IMAGE INVARIANTS. Andrej Košir, Jurij F. Tasič
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1 IMAGE INVARIANTS Andrej Košir, Jurij F. Tasič University of Ljubljana Faculty of Electrical Engineering Tržaška 25, Ljubljana, Slovenia ABSTRACT A pre-processing technique for image features extraction is presented in the paper. It is based on algebraic properties of image transform ψ that are invariant to a given geometrical transformation of an image. A vector space H of digital images is introduced. Image invariant is represented by a family of linear maps R. A linear transform ψ defined on a space of images H pertained to a given image invariant R is constructed. The main idea of this paper is contained in a property of map ψ, that maps all images sharing the same feature described by a family R into a single point. Gained information of the digital image extracted by the transform ψ is suitable for further analysis of image features. Some examples of families R are presented. An example of an introduced transform ψ pertained to a given family R is described. 1. INTRODUCTION An important part of image feature extraction is development of pre-processing techniques. Feature extraction can be seen as a controled loosing of information brought by a picture. In order to do so, we first have to disregard an irrelevant information and preserve an important part of it. This is the main goal of our proposed pre-processing technique. We describe a transform that maps all images sharing the same feature into a single point. Image invariants are object descriptors that remain unchanged under different image transformations, which appear in image acquisition and digital image processing. In this paper they are described in terms of a family of mappings R. Theoretical consideration and construction of a map ψ : H C for particular image invariants extraction is presented in the paper. A solution presented in the paper is construction of a linear map with desired properties which then can be applied as an on-line pre-processing algorithm. An important part of this paper is introduction of a vector space of images H that allows us theoretical consideration as well as comfortable implementation of the proposed technique. 2. VECTOR SPACE OF IMAGES In order to analyse image invariants we define a vector space H of digital images. A model for definition of a space H introduced here is a mechanism image intensity adjustment as implemented in a human eye. A computer representation of a digital image is n m matrix of intensity values ranging in an interval I. The difficulty here is that a vector space structure can not be simply added to a set I n m. We overcome this problem by introducing an equivalent relation, which defines a vector space of images, appropriate for theoretical consideration and computer representation. In order to achieve it we introduce a set l c ( Z 2 ) = {s : Z 2 IR; s(i, j) = 0 a.e.} of all functions on Z 2 with nonzero values at only finally many points (i, j) Z. We also introduce a map where Λ : l c ( Z 2 ) L 2 (IR 2 ) s s Sink, Sink(x, y) = c sin(ω 1x) sin(ω 2 y) x y and f g denotes a convolution of functions f and g. It is known that the map Λ is injective and though bijective on its image ImΛ L 2 (IR 2 ). It is desirable all images we are dealing with should have the same resolution. Let us now chose arbitrary but fixed m, n IN as an image resolution and define W R = [1, n] [1, m], W Z = W R Z 2. Since a map Λ is injective any image s l c ( Z 2 ) can be mapped into Λ(s) L 2 (IR 2 ) and then resample to a fixed resolution m n. If a procedure of resampling is denoted by Rs : l c ( Z 2 ) l c ( Z 2 ), a map P 1 = Λ Rs
