, j,k q = µsx i, j,k q, nd the covrince mtrix of the rndom stencil vector Σ s hs components Σ sxi,j,k m,n = σsx i, j,k m σsx i, j,k n ρsx i, j,k m,sx i, j,k n. From the given men grdient nd covrince mtrix, the trivrite probbility density function of Y for vector g is then derived s p g = exp.5g µ π 3 Σ detσ g µ, where expnding the determinnt of the grdient covrince mtrix yields detσ = σx σ y σ z + ρ xy ρ xz ρ yz ρxy + ρ xz + ρ yz, while the inverse of the mtrix cn be written s σ y σ z Σ = σ xσ y σ σ x ρyz σ z ρ xz ρ yz ρ xy σ y ρ xy ρ yz ρ xz z σ σ detσ z ρ xz ρ yz ρ xy x σ z σ y ρxz σ x ρ xy ρ xz ρ yz σ σ y ρ xy ρ yz ρ xz σ x ρ xy ρ xz ρ yz x σ y σ z. ρxy A similr procedure cn be performed for other derived vector fields tht cn be obtined by pplying liner opertor. In the formul bove, p describes the probbility tht the grdient tkes on given mgnitude nd direction, comprising the uncertinty of the grdient in both strength nd orienttion. For extrcting only the direction uncertinty, the density function is trnsformed into sphericl coordintes p θ,φ,r = r sinθ π 3 detσ expeθ,φ, θ [,π], φ [,π], r [, [ Eθ,φ = sinθ cosφ r sinθ sinφ µ cosθ Σ sinθ cosφ r sinθ sinφ µ. cosθ The probbility density function for the grdient direction is given by the θ,φ-mrginl θ,φ = p θ,φ,rdr, θ [,π], φ [,π]. For every pir of ngles θ,φ, probbility vlue is obtined by integrting the trivrite grdient density long line from r = to r =. In order to derive the formul for the probbility density function of the grdient direction, we first rewrite the θ,φ-mrginl s θ,φ = C r exp r br + c dr,

3 with the constnts = sinθ σz cosφ σy detσ ρ yz + sinφ σx ρ xz + cosθ σx σ y ρ xy + σ xσ y σ z sinθ cosθ cosφσ y ρ xy ρ yz ρ xz + sinφσ x ρ xy ρ xz ρ yz + sinθ sinφ cosφσ z ρ xz ρ yz ρ xy, b = σ detσ x σ y σ z sinθσ z ρ xz ρ yz ρ xy sinφ µ x + cosφ µ y + σ y ρ xy ρ yz ρ xz sinθ cosφ µ z + cosθ µ x + σ x ρ xy ρ xz ρ yz cosθ µ y + sinθ sinφ µ z + cosθ µ z σx σ y ρ xy + sinθσ z cosφ µ x σy ρyz + sinφ µ yσx ρ xz, c = σ detσ x σ y σ z µ x µ y σ z ρ xz ρ yz ρ xy + µ x µ z σ y ρ xy ρ yz ρ xz + µ y µ z σ x ρ xy ρ xz ρ yz + µ x σ y ρ yz + µ y σ x ρ xz + µ z σ x σ y ρ xy, σ z C = sinθ π 3 detσ. It follows tht θ,φ = C r exp r br + c b dr = C exp c r exp whereby the chnge of vribles r b/ = u gives r b dr, θ,φ = C b exp c u + b exp u du = C b b exp c I + b I + b I 3, the three integrls I, I, nd I 3 resulting from expnding the qudrtic expression: I = b u exp u du =.5 I = uexp u du = exp u b b = exp π I 3 = exp u du = b Putting everything together yields θ,φ = C b b exp c 3 Illustrtion uexp u b I 3 =.5 b exp erf b. exp b, b I 3, b + + b π erf b. This section illustrtes by mens of severl exmples the previously derived formul. The following figures hve been done using Mtlb. We strt with the simplest cse, tht of null men vector nd identity covrince mtrix, s shown by the unit sphere centered t the origin in Figure. In Crtesin coordintes, this results in uniformly distributed grdient. However, in sphericl coordintes, due to the sinθ term present in the Jcobin when performing the coordinte trnsformtion, the probbility density function of the grdient yields the prboloidl shpe in Figure, where the plotting is done with respect to the ngles θ nd φ. A more suitble visuliztion for the mrginl probbility when expressed in sphericl coordintes is by mens of the unit sphere, s indicted in Figure b. The figure lso hints why the probbility density function is no longer uniform, since the surfce elements on the sphere do not hve n equl re, s for regulr grids, but re lrger t the equtor thn t the poles. 3

