4. Gaussian derivatives

Transcription

1 04 GaussianDerivatives.nb In[3]:= << FEVinit`; << FEVFunctions`; 4. Gaussian derivatives 4. Introduction We will encounter the Gaussian derivative function at many places throughout this book. Therefore we discuss this function in quite some detail in this chapter. The Gaussian derivative function has many interesting properties. We will discuss them in one dimension first. We study its shape and algebraic structure, its Fourier transform, and its close relation to other functions like the Hermite functions, the Gabor functions and the generalized functions. In two and more dimensions additional properties are involved like orientation (directional derivatives) and anisotropy. We discuss these at the end of this chapter. 4. Shape and algebraic structure When we take derivatives to x (spatial derivatives) of the Gaussian function repetitively, we see a pattern emerging of a polynomial of increasing order, multiplied with the original (normalized) Gaussian function again. Here we show a table of the derivatives from order 0 (i.e. no differentiation) to 3. In[5]:= Unprotect@gaussD; gauss@x_, σ_d := σ è!!!!!!!!! ExpA x π σ E; In[6]:= Table@Factor@Evaluate@D@gauss@x, σd, 8x, n<ddd, 8n, 0, 3<D Out[6]= x σ x x x 9 è!!!!!!!! π σ, σ x è!!!!!!!! π σ, σ Hx σl Hx + σl 3 è!!!!!!!!, σ x Hx 3 σ L π σ 5 è!!!!!!! = π σ 7 The function Factor takes polynomial factors apart. The function gauss[x,σ] is part of the standard set of functions (FEVFunctions.m) with this book, and protected. If we want to modify the function, we must first Unprotect it. The zeroth order derivative is indeed the Gaussian function itself. This is how the graphs of Gaussian derivative functions look like, from order 0 up to order 7:

2 04 GaussianDerivatives.nb In[7]:= D, 8x, n<dd, 8x, 5, 5<, PlotLabel > DisplayFunction > IdentityD, 8n, 0, 7<D, 4D, ImageSize > 550D êê Show; Order= Order= Order= Order= Order= Order= Order= Order= Figure 4.. Plots of the Gaussian derivative function for order 0 to 7. The even order (including the zeroth order) derivative functions are even functions (i.e. symmetric around zero) and the odd order derivatives are odd functions (antisymmetric around zero). Note the marked increase in amplitude for higher order of differentiation. The Gaussian function itself is a common element of all higher order derivatives. We extract the polynomials by dividing by the Gaussian function: In[8]:= D@gauss@x, σd, 8x, n<d TableAEvaluateA E, 8n, 0, 4<E êê Simplify gauss@x, σd Out[8]= 9, x σ, x σ σ 4, x3 3 x σ σ 6, x4 6 x σ + 3 σ 4 σ 8 = These polynomials have the same order as the derivative they are related to. Note that the highest order of x is the same as the order of differentiation, and that we have a plus sign for the highest order of x for even number of differentiation, and a minus signs for the odd orders. These polynomials are the Hermite polynomials, called after Charles Hermite, a brilliant French mathematician (see figure 4.). They emerge from the following definition: n e -x x = H L n H n n HxL e -x. The function H n HxL is the Hermite polynomial, where n is called the order of the polynomial. When we make the substitution x x ê σ, we get the following relation between the Gaussian function GHx, σl and its derivatives: n GHx,σL x = H L n n σ H n n H x σ L GHx, σl.

3 04 GaussianDerivatives.nb 3 In[9]:= Show@Import@"Hermite.gif"D, ImageSize > 55D; Figure 4.. Charles Hermite (8-90). In Mathematica the function H n is given by the function HermiteH[n,x]. Here are the Hermite functions from zeroth to fifth order: In[0]:= Table@HermiteH@n, xd, 8n, 0, 5<D Out[0]= 8, x, + 4 x, x + 8 x 3, 48 x + 6 x 4, 0 x 60 x x 5 < The inner scale σ is introduced in the equation by substituting x x σ è!!!!. As a consequence, with each differentiation we get a new factor. So now we are able to calculate the -D Gaussian σ è!!!! derivative functions gd[x,n,σ] directly with the Hermite polynomials, again incorporating the normalization factor σ è!!!!!!! : π In[]:= gd@x_, n_, σ_d := i y j k σ è!!!! z { n HermiteHAn, x σ è!!!! E σ è!!!!!!!! π ExpA x σ E Check: In[]:= FullSimplify@gd@x, 4, σd, σ > 0D Out[]= x σ Hx 4 6 x σ + 3 σ 4 L è!!!!!!!! π σ 9 In[3]:= FullSimplifyADA σ è!!!!!!!! ExpA x E, 8x, 4<E, σ > 0E π σ Out[3]= x σ Hx 4 6 x σ + 3 σ 4 L è!!!!!!!! π σ 9