2 is a projection on l c ( Z 2 ). A relation P1 2 = P 1 is obviously valid. From the construction of a projection P 1 we can see for any s l c ( Z 2 ) and (i, j) W Z the following relation holds, (P 1 s)(i, j) = 0. It is reasonable an image ImP 1 is denoted by ImP 1 = l(w Z ) = {s : W Z IR}. Now we can focus on suitable image intensity normalization. It is accomplished as follows. All intensity values s(i, j) of an image s l(w Z ) are linearly interpolated between their minima min s(i, j), (i, j W z ) and their maxima max s(i, j), (i, j W z ). Therefore a projection P 2 : l c ( Z 2 ) l c ( Z 2 ) defined by ( (P 2 s)(i, j) = ) {s(k, l)}s(i, j) + min{s(k, l)}(1 s(i, j)). max k,l i,j Clearly ImP 2 = l(w Z ). A construction of P 1 and P 2 shows that P 1 normalises a resolution of the digital image and P 2 normalises its intensity. Another important feature we have achieved is (P 2 s)(i, j) I for any s l c ( Z 2 ) and I is a bounded interval. From this we conclude it is suitable to project a set of images l c ( Z 2 ) by a projection P = P 2 P 1. A definition of a vector space of digital images we are introducing in this paper is based on this projection. First we point out that a set l c ( Z 2 ) has a canonical vector space structure over a field IR. We also observe that a relation s 1 s 2 d P (s 1 ) = P (s 2 ) is an equivalent relation on l c ( Z 2 ). Finally we define a vector space of images H = l c ( Z 2 )/. Its members are equivalent classes [s] = {s 1 l c ( Z 2 ) : s 1 s}, denoted by capital letters A, B,.... Space H inherits a vector space structure of l c ( Z 2 ) over a field IR and we have λ[s] = [λs], s l c ( Z 2 ), [s 1 ] + [s 2 ] = [s 1 + s 2 ] This concludes a definition of a vector space of images H. The obtained space is suitable for theoretical consideration. On the other side, a computer representation of a real matrix s l c ( Z 2 ) equals to P (s) H. From the equality P (E ij ) = E ij, where {E ij : ij} is a standard base of l(w Z ), follows that a base of the space H is again {E ij : ij} and its dimension equals to the dimension of L(W Z ). Note that multiplication by λ > 0 of an image A H results to the same image s and multiplication by λ < 0 results to a negative of an image A. This is rather strange but desirable property of introduced vector space of images H. 3. IMAGE INVARIANTS DESCRIPTION AND DECOMPOSITION OF H Image invariants are descriptors that remain unchanged under different geometrical image transformation which arise in image acquisition and digital image processing. We describe them in terms of a family of mappings on a vector space H and in terms of subsets of a space H. The most important examples of geometrical transformations are translation, rotation and scaling. First we introduce a parameter set T and denote a family of transforms on space H by R = {R(t) : H H; t T }. For example, when rotation is considered, for a given image A H a set of images {R(t)A; t T } is simply a set of rotated images A. About our notation, for s l c ( Z 2 ), A = P (s) we write R(t)A = R(t)[s] = [R(t)s]. An elegant way of describing family R is by embedding a set of image pixels indices {1,..., n} {1,..., m} into complex numbers C. A well known structure of Mobious transforms can be applied for description and for analysis of different image transforms. An aim of image invariants description is a decomposition of space H suitable for further image feature analysis. For obvious reasons we wish such a decomposition would be a partition, H = H 1... H n. Subsets H i are disjoint and should be generated by the introduced family R describing chosen geometrical transformations. Let us choose a family of maps R. There is a subset L A H pertained to a given image A H defined by L A = {R(t)A; t T }. We wish a subset L A be a vector subspace of H. Unfortunately, this is usually not possible to achieve. We still
3 consider this situation here in order to examine desired properties of such decomposition. To do so we have to guarantee a conclusion λ 1 A 1 + λ 2 A 2 L A. For any A 1, A 2 L A and λ 1, λ 2 IR, there must exist such t T and λ IR that λr(t)a = λ 1 R(t 1 )A + λ 2 R(t 2 )A, where R(t 1 )A = A 1 and R(t 2 )A = A 2. In order to satisfy the above condition, family R should meet the following property, for any λ 1, λ 2 IR and t 1, t 2 T there must exists λ IR and t T such that λ 1 R(t 1 ) + λ 2 R(t 2 ) = λr(t). (1) When the above conclusion is valid for a chosen family R, we can define a subset L R H independently from a chosen image A H, and binary operation on H satisfying a relation where L A = L R A, L R A = {B A; B L}. An example of operation could be matrix multiplication. We have got another description of image invariant in terms of a subsets of a space H given by L R H when L A is a subset in H for any A H 