Scale and the Differential Structure of Images

Scale and the differential structure of images

Luc M J Florack, Bat M ter Haar Romeny, Jan J Koenderink and Max A Viergever

capture the crucial observation of the inherently multi- Why and how one should study a scale-space is scale character of image structure. prescribed by the universal physical law of scale For some time there has been discussion on the invariance, expressed by the so-called Pi-theorem. question of how to generate a scale-space, the con- The fact that any image is a physical observable with an tinuous analogue of the pyramid, in a unique way, as inner and outer scale bound, necessarily gives rise to a there seemed to exist no clear way to choose among the 'scale-space representation', in wphich a given image is many possible scale-space filters' *. One obviously represented by a one-dimensiond family of images needed a set of natural, a priori scale-space constraints. representing that image on vurious levels of inner spatial A fundamental approach was adopted by scale. An early vision system is completely ignorant of Koenderink', Witkin" and Yuille and Poggio', who the geometry of its input. Its primary task is to establish formulated an a priori conttraint in the form of a this geometry at any available scale. The absence of causality requirement: no 'spurious detail' should be geometrical knowledge poses additional constraints on generated upon increasing scale. This, together with the construction of a scale-space, notably linearity, some symmetry constraints, unambiguously established spatial shift invariance and isotropy, thereby defining a the Gaussian kernel (i.e. the Green's function of the complete hierarchical family of scaled pariial differential isotropic diffusion equation) as the unique scale-space operators: the Gaussian kernel (the lowest order, filter. Its width r can be identified with spatial scale. rescaling operator) and its linear partirzl derivatives. One can model an image as a scalar field on a finite- They enable local image analysis through the detection dimensional manifold and apply fundamental mathe- of local differential structure in a robust wuy, while at the matical operations, like differentiations, to reveal local same time capturing global features through the extra image structure. There exist many useful and rather scale degree of freedom. In this paper we show why the well-established mathematical disciplines, notably operations of scaling and differentiation cannot be differential geometry, tensor calculus, invariants separated. This framework permits us to construct in a theory, all of which have an increasing impact on systematic way multiscale. cartesiarl differential nowadays image structure analysis. invariants, i.e. true image descriptors that exhibit In this paper we discuss the fundamental concept of manifest invariance with respect to a change of cartesian scaling as well as some natural constraints of a front- coordinates. The scale-space operators closely resemble end visual system, and show that a complete hier- the receptive field profiles Jhrind in mammalian front- archical set of .scaled differential operators follows from end visual systems. these considerations. The lowest order kernel is the isotropic Gaussian. The higher order kernels are the Keywords: scale-space, Gaiwsian kerfiiel, Gaussian scaled Gaussian derivatives, which constitute the derivatives, differential invariants natural differential operators on a given scale. With this set we can study local image geometry to any desired order. To this end we will introduce the Over the last few years there has been an increasing conce t of a local jet of order N, JN[L(P)],also called tendency in the image analysis literature towards a N-jet ',defined as the equivalence class of functions L multiscale approach. A historical contribution to such which share the same N-truncated l'aylor expansion at an approach was the introduction of the pyramid I. a given point P. In other words, all images in a given N- Though being based on a rather ad hoc method of jet are locally indistinguishable modulo higher order averaging neighbouring pixels. this first model did differences. Such a local A.'-jet can be represented with respect to a cartesian coordinate system by the set of partial derivatives up to Ntli otdet, evaluated at the Computer Vision Research Group, Utrccht University hospital, point P, so: Heidelberglaan 100, 3584 CX Utrccht. The Nethcrlands received: February 1992 Paper 7 J"L(P)I = .(L,' . . JP)}L, (1) 0262-8856/92/006376-13 0199 12 Butterworth-Heinemann Ltd 376 image and vision computing 'The lower spatial indices attached to L all have values necessity of a multiscale approach and to derive the within the range 1 . . . 13, where D is the dimension of unique scale-space operators for arbitrary dimensions the image domain. and denote differentiation with n>1. respect to the associated spatial variable. Derivatives of arbitrary order are generally well- defined and robust provided they can be calculated on a Basic front-end vision constraints sufficiently high scale (relative to pixel scale and noise correlation width), and provided we have a sufficient Many interpretations of a front-end vision system are resolution of intensity values (dynamic resolution, possible. We assume that its sole task is to establish a noise). We will not present a detailed discussion on representation of a given observable in a convenient these trade-offs here. but refer to Blom er al.". In this forrnat. The interpretation is left to dedicated postpro- paper we will restrict ourselves to N d 3. cessing routines, which read out the formatted data The approach is valid in D dimensions. whereas represented by the front-end (cf. the 'sensorium' in much of the literature is limited to 1 or 2 Koenderink 13). By definition, a front-end vision system dimensions I. 2.4. s is assumed to be completely ignorant of any a priori geometry of its input. This lack of LI priori geometrical knowledge argues for an a priori symmetric sampling and preprocessing of its input. Hence it is quite natural THEORY to define a front-end vision system by formulating a set Physical versus mathematical operators of plausible symmetries. We propose the following set": The only way to obtain structural information about a physical scene is to extract oh.servab1e.s (i.e. images) 0 linearity: allowing for superposition of input with the help of some measuring apparatus. We stimuli. inevitably have to face the problem of fixing the proper 0 spatial shift invariance: implied by the absence of a scale. because observables are always characterized by perferred location. an intrinsic, finite scale range. Its lower bound is 0 isotropy: implied by the absence of a preferred determined by the sampling characteristics of the direction. device, whereas the upper bound is limited by the scope scale invariance: implied by the absence of a of the field of view. preferred scale. The very fact that an image is a physical observable makes it subject to an extra constraint imposed by the These basic symmetry requirements are rather weak, iiniversul lm*of sculr invariunce, which governs all laws because we do not want the front-end system to commit of physics. There is no such scaling constraint on a itself to any specific task beyond representation. Note mathematical, i.e. a dimensionless scalar field, defined that none of these symmetry constraints are strictly on a dimensionless manifold. but it is instructive to necessary for the sole purpose of data representation, observe how mathematicians alternatively constrain it but they do significantly decrease the burden on by imposing convenient regularity conditions: a mathe- interpreting routines that address the front-end, since matical function is typically assumed to be 'sufficiently these will now be refrained from the overhead of smooth', say a Ch'(LO)-function on a D-dimensional having to reconcile the data with the symmetries of the domain $1, with A' sufficiently large to justify the environment that are known in advance anyway: the operations performed on it. For a physical observable front-end system will make this a priori knowledge of we cannot pose such smoothness constraints. the environment manifest. In this precise sense, the Clearly, it makes no sense to define a derivative of a front-end postulates will make up for a convenient sampled image in the strict mathematical sense (this format. would require the existence of an infinitesimal neighbourhood as well as a smoothness constraint on neighbouring image values). One usually circumvents this problem by considering neighbouring pixels instead Scale invariance of infinitesimal neighbourhoods in the definition of a Let F(xl,. . ., xu) be some physical observable, e.g. derivative. A well-known example of this is the 5-point the image luminance as a function of spatial coordin- Laplacean kernel". 