4 04 GaussianDerivatives.nb 4 The amplitude of the Hermite polynomials explodes for large x, but the Gaussian envelop suppresses any polynomial function. No matter how high the polynomial order, the exponential function always wins. We can see this graphically when we look at e.g. the 7 th order Gaussian derivative without (i.e. the Hermite function, figure left) and with its Gaussian weight function (figure right). Note the vertical scales: In[4]:= BlockA8$DisplayFunction = Identity<, p = PlotA i y j è!!!! z k { p = PlotA i y j è!!!! z k { 7 HermiteHA7, 7 HermiteHA7, 8x, 5, 5<, PlotRange > AllE;E x E, 8x, 5, 5<, PlotRange > 8 800, 800<E; è!!!! x E ExpA x è!!!! E, Show@GraphicsArray@8p, p<d, ImageSize > 8450, 30<D; Figure 4.3. Left: The 7 th order Hermite polynomial. Right: idem, with a Gaussian envelop (weighting function). This is the 7 th order Gaussian derivative kernel. Due to the limiting extent of the Gaussian window function, the amplitude of the Gaussian derivative function can be negligeable at the location of the larger zeros. We plot an example, showing the 0 th order derivative and its Gaussian envelope function:

5 04 GaussianDerivatives.nb 5 In[6]:= << Graphics`FilledPlot`; n = 0; σ = ; BlockA8$DisplayFunction = Identity<, p = FilledPlot@gd@x, n, σd, 8x, 5, 5<, PlotRange AllD; gd@0, n, σd gauss@x, σd p = PlotA, 8x, 5, 5<, PlotRange AllEE; gauss@0, σd Show@8p, p<, ImageSize 830, 00<, DisplayFunction > $DisplayFunctionD; Figure 4.4. The 0 th order Gaussian derivative's outer zero-crossings vahish in negligence. Note also that the amplitude of the Gaussian derivative function is not bounded by the Gaussian window. The Gaussian function is at x = 3 σ, x = 4 σ and x = 5 σ, relative to its peak value: In[9]:= gauss@σ, D TableA, 8σ, 3, 5<E êê N gauss@0, D Out[9]= , , < and in the limit: In[0]:= Limit@gd@x, 7, D, x > InfinityD Out[0]= 0 The Hermite polynomials belong to the family of orthogonal functions on the infinite interval (-,) with the weight function e x, i.e. Ÿ e x H n HxL H m HxL x = è!!!!!!! π n! δ nm, where δ nm is the Kronecker delta, or delta tensor. δ nm = for n = m, and δ nm = 0 for n m. The Gaussian derivative functions, with their weight function e x 4 are not orthogonal. Other families of orthogonal polynomials are e.g. Legendre, Chebyshev, Laguerre, and Jacobi polynomials. Other orthogonal families of functions are e.g. Bessel functions and the spherical harmonic functions. Note that the area under the Gaussian derivative functions is not unity, e.g. for the first derivative:

6 04 GaussianDerivatives.nb 6 In[]:= Out[]= SetOptions@Integrate, GenerateConditions > FalseD; 0 è!!!!!!!! π gd@x,, σd x 4.3 Gaussian Derivatives in the Fourier Domain The Fourier transform of the derivative of a function is H i ωl times the Fourier transform of the function. For each differentiation, a new factor H i ωl is added. So the Fourier transforms of the Gaussian function and its first and second order derivative are: In[3]:= σ =.; FullSimplify@FourierTransform@ 8gauss@x, σd, x gauss@x, σd, 8x,< gauss@x, σd<, x, ωd, σ > 0D Out[3]= 9 σ ω è!!!!!!!!, σ ω è!!!!!!!! ω, σ ω ω è!!!!!!!! = π π π In general: F 8 n GHx,σL < = H iωl n F 8GHx, σl<. x n Gaussian derivative kernels also act as bandpass filters. The maximum amplitude is reached at ω = è!!! n : In[4]:= Out[4]= H I ωln n =.; σ = ; SolveAEvaluateADA è!!!!!!!! π 99ω 0 n =, 9ω è!!!! n =, 9ω è!!!! n == ExpA σ ω E, ωee == 0, ωe The normalized powerspectra show that higher order of differentiation means a higher center frequency for the bandpass filter. The bandwidth remains virtually the same.

