R[p ], Rip]. Axiomatic Quantification of Image Resolution. Xiao-ming Ma Center of Environment Science Peking University Beijing , China.

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1 Axiomatic Quantification of Image Resolution Ming Jiang* School of Mathematics Peking University Beijing 0087, China. Ge Wang Department of Radiology University of Iowa Iowa City, IA 52242, USA ovember, 200 Xiao-ming Ma Center of Environment Science Peking University Beijing 0087, China. Abstract In this paper, we generalize the axiomatic system for quantification of image resolution, prove that any resolution measure consistent with the generalized axiomatic system must be a homogeneous symmetric function of order //2 of the eigenvalues of the covariance matrix of the PSF. If the axiomatic system is augmented with an affine transformation axiom, the resolution measure is proved to be proportional to the squared root of the geometric means of the eigenvalues of the covariance matrix of the PSF. Introduction There are various kinds of imaging systems, such as camera, electronic and optic microscopes, medical and industrial tomographic scanners, and radars. Image resolution of an imaging system refers to the capability of discriminating two neighboring objects, and is the primary factor determining the system performance. Various criteria were introduced for quantification of image resolution, c.f. []. Using an axiomatic approach [2], it was recently proved that the resolution measure of a nonnegative one-dimensional spatially invariant imaging system should be proportional to the square root of the variance of the PSF. The result was later extended to multi-dimensional case in [3] if the axiomatic system is augmented with an orthogonal transformation invariance axiom. The resolution measure is proved to be the squared root of the arithmetic means of the eigenvalues of the covariance matrix of the PSF. However, the axiomatic system in [2, 3] is too restrictive. E.g., the affine transformation axiom (Axiom A in 4) is not consistent with it, though this axiom is in a rather general form. *Currently, Ming Jiang is a visiting professor with Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, IA 52242, USA. His work was supported in part by an KBRSF grant. In 2, a general axiomatic system is constructed by replacing the strong cascade axiom (i.e., the combination axiom in [2, 3]) with a weak cascade axiom. In 4, it is demonstrated that any resolution measure consistent with the generalized axiomatic system must be a homogeneous symmetric function of order /2 of the PSF eigenvalues. In 4, after the generalized axiomatic system is augmented with an affine transformation axiom it is proved that the resolution measure should be proportional to the squared root of the geometric means of the eigenvalues of the covariance matrix of the PSF. 2 General Axiomatic System The imaging system is assumed to be nonnegative spatially invariant: o=p,i () where i is an input, o is an output, and p _> 0 is a PSF with finite first, second order moments, and positive definite covariance matrix. Let Rip] denote an image resolution measure of the system (). The convention is that the smaller Rip] is, the finer details the system can resolve. Let I be the x identity matrix, G ~ the -dimensional Gaussian distribution with zero mean vector and positive definite covariance matrix a: Ga(x ) = e_½zt,.a-~.z (2) (2~)-~ v/det (a) for x E R, where det(a) is the determinant of a. Our general axiomatic system consists of the following seven axioms that are postulated for Rip]. Axiom (onnegativity) Rip] >_ 0, a = RIG I] > 0, where c~ is finite constant. Axiom 2 (Continuity) R is continuous w.r.t the weak topology of measures in the following sense: if fapn'f "~ frp'f, Vf e c0(a ), then R[p ], Rip] X/0/$ IEEE 697