'This is. however. a non-robust and ates, time, etc. From a pure mathematical point of view rather ad hoc solution that crucially relies on imaging there is no restriction whatsoever on the form of the conditions, like grid size and pixel shape. Using this function F. But because we are dealing with a physical operational Laplacean amounts to the implicit assump- entity, the requirement of scale invariance imposes a tion that the structures of interest have a spatial extent restriction on the form of F only those functions are close to pixel scale. Moreover, it assumes that the allowed that 'scale properly'. The precise meaning of structures of this scale are meaningful, which is this statement is expressed by the following generally not the case (think of pixel-correlated noise or dithered images). Disregarding the intrinsic dinlensionality of an image *There may be asynimetries in the external environment the system or, in other words, the scaling degree of freedom, is the has to operate in. like gravity, etc.. which might argue for a less cause of the failure of naively applying differential symmetric front-end vision model. On the other hand, this does not really limit the usefulness of these syrnmetries, since they can always methods in image analysis. Dimensional analysis is a be broken in a postprocessing stage. In addition, more restrictive well recognized concept in physics, and its precise (usually task-specific) symmetries may be imposed U posteriovi as mathematical formulation will be used to argue for the well. vol10 no 6 july/august I992 377 Universal law of scale invariance: that are presented in each point of the image, depend- Physical laws must be independent of the choice of ing on its position relative to the boundary. The further fundamental parameters. away from the nearest boundary, the larger the largest scale that is locally represented. On the boundary itself This is equivalent to: there is no scale information left. Reliable local Dimensional analysis: geometry on a given spatial scale can only be found a certain minimal distance proportional to that scale A function relating physical observables must be inde- away from the boundaries. The boundary problem as pendent of the choice of dimensional units. such, however, is clearly a scale-independent problem. similar on all scales, and consists of formulating the Important are those quantities that do not change trade-off between the (scale-independent) propor- under the given scalings. These are called dimension- tionality constant and accuracy together with a rigorous less. It is a necessary requirement to be able to express accuracy quantification. a physical relation in a unit free form. Hence, simple We will not give a rigorous solution for this notorious dimensional analysis will reveal scale invariance. boundary problem. but merely give a qualitative Remember, though, that it is the very notion of scale, indication of how to deal with it. To this end, we may in relation to the law of scale invariance, that justifies resort to the physiology of our own front-end. Here, the method of dimensional analysis. The rigorous way multiple scaling is essentially achieved already at the of formulating dimensional analysis is through the Pi- acquisition stage (due to the many RF sizes), rather theorem (for a detailed discussion see 0lverlh). than by a postprocessing of a fixed-scale sampled image (the output of individual rods and cones is ignored as Inner and outer scale such). Since RF’s never overlap with the ‘boundary’, the boundary problem simply does not arise in our An image is just another physical observable, with visual system. In this operational sense the boundary is inner scales limited to a finite range determined by the non-physical resolution of the sampling device (grid size) and by the These simple though important scale observations field of view. should suffice to support the claim that a multiscale In image analysis, there is a widespread concern with description of image structure is an indisputable neces- discretization effects. Strictly speaking, when we are sity in front-end image analysis. interested in local image structure on the sampling We will introduce a continuous scale parameter a to device’s inner scale, we are facing an apparent under- account for the spatial scaling freedom. It has the sampling problem, from which there is only one escape: dimension of a length and is used to define the notion zooming in on the scene or resorting to a higher of an ‘immediate neighbourhood’ of a point P on scale resolution acquisition. a as the ‘fuzzy’ set of points within a sphere of radius Once having fixed the inner scale, all smaller scale Y(U)0~ U centred at P, i.e. the smallest spatial image geometry has been destroyed and can by no ‘volume’ (a length in 1D. an area in 2D) within which means be reconstructed. This ‘catastrophical’ destruc- the image structure at that scale varies ‘neither too tion is of an intrinsically irreversible nature. One much nor too little’. cannot expect things to be geometricallj correct at the Usually, of course:, if U is larger than the pixel size, limiting lower scale boundary, where we will have the image structure does vary significantly over a ‘spurious detail’. The scales of interest should therefore distance U, because (of irrelevant small scale details. So be relatively large compared to the imaging device’s then U cannot denote inner scale. To reveal the ‘pure’ sampling width. Local image analysis cln these scales a-scale structure of the image. we have to suppress will then be of a continuous rather than a discrete those irrelevant details. This is most easily done in nature (for discretization issues, see also Lindeberg4 . the Fourier domain by suppressing ‘high’ spatial It pays to study the human front-end visual system’ 4, frequences: when interested in an inner scale of order being an astonishingly well performing device, having a we need a cut off frequency of order W(U) =( l/a. evolved over many millions of years. The front-end The question then arises of how to do the cut off. focuses on scales which are considerably larger than the Danielsson and Seger2’ use ‘constant plateau’ filters, eye’s true inner scale: the scale of a typical rod or cone. but they use the presumption that the sampled signal is It is not the output of individual rods and cones that is bandlimited and ignore the scale degree of freedom. transferred, but only a weighted sum over typically From our operational point of view, bandlimitedness is several hundreds of them, making up a receptive field irrelevant and without this presumption, as WitkinX (RF). The profile of such a RF takes care of the small- and Koenderink’ showed, there is essentially only one scale ‘spurious detail’ generated by the individual rods sensible way to do it. Their derivation relies on an and cones by scaling up to a larger inner scale in a very assumption that can be phrased as ‘prohibition of specific way. Only these larger scales are subject to spurious detail’, the interpretation of which has led to further analysis. Indeed. numerous physiological some confusion in the literature. We will not use this support the theory that RF profiles argument, but show that the simple front-end symmet- can be modelled by Gaussian filters of various widths or ries we proposed have exactly the same consequences, their partial derivatives2”, which, as we will prove, and we will emphasize on how they may set the stage precisely turns out to provide a complete solution to for local image analysis in D dimensions. our front-end requirements. We also often encounter problems related to the Natural scaling operator in scale-space device’s limited field of view. Finiteness of the image domain poses restrictions on the largecst inner scales In this section we will derive the unique scaling strategy