4. GROUP ACTION AND FAMILY R In previous section we have shown that the introduced family R brings a desirable decomposition of a vector space of images H. If a set L A is not a subspace in H, we have to apply new techniques for describing a decomposition of H. In this paper we propose to use a group action technique. First we briefly introduce it, for more information see [4]. Let (G, ) be a group and let M be a set. A group action of a group (G, ) on set M is a map G M M (g, u) g u for which the following is valid: (i) e u = u for group identity e and u M; (ii) g (h u) = (g h) u for any g, h G and u M; There are two sets playing the major role here, a stabilisator of u M S(u) = {g G : g u = u} and an orbit of u M O(u) = {g u : g G}. Their meaning derives from the following facts. S(u) is an invariant subgroup of a group G and in the case G is finite an index of a subgroup S(u) equals to the number of elements in orbit O(u), we write [G : S(u)] = O(u). Since orbits are disjoined sets and any element of M is contained in some orbit, a space M can be decomposed, M = O(u). u M We apply a group action to describe a decomposition of the space H regarding a family R = {R(t) : t T } by introducing a group structure to a parameter set T. If a group operation is again denoted by, then a group action of a group (T, ) on a space of images H is defined by t A = R(t)A, t T, A H. An aim of such decomposition is exactly the same as in described in the previous section. In the case of different image analysis like object recognition the same property should be assigned to all images from an orbit S(u). A presence and absence of one particular object of the digital image does not depend on its rotation, for instance. We can introduce an equivalent relation to a space of images H regarding a family R, A R B. It is reasonable to define an object extraction map not on H but on a quotient space H/ R. We will consider such definition later. 5. DESCRIPTION OF A FAMILY R In this section a description of image invariants as a family of maps R is presented. We can not simply describe it as a map on a set of indices Z 2. For instance, in the case where a rotation is considered, it may happen that for (i, j) Z 2 a rotated index (i cos(t) + j sin(t), i cos(t) + j sin(t)) is not a point of a set Z 2. This is the reason why we have to move from a set of indices Z 2 to a IR 2, perform a transformation on IR 2 and then project a result back on Z 2. Let g : IR 2 IR 2 be a bounded function (both coordinates are bounded in IR). The following diagram shows how a function g defines an image transformation on l c ( Z 2 ), l c ( Z 2 ) Λ L 2 (IR 2 ) Γ h L 2 (IR 2 ) Rs l c ( Z 2 ),
4 where for w L 2 (IR) we write (Γ h w)(x, y) = w(h(x, y)). When a function h is given, a pertained image transformation R is defined by a compositum R = Rs Γ h Λ : l c ( Z 2 ) l c ( Z 2 ). A transformation on a space H is then given by a commutative diagram l c ( Z 2 ) Λ L 2 (IR 2 ) P H 6. EXAMPLES OF INVARIANTS The most important image invariants are rotation, translation and scaling. Since we will need their definition later we introduce a pertained functions on IR 2 describing them in terms of a previous section. To a given function h : L 2 (IR 2 ) L 2 (IR 2 ) a parameter t T must be added in order to describe a family of image invariants R = {R(t) : t T }. So we have h : L 2 (IR 2 ) T L 2 (IR 2 ). Since the family of invariants is uniquely defined by a family of maps h all is left to be described is such a family of maps ROTATION For the case of rotation we chose T [0, 1] and define h 1 R(x, y, t) = x cos(2πt) + y sin(2πt), h 2 R(x, y, t) = x sin(2πt) + y cos(2πt) TRANSLATION If a maximum possible shift toward main axes are chosen N x, N y respectively, we chose T [0, 1] 2 and h 1 H(x, y, t 1, t 2 ) = x + t 1 N x, h 2 H(x, y, t 1, t 2 ) = y + t 2 N y SCALING A parameter space for scaling is chosen T [0, 1]. If a maximum scaling factor equals to N we define h 1 S(x, y, t) = tnx, h 2 S(x, y, t) = tny. 7. MAP ψ Up to this point we have prepared all what is needed for description of properties of a map ψ : H H. It is said a family R = {R(t) : t T } is invariant for a map Ψ if for any A H the following is valid, ψ(a) = ψ(r(t)a), t T. The