2 Axiom 3 (Translation Invariance) For every x0 E R, let pxo(x) = p(x- xo), Vx E R, then R[~o] = Rip]. Axiom 4 (Rotation Invariance) For every orthonormal matrix T, let pt(z) = p(t. x), Vx E R, then R~T ] -- R~9]. Axiom 5 (Luminance Invariance) For every c > 0, let p[c] = c. p(x), Yx e a, then Rip [c]] = Rip]. Axiom 6 (Homogeneous Scaling) For any finite/~ > 0, let p[/3](x)= p(flx) then then Rip[/3]] = RIp] /3 " Axiom 7 (Weak Cascade) There exists a function J such that for any PSF p, the resolution measure of the imaging system with the PSF q = p, p, which is a composite system consisting of two identical systems p in serial, only depends on the resolution measure of the system p: R[q] = Rip p] = Y(R[p]). (3) Axioms -6 are the same as their counterparts in [2]. Axiom 7 is new, which is less restrictive than the corresponding combination axiom in [2], which states that Axiom S (Combination or Strong Cascade) There exists a function F such that for any two imaging systems with PSFs Pl and P2, respectively, the image resolution measure of the composite system by serial connection of the two systems pl and p2 is R~] = R[p~ ;~.] = FIRe,I, RIP, I]. (4) 3 General Measure First recall that the covariance matrix a of a PSF p is defined by fr~(~j - ~j)(~k - ~k). ;(x) dx ~rj,k = fr p(x) dx (5) where # = (Pl," "", ~) is the mean value vector, with #k -- fan xk " p(x) dx fr p(x) dx (6) Lemma 3.. For any integrable function p(x) with finite first moment, under Axioms 3 and 5, Rip] = R[pl], where pl(x) = fr p(x)dxp(x- #) and # is the mean value vector defined as in (6). Thus, it is sufficient to consider PSFs that are probability density functions with mean zero. We will make such an assumption for all PSFs in the following. Lemma 3.2. If R[.] is a resolution measure satisfying Axioms, 5, 6, then for a Gaussian PSF G ~'I with covariance matrix ~. It ~ > 0, we RIO ~~] = o~. v~, (7) where a is the positive constant in Axiom. Proof. ote G ~'I (x) = c.gi(~x), for some positive c. By Axioms 5 and 6, R[G ~'I] = x/~" R[GI] (7) follows immediately. V] Lemma 3.3. Let J be the function in Axiom 7, if R[.] is a resolution measure satisfying Axioms, 5-7, then J(r) = v~r, for r > 0. (8) Proof. By Axiom, a = R[G I] > 0. For r > 0, let p be the Gaussian PSF defined by p = G(~) 2"I. Then, r by Lemma 3.2, Rip] - a. -~ = r. Because ;,; = G((~):+(~)~) ' =G(@) ~', (9) we Rip, p] = v/-2r. Hence, by Axiom 7, J(r) = J(R[p]) = Rip, p] = x/~r Lemma 3.4. If R[.] is a resolution measure satisfying Axiom 7, there exists a function K such that for any PSF p the resolution measure of the imaging system with a PSF q = p, p,..., p, where n = 2 k k = l... n terms only depends on the resolution measure of the system p and n: J R[q] = R~ p,... p] = K(R[p], n). (0) Y n terms Furthermore, if R[.] is a resolution measure satisfying Axioms, 5-7, then for n = 2 k k =,... K(r,n) = v~r, for r > 0. () Proof. Equation (0) is obtained after recursive use of Axiom 7. In fact, it is easy to get K(~, ~) = J o J...o 4(~). -v- k terms (I ) follows easily from Lemma 3.3. rq [5 698

3 Lemma 3.5. Let R[.] be a resolution measure satisfying Axioms, 5-7. For any PSF p, n= 2 k k= l... let qn(x) = (V~) " p(v/-n x) (2) Corollary 3.7. Under the same assumptions of Theorem 3.6, if R[.] further satisfies Axiom ~, then g(a) a f((7) = R~9] is a homogeneous symmetric function of order -~ of the eigenvalues of (7, and then gn - qn * qn *"" * qn, (3) y J n terms R[gn] = RIP]. (4) g(ta) = v~g(a). (9) Proof. With an orthonormal transformation y = Tx, a PSF p(x) is mapped to p(t try), and the new covariance matrix is D = T(TT tr. By Axiom 4 and Theorem 3.6, Proof. By Axioms 5 and 6, R[qn] = R[p] v~" Lemma 3.4, the conclusion follows immediately. K] Theorem 3.6. If R[.] is a resolution measure satisfying Axioms -3, 5-7, and p is a PSF with finite first and second order moments and positive definite covariance matrix (7, then R[;] = Moreover, the function is homogeneous of order ~, By R[c ] (6) f(t(7) = v/'tf((7), for t > 0. (7) Proof. Let qn and gn be constructed as in Lemma 3.5. By Lemma 3.5, for n = 2 k, k =,..., R[gn] = R~]. Since the covariance matrix of p is positive definite, by Theorem of [4] (p. 22), for any w e R, gn converges to G vaguely in the sense of measure as measure density functions. Since all the gn has the same resolution Rip], by Axiom 2, Rip] = R[G~]. Finally, for any t > 0, by Axiom 6, f (t. (7) -- R[G t~] - R[G7~,] = vqr[a = For a positive definite covariance matrix (7, there exists an orthonormal matrix T such that Fq (7 = Ttr DT, (8) where D is a diagonal matrix with positive diagonal elements being the eigenvalues of (7, A,...,A. Let us define A = (A,..-, A)tr. Rip] = f((7) = R[PT, r ] = f(d). The order of A,...,,~ can be freely altered by an appropriate orthonormal matrix T, hence f((7) is a symmetric function of A,...,A. The property of g can be easily verified. [2] Corollary 3.8. Under the same assumptions of Corollary 3. 7, if =, then Rip] = where (7 is the variance of p. (2o) Proof. By Corollary 3.7, Rip] = f(e). Since f is homogeneous, by (7), f((7) - x/'ef(). By Axiom, f() = a, then the conclusion is obtained, which is the same as the finding by Wang and Li [2]. E] Corollary 3.9. Under the same assumptions of Corollary 3. 7, if all eigenvalues of the covariance matrix of p are the same as ~ > O, then Rip] = o~. V/'~. (2) Proof. This is obtained by Corollary 3.7 and Axiom. E:] If a PSF is radially symmetric, i.e., p(x) = P(lixll), x E R, its covariance matrix must be diagonal with the same diagonal elements By Corollary 3.9, we = - J[x[J2P([JxJ])dx. (22) Corollary 3.0. Under the same assumptions of Corollary 3. 7, for a radially symmetric PSF p, RE.] = (23) 699