378 image and vision computing for D-dimen\ional images (D> 1 ), using its semigroup '8' need not coincide with the familiar additive nature in combination with the front-end vision operator '+'. All that is required by consistency is that 4ymmetries. the set {RJ; S}constitutes a commutative semigroup Linear shift invariance implies that a rescaled image isomorphic to the commutative semigroup of image must be a convolution of the original image by some rescalings: kernel C(x:a)*. so:

L(x:o)= {L,,JG(.:u-)}(x:u) (2)

It is especially attractive to consider this property in the Fourier domain, in which the kernel becomes diagonal: (2) then becomes an algebraic relation:

Y (0:CT) = Y,)(0)'3(0:U) (3) The Pi theorem states that because of conventional scale invariance there are only two independent dimensionless variablesd!? this case. We may take these to be 94 = %/2(,and L? = a w. Let us therefore define: Natural frequency (spatial) coordinates: Natural ,frequency (spatial) coordinates are defined as The consistency requirement that there is a one-to-one correspondence y(~) manifests itself mathe- the dimensionless numbers (X) associated with the +-+a R matically by the existence of an isomorphism { y(cr), frequency (spatial) coordinates w at scale-space (x) = {R;; i.e. a one-to-one map between these level a > 0 through: o} S}, two semigroups preserving the semigroup structure: X L? = U 0 X = - respectively (4) y((~)oy(6= y(aS6) (9) i (7 1 According to the Pi theorem we may thus write the This isomorphism poses a very strong constraint on the kernel %(m; (T) as a function of 0: form of the scale-space kernels. We will now derive an explicit formula for the %(0:a)=2/Y%(f2) (5) semigroup operation 8. On dimensional grounds (manifest scale invariance), any allowable reparametri- For a scalar function. spatial isotropy implies that % sation of a must be homogeneous, i.e. it must have the depends only on the magnitude (Euclidean length) of form D + -AmP for some dimensionless parameters the vector f2: A > 0 and p + 0. Without loss of generality we may put A = 1, since it is merely a scaling factor and (9) must (4(a) = '5 (R) (6) be insensitive to the choice of units. So any allowable reparametrisation of the scale-parameter a can be dct my--7 with L(2 = I X,=lll;. realized by an automorphism 9: Let us choose a to be such that for fixed w the hypothetical zero-scale limit U 1 0 will leave the initial Y:{R,+;s}-,{R,+; +}:aHD~ (10) image unscaled, so: Its inverse is given by: %(L(2)+ 1 asfl 1 0 (7) This means that we include the identity as a limiting, zero-scale kernel. If we assume that ordinary addition applies to {RJ; Also, we require the infinite-scale limit a+ m to +}, then the following identity holds (see (9)): give us a complete spatial averaging of the initial image: y(r)Oy(6) =y(9'-'(9a++G)) (12) %(lZ) 1 0 asfl- x (8) Note, however, that (12) still makes sense in the Performing several re4calings in succession should be limiting case p+f w, for which (10) and (11) by consistent with performing a single, effective rescaling. themselves have no meaning. It is easy to see that this More specifically, if ul, U? are the scale parameters singular case corresponds to the singular idempotent associated with two rescalings %(al),%(a2) respec- semigroups { R; ; max} and { R; ; min} defined by: tively, then the concatenation of these should be a def rescaling '8 (a,)corresponding to an effective scale a,0a2 = max(cr,, D?) Val,a2 E R: (13) parameter a3= a?8 al, in which the additive operator '8' relates the effective scale parameter u3 and: to the parameters U,, U?.It is important to note that drf aI8 cr2 = min (cl,u2) Vul,u2 E R: (14) 'XThevector notation for x and o is legitimate, mice having chosen an arhitrary. fiducial origin for rcfcrcncc. respectively, which emerge as limiting cases from the voll0 no 6 july/augu.~t1992 379 regular monotonic semigroup defined by: To single out a unique scale-space kernel, we need a final constraint on the parameter p. For a consistent def Ii o,Q#2= V.-l;+CT! interpretation of %(O)as a spatial rescaling it is natural to impose the condition of separability in D > 1 dimen- We stress the fact that the null elements of these sions: semigroups arise from non-physical limiting proce- dures: for p < 0 we have U,)= x, whereas for p > 0 we have cro = 0. Note that we have already decided on the null element = 0 by our limiting requirement (7), hence only positive p-values will be of interest ot us”. in which is given by the magnitude of the projection We now turn to the derivation of scale-space kernels vector (fl.6,) t,.This condition states that an isotropic compatible with (12). It is convenient to consider the rescaling can be obtained either directly through %(Cl) frequency representation: if we define: or through a concatenation of rescalings %(ill)by the same amount in each of the independent spatial %(n)%(n.) (16) directions et. i= 1 . . . D separately. Indeed, only in this way we can think of (T as a natural length unit in an we get from (12): isotropic sp:~~. The separability requirement fixes p = 2, so s = r2, not .- itself, is the ‘additive’ 4i (a)4i (h) = G (a+ 6) (17) parameter: The general solution to this cons1 raint is a normalized exponential function: Note that the idempotent kernel (20) is not separable. 4i (a)= exp (an) (18) A convenient choice for a is obtained by letting scale coincide with Gaussian width in the spatial domain, so or: that = - 1 /2. So we have finally established the unique scale-space %(Cl) = exp(aW) (19) kernel. In the Fourier domain it is given by: in which (Y is an arbitrary, negative constant (see (8)), whose absolute value can be absorbed into the definition of the scale parameter. For the limiting case we have $(n)=lim,,, or, in dimensionful coordinates: %(@’), i.e. %(O) if OSCk1. Together wib9 the limiting conditions (7) and (8) (and 1 taking %(1) = limnt %(a)for definiteness), we thus % (w ; #) = exp (-;; a22) find the following idempotent kernel: In the spatial domain it is given by the normalized convolution kernel: in which xIis the indicator function defined by:

Note that in the spatial domain the Gaussian kernel In dimensionful coordinates this becomes: C(x;.-) has a scale dependent amplitude. It has the dimension of an inverse D-dimensional volume: we may write it as a product of an explicit volume factor %(m: =x(o.l/u](4 (22) U) and a dimensionless, scaled Ciaussian: i.e. an ideal low-pass filter with cut-off l‘requency w = l/#. In his article, Mallat proposes such iin idempotent (29) semigroup requirement as it starting point for a so- called ‘multiresolution approximation’h; the operator which approximates a given signal at a resolution U is a with: linear projection, satisfying (9) and (13) The general, regular case comprises a so-called (strongly) continuous semigroup of operators for each value of p, as opposed to the idempotent semigroup (22): Therefore:

%(w:c) = exp(aa”w”) (23) df’XX(X) = d”xG(x; U) (31)

*This is not a rcstriction: a reflection p H-p rncrely amounts to all is a scale invariant measure. interchange uf the comp’lementary concepts of scale and resolution, The prefix semi in ’semigroup’ expresses the intrinsi- i.e. inverse scale. The reader may verify that by expressing all physical requirements in terms ol resolution will yield thc same cally irreversible nature of rescaling. Put differently, result. rescaling gives rise to irreversible catastrophes in the

380 image and vision computing topological structure ot the original image. In ‘lorward symmetries, as well as some additional constraints, direction’. i.e. when increasing scale, there are no noticeably the concatenation or semigroup requirement ‘acausal’ bifurcations (no creation of spurious detail). (9) and the separability condition (24). But it is It is important to stress that filtering a given image important to stress that the very assumption that it has with (;(.xi a)does tzot yield an image with an inner to be a scalar subject to these scaling properties has scale a, but with an inner scale a8ao,if a,, is the been explicitly added by our desire to find a filter that inner scale of the original image. Each layer L(x; a) merely scales its input, but is not part of our funda- in scale-space can in turn be regarded as an initial mental front-end vision requirements. Indeed, it is only image with an inner scale equal to rr8a0, and it is by virtue of these extra constraints that we were able to only when U %- a. that the inner scale of L(x;a) single out the Gaussian as the unique solution. approximately equals (7. This observation is especially In the next section we show that the Gaussian scale- important if one is interested in scales only slightly space kernel is merely the lowest order member of a larger than one pixel, for we can at most associate some complete, hierarchically ordered family of scale-space ‘effective’ inner scale with the originally sampled filters, all of which are compatible with our front-end image. but we cannot expect the scale-space require- vision requirements. ments to be nicely fulfilled near pixel scale‘. Having defined nalural length units it seems rather trivial to remark that we now have a natural distance Complete hierarchical family of higher order mwsure for the separation of two points XI and X, on operators a given level (T: Although in principle the one-parameter Gaussian kernel is all one needs to generate a scale-space, it is (32) highly insufficient for a complete, local description of image structure. In fact, this filter is the physical Note its singularity at the highest (fictitious) resolution counterpart of the trivial mathematical identity oper- U = 0. When viewed with an infinite resolution, two ator in the sense that it extracts a scaled copy of a given distinct points are al~ays‘infinitely’ far apart. since input, representing the same scene merely on a there can be an arbitrarily large amount of structure different inner scale. inbetween. In this section we show that the front-end vision Significant changes due to rescaling will occur only requirements sec admit many more scale-space opera- when we increase scale by an order of magnitude rather tions beyond mere scaling. We derive a complete. than by some absolute amount. Hence it is more hierarchically ordered family of n-th order tensorial natural to reparametrize our scale parameter, thus scale-space filters {yI, l,,(a)}~=o(in both spatial and removing the artificial singularity at (T = 0: Fourier representation), and discuss their role in front- end image analysis. The previously established Gaus- sian scale-space kernel naturally fits into this family as Natural scale parameter: just the zeroth order, scalar member. A natural, dimensionless scale parameter T is obtained Since the kernels are diagonal in the Fourier domain, by the followirig repararnetrization of a : it is easiest to consider their Fourier representations. It is a common misconception to think that rotational cr=~exp{~}or r=In{v/~} TE(--,++) (33) invariance of the kernels implies that they only depend on the length 11011 of the vector o. This only holds Note that we are forced to introduce, on dimensional for scalar kernels. It is easy to construct other, tensorial grounds, a ‘hidden scale’ F, which carries the dimen- kernels within the isotropy constraint. In fact. any sion of a length. It is a property of the image, not of the tensorial kernel must be proportional to a tensor universal scale-space kernel. An intrinsic scale inherent product containing n factors o,with n = 0, 1. 2. . .. to any imaging device that presents itself is the since o is the only independent vector available. The sampling width or pixel width. Now we have a proportionality constant must be a scalar. Putting in a dimensionless scale parameter T that indicates in a scalar multiplier %(w: a) to account for proper scale continuous manner the order of magnitude of scale fixing we can formulate the following claim: relative to F and that can take on, at least in theory. any real value. If we take F to be the sampling width: Claim 1 A complete, hierarchically ordered family of then T = 0 corresponds to a resolution, where the multiplicative scale-space kernels is given in the Fourier width of the blurring kcrncl is of thc same order of representation by the set: magnitude as the pixel width F, i.e. the inner scale of the original image. This sets a practical lower limit to the kernel widths, at which discretization effects will start to contribute to a significant degree. The range Alternatively, in the spatial representation, by the set of T E (- x, 0) corresponds to subpixel scales that are convolution filters: not represented in the image and in which all structure has been averaged out. When building up a scale-space {Gi,.. (35) it is most natural to use an equidistant sampling of T, because it is this parameter that precisely formalizes the Note that the zeroth order kernel G is the only physical notion of scale. An equidistant sampling of scalar kernel. All higher order kernels are tensorial absolute scale (7 would violate scale-invariance. quantities. For example, the first order kernel, i.e. the In this section we have derived the unique scalar gradient, is a vector. scale-space kernel that satisfies all our front-end vision The proof of this claim is given below. where we vol I0 no 6 julyiaugust I992 381 show that this cartesian family is sufficient for a degree of freedom. 'rhus is is also a minimal set. The complete determination of local image !structure. Con- zeroth order member of the Cartesian family represents sider a given image L(x;a) at a fixed scale. If we are the scaled identity operator, the higher order members interested in the geometrical structure of this image in constitute the physical. scaled counterpart of a the neighbourhood of some fixed point Y E RD we may complete family of mathematical linear dijf'erential consider its Taylor approximation up to a sufficient operators. In Figure 1 the spatial profiles of a number order N of operators are shown. An alternative, but equivalent way of looking at the L(x+6x; a)= completeness of this set of filters has been given by J. Koenderink and A. Van Doorn, who took the isotropic N1 diffusion equation as a fundamental starting point for 7L;,, . ,in (x;a) 6Xi, . . . 6x;,,+ 0((?)"+I) (36) the derivation of the complete family of scale-space ,,=o n. filters or local neighbourhood operators'-',2",since this equation uniquely prohibits the generation of 'spurious Note that a scaled image L( . ; U)'!?I,,, * G( . ; a) is a detail' in scale-space '. smooth function for all U > 0, no matter how 'dirty' the We end this section by noting that, although in theory initial condition Lo may be (within certain very weak nth order derivatives of a scaled image are all well- restrictions). Digitized images are, by the very fact of defined, there is only one operational way of calculat- being digitized, always 'dirty' in the differential ing them, viz., by a convolution of a lower scale image