7 04 GaussianDerivatives.nb 7 In[5]:= H I ωl n è!!!!!!! ExpA σ ω E π σ = ; p = TableAPlotAAbsA E, H I n ê L n è!!!!!!! ExpA σ n E 8ω, 0, 6<, DisplayFunction > IdentityE, 8n,, <E; Show@p, DisplayFunction > $DisplayFunction, PlotRange > All, AxesLabel > 8"ω", ""<D; π w Figure 4.5. Normalized power spectra for Gaussian derivative filters for order to, lowest order is left-most graph, σ =. 4.4 Zero Crossings of Gaussian Derivative Functions How wide is a Gaussian derivative? This may seem a non-relevant question, because the Gaussian envelop completely determines its behaviour. However, the number of zero-crossings is equal to the order of differentiation, because the Gaussian weighting function is a positive definite function. It is of interest to study the behaviour of the zero-crossings. They move further apart with higher order. We can define the 'width' of a Gaussian derivative function as the distance between the outermost zerocrossings. The zero-crossings of the Hermite polynomials determine the zero-crossings of the Gaussian derivatives. When we plot all zeros of the first 0 Hermite functions as a function of the order we clearly see the increasing largest zero as an indication for the 'width' of the kernel:

8 04 GaussianDerivatives.nb 8 In[7]:= gd@x_, n_, σ_d := i y j k σ è!!!! z { n x HermiteHAn, σ è!!!! E σ è!!!!!!!! π ExpA x σ E; nmax = 0; σ = ; ShowAGraphicsAFlattenATableA8PointSize@0.05D, x<d< ê. x SolveAHermiteHAn, E == 0, xe, 8n,, nmax<e, EE, è!!!! AxesLabel > 8"Order", "Zeros of\nhermiteh"<, Axes > TrueE; Zeros of HermiteH Order order. Each dot is a zero- Figure 4.6. Zero crossings of Gaussian derivative functions to 0 th crossing. Note that the zeros of the second derivative are just one standard deviation from the origin: In[9]:= σ =.; Simplify@Solve@D@gauss@x, σd, 8x, <D == 0, xd, σ > 0D Out[9]= 88x σ<, 8x σ<< An exact analytic solution for the largest zero is not known. The formula of Zernicke (93) specifies a range, and Szego (939) gives a better estimate:

9 04 GaussianDerivatives.nb 9 In[30]:= BlockA8$DisplayFunction = Identity<, p = PlotA SqrtAn è!!!!!!!!!!! 3 n + E, 8n, 5, 50<E; H Zernicke upper limit L p = PlotA SqrtAn +.5 è!!!!!!!!!!! 3 n + E, 8n,, 50<E; H Zernicke lower limit L p3 = PlotA è!!!!!!!!!!!!!! n í è!!!!!!!!!!!!!! 6 n +.5, 8n,, 50<, PlotStyle > Dashing@8.0,.0<DEE; Show@8p, p, p3<, AxesLabel > 8"Order", "Width of Gaussian\nderivative Hin σl"<d; Width of Gaussian derivative Hin sl Order Figure 4.7. Estimates for the width of Gaussian derivative functions to 50 th order. Width is defined as the distance between the outmost zero-crossings. Top and bottom graph: estimated range by Zernicke (93), dashed graph: estimate by Szego (939) For very high orders of differentiation of course the numbers of zero-crossings increases, but also their mutual distance between the zeros becomes more equal. In the limiting case of infinite order the Gaussian derivative function becomes a sinusoidal function: lim nz n G n x Hx, σl = Sin Jx "#################### σ H n+ L N. 4.5 The correlation between Gaussian Derivatives Higher order Gaussian derivative kernels tend to become more and more similar. Compare e.g. the 0 th and 4 nd derivative function:

10 04 GaussianDerivatives.nb 0 In[3]:= Block@8$DisplayFunction = Identity<, g = Plot@gd@x, 0, D, 8x, 7, 7<, PlotLabel > "Order 0"D; g = Plot@gd@x, 4, D, 8x, 7, 7<, PlotLabel > "Order 4"DD; Show@GraphicsArray@8g, g<, ImageSize > 8480, 40<DD; Order 0 Order Figure 4.8. Gaussian derivative functions start to look more and more alike for higher order. Here the graphs are shown for the 0 th and 4 th order of differentiation. This makes them not very suitable as a basis. But before we investigate their role in a possible basis, let us investigate their similarity. In fact we can express exactly how much they resemble each other as a function of the difference in differential order, by calculationg the correlation between them. We derive the correlation below, and will appreciate the nice mathematical properties of the Gaussian function. Because the higher dimensional Gaussians are just the product of D Gaussian functions, it suffices to study the D case. The correlation coefficient between two functions is defined as the integral of the product of the functions over the full domain (in this case - to +). Because we want the coefficient to be unity for complete correlation (when the functions are identical by an amplitude scaling factor) we divide the coefficient by the so-called autocorrelation coefficients, i.e. the correlation of the functions with themself. We then get as definition for the correlation coefficient r between two Gaussian derivatives of order n and m: r n,m = Ÿ g HnL HxL g HmL HxL x "########################################################################### HnL HxLD x HmL HxLD x () with g HnL HxL = n g HxL. The Gaussian kernel ghxl itself is an even function, and, as we have x n seen before, g HnL HxL is an even function for n is even, and an odd function for n is odd. The correlation between an even function and an odd function is zero. This is the case when n and m are both not even or both not odd, i.e. when Hn ml is odd. We now can see already two important results: r n,m = 0 for Hn ml odd; r n,m = for n = m. The remaining case is when Hn ml is even. We take n > m. Let us first look to the nominator,ÿ g HnL HxL g HmL HxL x. The standard approach to tackle high exponents of functions in integrals, is the reduction of these exponents by partial integration:

11 04 GaussianDerivatives.nb Ÿ g HnL HxL g HmL HxL x = g HnL HxL g HmL HxL» Ÿ g Hn L HxL g Hm+L HxL x = H L k Ÿ g Hn kl HxL g Hm+kL HxL x when we do the partial integration k times. The 'stick expression' g HnL HxL g HmL HxL» is zero because any Gaussian derivative function goes to zero for large x. We can choose k such that the exponents in the integral are equal (so we end up with the square of a Gaussian derivative function). So we make Hn kl = Hm + kl, i.e. k = Hn ml. Because we study the case that Hn ml is even, k is an integer number. We then get: H L k Ÿ g Hn kl HxL g Hm+kL HxL x = H L n m Ÿ g H n+m L HxL g H n+m L HxL x The total energy of a function in the spatial domain is the integral of the square of the function over its full extent. The famous theorem of Parceval states that the total energy of a function in the spatial domain is equal to the total energy of the function in the Fourier domain, i.e. expressed as the integral of the square of the Fourier transform over its full extent. Therefore H L n m H L n m Parceval Ÿ g H n+m L HxL g H n+m L HxL x = n+m π Ÿ HiωL ĝ HωL ω = H L n m π Ÿ ω n+m ĝ HωL ω = H L n m We now substitute ω' = σω, and get finally: H L n m π π Ÿ ω n+m e σ ω ω σ Ÿ n+m+ ω' Hn+mL e ω' ω '. This integral can be looked up in a table of integrals, but why not let Mathematica do the job (we first clear n and m): In[34]:= n =.; m =.; x m+n e x x Out[34]= H + H Lm+n L GammaA H + m + nle Log@eD H m nl The function Gamma is the Euler gamma function. In our case Re[m+n]>-, so we get for our correlation coefficient for Hn ml even: r n,m = H L n m π σ n+m+ ΓH m+n+ L π σ n+ ΓH n+ L π σ m+ ΓH m+ n m "#################################################################################### = H L L L "######################################## ΓH n+ L ΓH m+ L ΓH m+n+ With this function we can study the correlation behaviour for large n and m quite easily. Let's first have a look at this function tabulated for a range of values for small n and m (0-7):

12 04 GaussianDerivatives.nb In[35]:= m_d := H L n m GammaA m + n + E ì $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GammaA n + E GammaA m + E ; Table@NumberForm@r@n, md êê N, 3D, 8n, 0, 5<, 8m, 0, 5<D êê MatrixForm Out[36]//MatrixForm= i y j z k { In[37]:= ListPlot3D@Table@Abs@r@n, mdd, 8n, 0, 5<, 8m, 0, 5<D, Axes > True, AxesLabel > 8"n", "m", "Abs\nr@n,mD"<, ViewPoint > 8.348,.540,.8<D; n Abs r@n,md m 5 Figure 3.9. The magnitude of the correlation coefficient of Gaussian derivative functions for 0 < n < 5 and 0 < m < 50. The origin is in front. The correlation is unity when n = m, as expected, is negative when n m =, and is positive when n m = 4, and is complex otherwise. Indeed we see that when n m = the functions are even but of opposite sign: In[38]:= Block@8$DisplayFunction = Identity<, p = Plot@gd@x, 0, D, 8x, 5, 5<, PlotLabel > "Order 0"D; p = Plot@gd@x,, D, 8x, 5, 5<, PlotLabel > "Order "DD; Show@GraphicsArray@8p, p<, ImageSize > 8480, 40<DD; Order Order