4 Figure : 3D Gussin uniform probbility density of the grdient vector. The men vector is zero nd thus not displyed, while the yellow sphere indictes the confidence region. Nonetheless, Figure shows tht, but for the sinθ term, the probbility density function in sphericl coordintes would lso be uniform. Dividing ech point t which the function is given by the corresponding sinθ vlue, gives indeed uniform probbility density function, s illustrted in Figure c nd d. The following exmples will only present the probbility density function mpped onto the sphere fter the division by sinθ. The second exmple trnsltes the men with one unit to the right on the x-xis nd modifies the digonl of the covrince mtrix, so tht σ x becomes.5 nd σ y becomes.5. This is reflected in Figure 3, where the sphere hs turned into n ellipsoid tht is centered on the x-xis, rther thn t the origin, nd hs the lrgest semi-xis on the x-xis, where the stndrd devition is lrgest. The probbility density function, shown in Figure 3 b, is symmetric nd hs strong mode t θ = π/ nd φ =, i.e., in the direction of the men, on the x-xis. The lst exmple trnsltes the men with one unit in both the x- nd z-xes, nd modifies the covrince mtrix to hve σ y = on the digonl. Then, the covrince mtrix hs further non-digonl elements different from zero, tking correltions into ccount by setting ρ xy =.5 nd ρ yz =.7. Figure 4 shows the corresponding ellipsoid. The effect of correltions cn be noticed in the chnge in orienttion of the ellipsoid, whose semi-xes re no longer prllel to those of the Crtesin grid. Figure 4 b revels n lmost symmetric bimodl density function, with right slightly stronger mode, nd left slightly lower mode. The direction of the men is rther unlikely. 4 Conclusion In this report, we introduced mthemticl frmework to compute probbility density functions for directions of vectors in Gussin distributed uncertin 3D sclr fields tht cn be obtined by pplying liner opertor, nd derived pproprite formule for grdients t Gussin distributed 3D sclr fields. We showed tht performing coordinte trnsformtion from Crtesin to sphericl coordintes nd computing in the sphericl coordinte system is better suited to convey the probbility density for grdient directions, nd illustrted vrious direction probbility density functions in severl exmples. 5 Acknowledgments This work ws supported by the Europen Union under the ERC Advnced Grnt 937 SferVis - Uncertinty Visuliztion for Relible Dt Discovery. 4

5 Figure : The previous probbility density expressed in sphericl coordintes on regulr grid hving ngles θ nd φ s coordintes. b The sme probbility density s in, mpped onto the unit sphere. It cn be noticed tht the probbility density is no longer uniform, s the surfce elements on the sphere no longer occupy the sme re. c The probbility density fter division expressed in sphericl coordintes on regulr grid hving ngles θ nd φ s coordintes. d The probbility density fter division, mpped onto the unit sphere. It cn be noticed tht the probbility density is now uniform. 5

6 Figure 3: 3D Gussin probbility density of the grdient vector, with the men vector drwn in mgent long the x-xis, nd the yellow ellipsoid indicting the confidence region. The three ellipses colored in red, green, nd blue, respectively, re the corresponding ellipses hving s semi-mjor nd semi-minor xes the semi-xes of the ellipsoid. b The symmetric probbility density mpped onto the unit sphere, where the mgent-colored line extending to infinity hs the direction of the men vector. Figure 4: 3D Gussin probbility density of the grdient vector, with the men vector drwn in mgent in the XZ-plne nd the tilted yellow ellipsoid indicting the confidence region. b The bimodl probbility density, mpped onto the unit sphere, where the mgent-colored line hs the direction of the men vector. 6

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