geometric sense (even in the absence of noise). In with their corresponding tensor components G,, , I,, theory, Lo may even be everywhere discontinuous. (x; a) (cf. (37)). This brings us to another important Scaled differentiation, as opposed to ordinary ('un- result: scaled') differentiation, is well-posed by nature. Note the following identity: Result 2 The operations of sculing and differentiation are intrinsically related. (37) Because of the one-to-one correspondence between

In other words, we have obtained the following the Gaussian kernels yI,,, .l,, (a) (on a fixed scale: important result: 'horizontal image structure') and the Cartesian partial differential operators, it is straightforward to invoke Result 1 To obtain the Cartesian partiar' derivatives of the powerful machinery of well-established mathe- order n of a rescaled image L(x; a)on(' only needs to matical disciplines, like differential geometry, tensor convolve the original image L,,(x) with the correspond- calculus and invariants theory in a robust way. This ing partial derivatives of the zeroth-order Gaussian enables us to study the visual system as a 'geometry G(x;U). engine"2. The calculation of derivatives is most easily done in the Fourier domain, in which the filters are diagonal: Differential invariants in scale-space Once we have calculated the N-jet, we are provided with all partial derivatives of the image up to and Since (36) represents the image's local geometry at x including Nth order. However, one such derivative, L, and at scale U up to any desired order of precision N, say, does not represent any geometrically meaningful we have proven the completeness of the constructed property, since the choice of the coordinate axes is Cartesian kernel family. Each essential kernel compo- completely arbitrary. If we restrict ourselves to an nent in the family corresponds to an independent orthonormal basis, we still have the possibility of

Figure I. Some Gaussian derivative projVes: G,, G,, and GXYY

382 image and vision computing rotating a given coorclinatc frame over any angle. Indeed, this suffices to obtain a 2-component tensor Clearly, such a choice does not have anything to do { L,, L.,,},since these two components do transform in a with the image. On the other hand, it is also clear that closed way, Their transformations can be written in any property that is invariant under such a coordinate matrix form: transformation, must be connected to the image ‘itself‘ and therefore can be given a geometric interpretation. The reverse is also true: every image property can be expressed through an invariant function (or ‘invariant’. in short). This shows that there is an intimate relatioi It is clear that also the coordinates {x. y},taken as a between invmriants theory and differential geometry. pair relative to some fixed origin, constitute a tensor. Although it is possible to give a coordinate-free Let RI,stand for the (i, j)th element of the transforma- description of image geometry in principle, we do need tion matrix, then the abovementioned transformation coordinates in actual calculations. To this end we can be conveniently written in condensed form as: simply choose any allowable coordinate system, but at the same time assure that the functions of interest are independent of that particular choice. Note that the Li = R,, L, (31 1 term ‘invariant’ always implies the existence of a group Because L, has only one free index it is called a of allowable transformations. In our case this is a rather l-tensor or vector. But we can also consider tensors ‘minimal’ group, viz. the product group of SO(D), the with more free indices (n-tensors). An example of a special ortliogorzal group of all rotations in D dimen- 2-tensor is the Hessian, i.e. the set of all second order sions (sometimes extended to 0 the full ortho- (D), partial derivatives: its transformation is given by: gonal group, by admitting reflections) and T(D), the translation group. This is a very basic group, which we believe is especially important in medical imaging. Li~=RikRjlLkl (42) Those special combinations of image derivatives that It has exactly three essential components, viz. L rA, exhibit such an invariance under cartesian coordinate L,, = L,,, L),,. This means that because of the transformations are called cartesian differential symmetry of the tensor, we cannot choose all its intwiants. To get a basic understanding of the theory of components independently. cartesian invariants, it is necessar to understand the 7x By now it may be obvious that all partial derivatives basics of carteJiun terisor culciilris- . In this section we of a given order n form the components of an n-tensor. briefly outline how to construct functions describing For each of its free indices its transformation law true image properties. contains a transformation matrix with one free and one It turns out that we can in fact construct an infinite contracted index: number of invariants in each point of the image. but we argue that there is only a small number of independent ones among these. This is to say that. to a given order (43) N, we can build any geometrically meaningful quantity These derivative tensors share the additional property as a function of those (typically very few) independent of being symmetric, i.e. we can freely inter-change or irreducible invariants. indices without any effect, e.g. L,, = L,,. Thus there is a significant reduction of essential components. Manifest invariant index notation Of great importance are the following two constant It is clear that we cannot form an invariant out of a tensors: the symmetric Kronecker tensor S,, , which is single derivative like L ,, whose value always crucially always a 2-tensor, and the antisymmetric Le‘vi-Civita relies on the choice of the x-axis and thus varies among coordinate systems. It is the subject of tensor calculus tensor E~,, , ,iD, which is a D-tensor in D dimensions. to describe the transformation behaviour of quantities These tensors have invariant components, independent of the choice of the coordinate axes (this property like L,, called tensor components. A ‘closed set’ of tensor components, although given with respect to makes them well-defined). They are defined as follows: some arbitrarily chosen coordinate system. does 1 ifi=j however constitute a coordinate independent object, 6ij = and called a tensor. The meaning of the word ‘closed’ in this { 0 otherwise context is that, after a change of coordinates, each 3 if (il . . . ir,) is even tensor component acquires a new value that can be expressed as some function of the old tensor compo- &i,. . .in ={ -1 if(i l...iD)isodd (44) nents. This function only depends on the transforma- 0 otherwise tion parameters involved. i.e. a set of rotation angles and translation components (such a smoothly para- When including reflections into the transformation metrized group of transformations is called a Lie group, the &-‘tensor’ is not a true tensor in the above group“). For example, in 2D the partial derivative L, sense anymore. Its significance still remains as a so changes. after a rotation over an angle a. according to called relative or pseudo-tensor. Its transformation law the following rule: is then slightly modified so as to render its components invariant again :