13 04 GaussianDerivatives.nb 3 Figure 4.0. Gaussian derivative functions differing two orders are of opposite polarity. and when n m = they have a phase-shift, leading to a complex correlation coefficient: In[40]:= Block@8$DisplayFunction = Identity<, p = Plot@gd@x, 0, D, 8x, 5, 5<, PlotLabel > "Order 0"D; p = Plot@gd@x,, D, 8x, 5, 5<, PlotLabel > "Order "DD; Show@GraphicsArray@8p, p<, ImageSize > 8480, 40<DD; Order 0 Order Figure 4.. Gaussian derivative functions differing one order display a phase shift. Of course, this is easy understood if we realize the factor H i ωl in the Fourier domain, and that i = e i π. We plot the behaviour of the correlation coefficient of two close orders for large n. The asymptotic behaviour towards unity for increasing order is clear. In[4]:= Plot@ r@n, n + D, 8n,, 0<, DisplayFunction > $DisplayFunction, AspectRatio >.4, PlotRange > 8.8,.0<, AxesLabel > 8"Correlation\ncoefficient", "Order"<D; 0.95 Order Correlation coefficient Figure 4.. The correlation coefficient between a Gaussian derivative function and its even neighbour up quite quickly tends to unity for high differential order. 4.6 Discrete Gaussian Kernels Lindeberg [Lindeberg990] derived the optimal kernel for the case when the Gaussian kernel was discretized and came up with the "modified Bessel function of the first kind". In Mathematica this function is available as BesselI. The "modified Bessel function of the first kind" BesselI is almost equal to the Gaussian kernel for σ >, as we see below. Note that the Bessel function has to be normalized by its value at x = 0. For larger σ the kernels become rapidly very similar.

14 04 GaussianDerivatives.nb 4 In[4]:= σ = ; x PlotA9 ExpA è!!!!!!!!!!!!!! π σ σ E, BesselI@x, σ è!!!!!!!!!!!!!! D ê BesselI@0, σ D=, π σ 8x, 0, 8<, PlotStyle 8RGBColor@, 0, 0D, RGBColor@0, 0, 0D<, PlotLegend 8"Gauss", "Bessel"<, LegendPosition 8, 0<, LegendLabel "σ = ", PlotRange All, ImageSize 8330, 66<E; s = Gauss Bessel Higher dimensions and separability Gaussian derivative kernels of higher dimensions are simply made by multiplication. Here again we see the separability of the Gaussian, i.e. this is the separability. The function gdd@x, y, n, m, σ x, σ y D is an example of a Gaussian partial derivative function in D, first order derivative to x, second order derivative to y, at scale (equal for x and y): In[44]:= gdd@x_, y_, n_, m_, σx_, σy_d := gd@x, n, σxd gd@y, m, σyd; Plot3D@gdD@x, y,,,, D, 8x, 7, 7<, 8y, 7, 7<, AxesLabel > 8x, y, ""<, PlotPoints > 40, PlotRange > All, Boxed > False, ImageSize > 850, 40<D; Figure 4.3. Plot of 3 GHx,yL. The two-dimensional Gaussian derivative function can be constructed x y as the product of two one-dimensional Gaussian derivative functions., and so for higher dimensions, due to the separability of the Gaussian kernel for higher dimensions. The ratio σx σ y is called the anisotropy ratio. When it is unity, we have an isotropic kernel, which diffuses in the x and y direction by the same amount. The Greek word ισος means 'equal', τροπος means 'direction' (τοπος means 'location, place').