reason for such definition is obvious. In order to extract a given feature of the digital image for which a rotation of an image is irrelevant, an information on rotation should be disregarded. To ignore such a property, all images containing the same object should be mapped into a single point. When different positions of an object of an image A H is described as R(t)A, the above definition gives us desired properties to a map ψ. In terms of group action, we introduce a group structure to a parameter space T. The above definition then says an image ψ(o(a)) of an orbit O(A) must be a single point. A group action techniques is then a useful tool for constructing maps on H possesing desired invariants. 7.1 AN EXAMPLE OF MAP ψ To conclude this section we briefly describe a construction of a map on H with invariants described above, i.e. translation, rotation and scaling. Let A H and A = P (s). A construction is evolved on an image r l c ( Z). Consider a rotated, scaled and translated replica s(x, y) = r(h S (h H (h R (x, y, α), x 0, y 0 ), ))) = r((x cos(α) + y sin(α)) x 0, ( x sin(α) + y cos(α)) y 0 ), where α is rotation angle, scale factor and x 0, y 0 transitional offsets. The Fourier transforms of r and s are related by S(u, v) = e iφ(u,v) 1 R(ρ 1 (u cos(α) + v sin(α)) x 2 0, 1 ( u sin(α) + v cos(α)) y 0 ), In order to derive a transformation possessing desired properties, we have to eliminate parameters describing geometrical transformations, i.e. rotation angle α, scale factor s and translation offsets x 0 and y 0. We observe that spectral magnitude S(u, v) is translation invariant. Let us denote r p (θ, ρ) = R(ρ cos θ, ρ sin θ) s p (θ, ρ) = S(ρ cos θ, ρ sin θ). Spectral magnitude is a periodic function of the polar angle θ. Since the original image is real, r p (θ + kπ, ρ) = r p (θ, ρ), k Z.
5 Half of the magnitude field is then sufficient to preserve required information. Trigonometric relations 1 (u cos(α) + v sin(α)) = ρ cos(θ α) 1 ( u sin(α) + v cos(α)) = ρ sin(θ α) lead us to s p (θ, ρ) = 1 2 r p(θ α, ρ/). Image rotation shifts the function s p (θ, ρ) along the angular axis and it can be reduced by using a logarithmic scale for radial coordinates. For λ = log(ρ) and κ = log() we define polar-logaritmic representations r pl (θ, λ) = r p (θ, ρ) s pl (θ, λ) = s p (θ, ρ) = 1 ρ 2 r pl(t α, λ κ). By Fourier transforming the above relations we obtain 9. REFERENCES [1] E. Rivilin, I. Weis Local Invariants for Recognition IEEE Transaction on pattern analysis and machine intelligence, vol. 17, No. 3, January [2] F. Heijden Edge and Line Feature Extraction Based on Covariance Models IEEE Transaction on pattern analysis and machine intelligence, vol. 17, No. 1, January [3] Q. Chen, M. Defrise, F. Deconinck Symetric Phase-Only Matched Filtering of Fourier-Mellin Transform for Image Registration and Recognition IEEE Transactions on pattern analysis and machine intelligence, vol. 16, No. 12, December [4] I. A. Faradžev Investigations in algebraic theory of combinatorial objects Kluwer & Co., Boston S pl (ζ, ξ) = 1 2 e 2πi(ζκ+ξα) R pl (ζ, ξ). We have acchived rotation and scaling now appear as phase shifts. It shows that a description r pl (θ, λ) is independent of parameters ρ, l, x 0 and y 0 and a map r(x, y) r pl (θ, λ) is then invariant to rotation, shift and scaling what was our goal. If we define ( ψr)(x, y) = r pl (ζ, ξ), a map ψ is obviously invariant on translation, rotation and scaling. Finally we define ψ(a) = [ ψs] = P ( ψs), where s l c ( Z 2 ) and A = [s] = P (s) H. 8. CONCLUSION We have proposed a pre-processing technique for digital image features extraction. It is based on an introduced map ψ that is invariant to a given geometrical transformation of a digital image. An example of such a map with desired properties is presented in a paper. It is defined on an introduced vector space of images H suitable for theoretical consideration as well as for a computer implementation. A description of image invariance is also included in the paper. An aim of a map ψ is an extraction an irrelevant information from a digital image and is used as a pre-processing for different image feature extraction, mainly object recognition.
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