L, =cosaL,+sincuL.,. (39) def E:,... in- (det RI-’ Ri,jl...Rir~j,]&j,...jr,= This shows that L, cannot be the single component of a tensor. We should at least add the component L, to it. E~,,,.~,]with det R= fl (45) vol10 no 6 julylaugust 1992 383 Once we understand the transformation behaviour of We call such a requirement a guuge condition, and the the derivatives, we can try and combine them into resulting coordinates gauge coordinates. It should (absolute or pseudo-) invariant combinations. This is, always be checked whether a gauge condition is in fact, very easy. Given a set of tensors, the way to admissible, i.e. realizable through a suitable transfor- form an invariant is by means of full contractions and mation provided by the transformation group at hand. alternations of indices in a tensor product. A contrac- The directional derivative operators applied to tion is a procedure that pairwise reduces the number of the image will yield invariants. Invariance becomes free indices in a tensor by performing a restricted manifest by writing differential invariants using these summation over them. More precisely, a contraction invariant differential operators. of i, j in L,, by definition yields L,, = L,,S,, (this To illustrate the use of differential geometry and at contraction is also referred to as the truce of L,,). the same time show the power of gauge coordinates An alternation of D tensor indices is defined as a full that are tuned to a particular problem, let us derive an contraction of these indices onto the D indices of the expression for the isophote' curvature K (cf. Clark"). &-tensoras. for example, (in 2D) in: The meaning of curvature of a planar curve may be intuitively clear; it is a measure for the local deviation 2' E,] L,Lk L,k = (L, - L;) I-, , t I,, I,, (15, \, - L,,) (46) from its tangent line". A useful definition is the following one: put a coordinate frame with its origin in Of course, functions of wveral invariants are them- the point P of interest on the curve. The x-axis should selves invariants. be tangent to the curve. The curve can then locally be If we consider reflections as well as rotations (after described by a function y (x) on an open interval around all, these also respect orthonormality of the coordinate x=O. In this system the curvature in the origin is basis), then the &-tensor become5 a relative tensor and defined as the second derivative y"(0).So in the (v, w)- invariants containing an odd number of these become system centred at P we have. by definition, K = w"(0), relative invariants, i.e. quantities that are invariant up in which w(v) denotes the function describing the to a possible minus sign (which shows up only when the isophote locally near P(i2=0, U' = 0). Now the orientation of the coordinate basis is reversed). In fact, isophote passing through the point P is implicitly given we can always write a relative invariant using exactly by the equation L = L/,. Taking first and second one &-factor. This follows from the fact that any tensor implicit derivatives of this equation with respect to v product of an even number of F-tensors can be written yields: in terms of &tensors. In 3D: L,.+L, w' = 0 and &ijEkl= azkajll- ailajk (47) L,,,+2L,,,,W'+ l.,l,M"~+ L,,w"=O (48) Similar relations hold in arbitrary dimensions. Relative invariants are related to oriented geome- In P we have. by our suitable choice of gaug%;,L,,(0)= trical objects. We will often speak of invariants, but 0, hence also w'(0) = 0. So in P we have K = w"(0) = silently admit relative invariants, too. The term abso- -LL,,,lLR,.In a similar way. one may proof that the lute invariant is then used to explicitly exclude relative flow line curvature ,U, i.e. the curvature of the integra\ ones. curves of the gradient vector field (the orthogonal trajectories of the isophotes), is given by the formula Gauge coordinates ,U = - L,,,/L,, . So we have the following result: In the index notation one refrains from choosing any particular coordinate frame. The invariance of a Func- tion then manifests itself through a full contraction of indices in the tensor products that make up the function. For this reason we speak of manifest invari- Because these invariants are closely related to simple ance when using this notation. isophote properties, they look simplest when written in Another way of forming manifest invariants is by this particular gauge. singling out one particular. geometrically meaningful It may come as a surprise to learn that, although we coordinate frame and using directional derivatives have calculated the isophote curvature in a simplifying along its axes. There are several ways to set up such a coordinate system, it takes hardly any effort to arrive at coordinate frame. One useful way is to require, in each the general expression in arbitrary coordinate systems. point of the image separately, one axis t'3 coincide with The method goes as follows: write down an invariant in the image gradient direction (called the w-axis hence- manifest index notation that reduces to the simplified forth). The other axis (v-axis) is then automatically expression evaluated in gauge coordinates (simply directed tangentially along the isophote. By its very guessing in combination with a modest amount of definition, L, vanishes identically. This is precisely the foresight usually does the trick in one go). By invari- motivation for this particular gauge. Hecause of the ance, the two expressions are guaranteed to represent rotation freedom, this kind of requirement is always the same geometrical property. The following index allowed, provided the image gradient does not vanish". notation for K and ,U (4Y) can he easily justified this *The (v, w)-gauge is ill-defined in points with a .vanishing gradient, way: but these points form a countablc wt. at least in generic images, irrrages tliat arc Lupulugically stable against local distortions. Blurring a nontrivial image to a certain levcl of resolution always yields 3 generic image (which may, howcvcr, bccomc obscured in a computer implementation by truncations ciuc to thc finite precision of L- values). 'An isophote is a contour of constant image valucs.