15 04 GaussianDerivatives.nb Directional derivatives and steerable filters 4.9 Other families of kernels The derivation given above required first principles be plugged in that essentially stated "we know knothing" (at this stage of the observation). Of course, we can relax these principles, and introduce some knowledge. When we want to derive a set of apertures tuned to a specific spatial frequency k in the image, we add this physical quantity to the matrix of the dimensionality analysis: In[46]:= m = 88,,,, <, 80, 0,,, 0<<; TableForm@m, TableHeadings > 88"meter", "candela"<, 8"σ", "ω", "L0", "L", "k"<<d NullSpace@mD Out[47]//TableForm= σ ω L0 L k meter candela Out[48]= 88, 0, 0, 0, <, 80, 0,,, 0<, 8,, 0, 0, 0<< Following the exactly similar line of reasoning, we end up from this new set of constraints with a new family of kernels, the Gabor family of receptive fields, with are given by a sinusoidal function (at the specified spatial frequency) under a Gaussian window. In the Fourier domain: GaborHω, σ, kl = e ω σ e i k ω, which translates into the spatial domain: In[49]:= gabor@x_, σ_d := Sin@xD x ExpA è!!!!!!!!!!!!!! π σ σ E; The Gabor function model of cortical receptive fields was first proposed by Marcelja in 980 [Marcelja980]. However the functions themselves are often credited to Gabor [Gabor946] who supported their use in communications. Gabor functions are defined as the sinus function under a Gaussian window with scale σ. The phase φ of the sinusoidal function determines its detailed behaviour, e.g. for φ = π ê we get an even function.

16 04 GaussianDerivatives.nb 6 In[50]:= gabor@x_, φ_, σ_d := Sin@x + φd gauss@x, σd; Block@8$DisplayFunction = Identity<, p = Plot@gabor@x, 0, 0D, 8x, 30, 30<, PlotRange > 8.04,.04<D; p = Plot@gabor@x, π ê, 0D, 8x, 30, 30<, PlotRange > 8.04,.04<D; p3 = Plot@gauss@x, 0D, 8x, 30, 30<, PlotRange > 8.04,.04<D; p3 = Show@p, p3d; p3 = Show@p, p3dd; Show@GraphicsArray@8p3, p3<d, ImageSize > 8480, 40<D; Figure 3.4. Gabor functions are sinusoidal functions with a Gaussian envelope. Left: Sin[x] G[x,0]; right: Sin[x+π/] G[x,0]. Gabor functions can look very much like Gaussian derivatives, but they are not the same: - Gabor functions have an infinite number of zero-crossings on their domain. - The amplitudes of the sinusoidal function never exceeds the Gaussian envelope. They can be made to look very similar by an appropriate choice of parameters: In[53]:= gauss@x_, σ_d := ExpA è!!!!!!!!!!!!!! π σ σ E; x p = Plot@ gabor@x, D, 8x, 4, 4<, PlotStyle Dashing@80.0, 0.0<D, DisplayFunction IdentityD; p = Plot@Evaluate@D@gauss@x, D, xdd, 8x, 4, 4<, DisplayFunction IdentityD; Show@8p, p<, DisplayFunction $DisplayFunctionD; Figure 3.5. Gabor functions can be made very similar to Gaussian derivative kernels. In a practical application then there is no difference in result. Dotted graph: Gaussian first derivative kernel.

17 04 GaussianDerivatives.nb 7 Continuous graph: Minus the Gabor kernel with the same σ as the Gaussian kernel. Note the necessity of sign change due to the polarity of the sinusoidal function. By relaxing or modifying other constraints, we might find other families of kernels (see e.g. [Pauwels995]). We conclude this section by the realization that the front-end visual system at the retinal level must be uncommitted, no feedback from higher levels is at stake, so the Gaussian kernel seems a good candidate to start observing with at this level. At higher levels this constraint is released. The extensive feedback loops from the primary visual cortex to the LGN may give rise to 'geometry-driven diffusion' [terhaarromeny994c], nonlinear scale-space theory, where the early differential geometric measurements through e.g. the simple cells may modify the kernels LGN levels. Nonlinear scale-space theory will be treated in chapter 9. ò [Task 3.] When we have noise in the signal to be differentiated, we have two counterbalancing effect when we change differential order and scale: for higher order the noise is amplified (the factor H i ωl n in the Fourier transform representation) and the noise is averaged out for larger scales. Give an explicit formula in our Mathematica framework for the propagation of noise when filtered with Gaussian derivatives. Start with the easiest case, i.e. pixel-uncorrelated noise, and continue with correlated noise. See for a treatment of this subject the work by Hans Blom et al. [Blom993a]. ò [Task 3.] Give an explicit formula in our Mathematica framework for the propagation of noise when filtered with a compound function of Gaussian derivatives, e.g. by the Laplacean G x + G y. See for a treatment of this subject the work by Hans Blom et al. [Blom993a].