384 image and vision computing '1'0 see how this work\, consider the isophote curvature freedom. In the (v, w)-gauge in which Id, =O. thew expression in (SO). Its denominator is the third power correspond to the set {L, I,,,. I,, ,. I,, ,,. I.,,,,}. of the gradient magnitude. i.e. Id: = (L,~lL,ll)"'.while Therefore, we might foresee the existence of a finite its numerator can be translated from index into gauge number of so called irreducible polynonticil in~~ri(iiit\. notation as follows: i.e. a set of basic polynomial invariant\ in term3 of which all other invariants can be expressed. However KL:= L,F,/L~~F~/(arbitrary system) = plausible this argument may seem the proof of it in the general case is far from trivial. A proof of existence was -I,, ,.I,: (gauge system) (51) established by Hilbert, although the mathematical literature does not seem to provide an algorithm for the (the last equality follows from L,;= 0, F,.,,.= actual construction of such an irreducible set. In the -~,,,,.=l and C~.,.=F~,.,,.=O,so that the only non- simple case of the 2-jet, however. such an irreduciblc trivial term is the one with indices i= 1 = M' and set can readily be givenz8. In 2D: i= k = v). Similar arguments hold for ,u in (SO). Evaluating the contractions in some Cartesian coordi- :j = {L,L,L,, L,L,,L,, I.,,, L,,L,,) (55) nate system will give us the explicit formulas for K and ,u in (SO): An example of reducibility is given by the tollowing identity: (52) (56) (L:- Lf) L,, +I,, I,, (L,,,- Lx,) P= (53) (id: + Ly dwhich can be verified most easily in the (j),q)-gauge, defined by the gauge condition fa,],, = 0. i.e. the It is clear that this explicit notation obscures the coordinate system in which the Hessian matrix of all Cartesian invariance property and can become very second order derivatives is diagonal. This gauge is cumbersome when there are many contractions to he admissible, since it can always be realized by a suitable performed. rotation. Because of invariance, this reducibility pro- Another example of a nianilest invariant is given by perty holds in arbitrary coordinate systems. the well-known Laplacean: In the next section we will take one or two examples for each of the lowest order jets (N = 0.. .3). For the sake of presentation we only consider the 2D case. but it must be stressed that the dimensionality does not in which we have used the expression for isophote pose a fundamental restriction to the concepts intro- curvature (39). This cxaniple shows that, in general. duced. In fact, in much of the previous theory we invariants can be interrelated. More specifically, the refrained from specifying the dimension of space (v, wv)-gauge nicely reveals the shortcomings of edge explicitly wherever this was irrelevant. detection methods based on Laplacean zero crossings often encountered in the literature'h. ". The term I,,,.L,, is the second order image derivative along the gradient APPLICATIONS direction, i.e. normal to the isophote. If we define an cdge as the locus of points of maximum gradient A trivial 0-jet example of a differential invariant is 1.. magnitude, which sccms it quite natural choice. then the local image intensity (in an implicitly given point P the zero crossings of AI- can only accurately describe and on an implicitly given scale U). A simple I-jet edges if the isophotes are sufficiently straight, so that example is = 1) V I, 1). the image gradient magni- the curvature term can be ignored. It is well-known that tude. It is most pronounced on edges. where there is a this condition ceases to hold near corners and this is relatively strong change of intensity values over a one deficiency of this zero crossings method. Another relatively short distance. Note that this is just the deficiency is the detection ofpl1rrt7tom edges'-', i.c. non- Canny edge detector'" (see also De Michcli et d7'').A edge points detcctcd by this zero crossings method. simple 2-jet example is the familiar Laplacean I.,, = Even if the isophotes are straight, the I,, I, zero AI,. Figure 2 shows some differential invariants ;is they crossings detect not only local maxima of I, ,,, (Irw were calculated for noisy test images on several scales. ctfgcs).but also local minima. which are the least likely We already pointed out that there are bnsically only candidate edge-points of :ill. two independent, pure second order 'irreducibles'. which can be taken as Ljjand Llild,,(55). Any other pure second order property can be expressed as some Complete sets of' differential invariants combination of these two. From ;I geometrical point of It may be evident that we can construct an infinite view this is clear, since a second order image property is number of invariants from any finite set of tenwrs by always related to its 'deviation from flatness'. The local means of tenwr multiplication\ and contractions. Rut it image intensity profile can deviate from its first order i\ also clear that the N-jet in ;I given point only has a behaviour in two directions independently. There are finite number of independent degree\ of freedom. For two principal directions, corresponding to the coordi- example, in 2 dimension\ local image \tructure up to nate axes of a system in which the mixed derivatives \rcond order i\ completely determined by five indepen- vanish (the Cp, q)-gauge). The invariants ancl I,,,, dent ?-jet component\, which i\ 1 less than the number can then be regarded as measures for the deviation of of essential component\ becauw of the gauge degree 01 flatness in these principal directions. This way the Figure 2. Some simple examples of invariants calculated for noisy test images on various scales. The additive gaussian noise imposed serves to illustrate the rclbustness of scale-space differentiation, (a) 1st order invariant L, for noisy straight edge: ‘edgeness’; (b)2nd order invariant LvvL$for noisy polygon (16 corners): ‘cornerness’; (c) 3rd order invariant LvvvLz- 3LvvL,,~ Lt for noi:iy inflexion: ‘bendedness’

Laplacean L,, = L,,+ L,, turns out to be twice the Figure 3 shows the light blobs of a NMR image on mean deviation from flatness, whereas the square root various scales. The complementary dark blobs and of L,, L,, = L;p + L:y is an absolute measure for the indifferent patches have been suppressed. total deviation from flatnessi3. The ordered light blob patches are reasonable Another geometrical property closely related to this primitives for a fixed-scale segmentation, and an across- ‘deviation from flatness’ property is the notion of light scale linkage algorithm might be set up using these and dark ‘blobs’ in the image. These ‘blobs’ can be ordered fixed-scale segments to define linkage criteria given an exact meaning by looking at the sign of the for a more realistic, multiscale image segmentation. following invariant, called umbilicity (U):

(57) DISCUSSION AND CONCLUSIONS In this paper we have shown that the fundamental Note that we have normalized U such that motivation for the construction of a scale-space is given -1s US+l. Dark and light blobs (or ‘hills’ and by the physical nature of images and the universal law ‘dales’) now correspond to patches with equally signed of scale invariance. Constraints arising from the lack of principal deviations, i.e. with positive U, whereas the a priori geometrical knowledge naturally lead to the complementary, indifferent (‘saddle-like’) patches have Gaussian kernel and its derivatives in D > 1 dimensions. negative U. The blobs are separated from the indif- The operations of scaling and differentiation are ferent regions by the zero-crossings of U. We can single essentially intimately related. The study of invariants out the light blobs by looking at the sign of the under a certain group of image transformations gives a Laplacean in addition: blobs with negative (positive) robust mathematical basis for the study of image Laplacean are light (dark) blobs. structure.

386 image and vision computing t

C Figure 3. Patch classification based on 2-jet differential invariants calculated for an NMR image on various scales. (a) Scaled original image; (b)light blobs, i.e. patches with positive umbilicity and negative Laplacean; note that these binary invariants are by their very nature unstable in (nearly) flat regions, since a small perturbation SL of L may easily cause thein to flip. This is .seen to occur in particular in the flat background; (c) as (b) but now the patches have been weighted by the zeroth order images from (a) (on corresponding scales) so as to obtain continuous invariants again. This trick preserve.s the patches and may be used to obtain a hierarchically labelled patch classification by assigning a priority number to each patch corresponding to the relative ranking of its average intensity value (this ranking is not shown here). Thc patch with the highest average intensity value will then be the ‘mostpronounced blob’ at that scale, etc. It ip clear that the unstable patches surrounding the skull will acquire a low priority label in this way

The theory allows for the good understanding of The resemblance between the complete family of many current available feature detection mechanisms. scale-space kernels and mammalian receptive field e.g. the Canny edge detector. Laplacean zero cros- profiles known from numerous neurophysiological data sings. isophote curvature. etc.. and puts these in the is encouraging: it suggests that our theory of differen- perspective of a broad class of differential invariants up tial invariants in scale-space is a promising attempt to some order. The theory is applicable in many areas towards a robust simulation of some of the successful of computcr vision. geometric routines actually working in the human This theory may be further developed by a more visual system. systematic study of the irreducible invariant ‘building This theory may have an important impact on various blocks’ up to an!’ order, by inclusion of the temporal topics in medical imaging, notably image segmentation. domain”, sterco, optic flow. and also by studying the classification and pattern recognition. ‘deep structure‘ in scale-space (,i.e. the structure across scales), incorporating our local theory into ;I global model. A particularly important, but still unanswered ACKNOWLEDGEMENTS question is also how to operationally gauge the local nieasurements and how to establish a (multilocal) This work was supported by the Dutch Ministry of connection. Economic Affairs Grant [VS-3DM]-S0249/89-01. vol 10 no 6 jul~~iriugir.st1902 387 REFERENCES 16 Olver, P J Application., of I.ie Groups to Dgfererz- fial Equations, Vol 107 of Graduate Texts in Mathematics, Springer-Verlag, Berlin (1 986) pp 1 Burt, P J, Hong, T H and Rosenfeld,, A ‘Segmenta- 21 8-22 1 tion and estimation of image region properties 17 Hubel, D H and Wiesel, T N ‘Receptive fields, through cooperative hierarchical computation’, binocular interaction, and functional architecture IEEE Trans. Syst. Murr. CG Cvbern., Vol 11 No 12 in the cat’s visual cortex‘. J. Physiokogy, Vol 160 (1981) pp 802-825 ( 1962) pp 106-1 53 2 Brunnstrom, K, Eklundh, 0 and Lindeberg, T 18 Young, R A ‘The gaussian derivative model ‘On scale and resolution in the analysis of local for machine vision: visual cortex simulation’, image structure’, 1st Euro. Con& Cimput. Vision, Karlsruhe, Gerrriany (1083) pp 1019-1023 Antibes, France (1990) pp 3--13 19 Young, R A ‘Simulation of human retina1 function 3 Korn, A ‘Toward a \ymbolic representation of with the gaussian derivative model’, Proc. IEEE intensity changes in images’, IEEE Tram. PA MI, CVPR Ch2290-5, Miami, FL (1986) pp 564-569 Vol 10 NO 5 (1988) pp 610-025 20 Koenderink, J J and van Doorn, A J ‘Receptive 4 Lindeberg, T ‘Scale-space for discrete signals’, field families‘, Biol. Cybern., Vol 63 (1090) pp IEEE Trans. PAMI. Vol 12 No 3 (1990) pp 233- 29 1-298 245 21 Danielsson, P-E and Seger, 0 ‘Rotation invariance 5 Babaud, J, Witkin, A P, Baudin, M and Duda, R 0 in gradient and higher order derivative detectors’, ‘Uniqueness of the gaushn kernel for scale-space C‘omput. Vision, Graph. CP: Image Process., Vol 49 filtering’, IEEE Tram. I’A MI, Vol 8 No I (1986) (1990) pp 198-221 pp 26-33 22 Koenderink, J J ‘The brain a geometry engine’, 6 Mallat, S G ‘A theory Tor multiresolution signal Psychol. Res., Vol 52 ( 1990) pp 122-127 decomposition: The wavelet representation’, 23 Kay, D C Tensor C~rlcirlu.~,Schaum’s Outline IEEE Trans. PAMI, Vol 11 No 7 (1989) pp 674- Series, McGraw-Hill. New York (1988) 694 24 Clark, J J ‘Authenticating edges produced by 7 Koenderink, J J ‘The structure of images‘, Bid. zero-crossing algorithms’, IEEE Trans. PAMI, Cybern., Vol 50 (1984) pp 363-370 Vol 11 (1989) pp 43-57 8 Witkin, A ‘Scale space filtering’, Proc. Int. Joint 25 Spivak, M A Comprehensive Introduction to Dif- Conf. on Artif. Intell., Karlsruhe, Germany (1983) ferential Geometry (vol. 1-V), Publish or Perish pp 1019-1023 Inc., Berkeley, CA (1970) 9 Yuille, A L and Poggio, T A ’Scalini; theorems for 26 Marr, D C and Hildreth, E C ‘Theory of edge zero-crossings’, IEEE Truns. PAMl, Vol 8 (1986) detection’, Proc. Roy. Soc. London B, Vol 207 pp 15-25 (1980) pp 187-2,17 10 Poston, T and Steward, I Catastrop,he Theory and 27 Torre, V and Poggio, T A ‘On edge detection’. its Applications, Pitman, London (1978) JEEE Tram PAMJ, Vol 8 No 2 (1986) pp 147- 11 Blom, J, ter Haar Romeny, B M and Bel, A Spatial 163 derivatives and the propagation of noise in Gaus- 28 Kanatani, K Group- Theoretical Methods in Image sian scale-space. Submitted, May 1091 Understanding, Vol 20 of Series in Information 12 Jain, A K Fundamerituls of Digital Image Proces- Sciences, Springer-Verlag, Berlin (1990) sing, Prentice Hall, NJ (1989) 29 Canny, J ‘A computational approach to edge 13 Koenderink, J J and van Doorn, A J ‘Representa- detection’, IEEE Trrrt1.c;. PAMI, Vol 8 No 6 (1987) tion of local geometry in the visual system’, Biol. pp 679-698 Cybern., Vol 5.5 (I 987) pp 367-375 30 De Micheli, E, Caprile, B, Ottonello, P and Torre, 14 Fourier, J The Anulytical Tlirory of Neut, Dover V ‘Localization and noise in edge detection’, IEEE Publications, New York (1955) Tram. PAMI, Vo1 10 No I1 (1989) pp 1106-1 117 15 Strutt, L R J W ‘The principal of similitude’, 31 Koenderink, J J ‘Scale-time‘, Biol. Cyhern., Vol Nature. Vol XCV (March 1915) pp 66-68. 644 58 (1988) pp 15‘+-162

--% 388 image arid vision computing