4 4 Arithmetic and Geometric Measures In the preceding section, a family of resolution measures has been characterized by our general axiomatic system. In practice, specific resolution measures can be determined among these candidate measures for measurement. In this section, two important measures, arithmetic and geometric measures are identified with the strong cascade axiom and an affine transformation axiom, respectively. The he justification of the arithmetic measure was already finished [3]. The focus of this section is to present the affine transformation axiom and justify the geometric measure. Let us summarize the result of the arithmetic measure in [3]. Theorem 4.. If R[.] is a resolution measure satisfying Axioms -6, and the strong cascade Axiom 8, then for any PSF p with finite first and second order moments and positive definite covariance matrix o, we... iv... E ~J i.e., the squared root of the the arithmetic means of )~,"",A (up to a positive constant), where o, )~I,''',)~ are the eigenvalues of or. A natural question is what the resolution measure of an imaging system could be if affine transformations can be applied in the imaging process. In the following, we present the affine transformation axiom in a very general form, and prove that a new measure can be derived if the general axiomatic system is augmented with the affine transformation axiom. Axiom A (Affine Transformation) There exists a function H such that for any non-singular matrix T and any PSF PT -- p(t x), x E R, w e R[pT] = H(R[p], T). Theorem 4.2. If R[.] is a resolution measure satisfying Axioms -7 and the ajfine transform Axiom A, then for any PSF p with finite first and second order moments and positive definite covariance matrix or, we Rip] = a i.e., the squared root of the the geometric means of ~,"",A (up to a positive constant), where o, ~,"",A are the eigenvalues of o. Proof. By Theorem 3.6 and Corollary 3.7, we )~j Rip] = RIP'I, where D is the diagonal matrix with positive diagonal elements )~,"",A. Let B = V~ be the diagonal matrix with diagonal elements v~l,"", ~X/X. Then, G D = cg I where c is a positive constant. Therefore, Rip] = H(c~,B). For any PSF q and any two non-singular matrices U and V, by Axiom A, H(R[q], U. V)= R[qvv ] = R[(qv)v ) = H(R[qv], U) = H (H(R[q], V), U). Specifically, if U is orthonormal, by Axiom 4, H(R[q], U. V) = H(R[q], V). Similarly, if V is orthonormal, (24) H(R[q], U. V) = H(R[q], U). (25) Hence, pre- or post-multiplication of an affine transformation T by an orthonormal matrix does not change the value of H(R[q], T). Furthermore, if Y is orthonormal, U and W are non-singular, then H(R[q], U. Y. W) = H (H(R[q], WV), U) = H (H(R[q], W), U) = H(R[q], U. W), where the second equality follows by (24). Consequently, if T is a product of non-singular matrices and orthonormal matrices, those orthonormal matrices can be removed from the product without changing the value of H[n[q], T]. Let C(i, j) be the matrix obtained by exchanging the i-th and j-th columns, R(i,j) the matrix by exchanging the i-th and j-th rows of the identity matrix, for < i, j <. For ~ > 0, let S(~) be the diagonal matrix with the first diagonal element being ~ and other diagonal elements being. Then, B can be decomposed into a product of S(v/~l),..-,S(Ax/~), and a finite number of C(i, j)'s and R(i, j)'s. More precisely, we D = S(X~I) H R(,j)s(x/~)C(,J) j-'2 Since each of C(i, j)'s and R(i, j)'s is orthonormal, we ' [... R~] = g(a, H S(X/~J)) = H(a,S(\I H Aj)). ~ 700

5 Therefore, the function g(a,"",)~) in Corollary 3.7 only depends on the product of ~,"",A. Let us define a function ~ for this dependency Since eta( H = = g(t ) = l-[ for t > 0, we Hence, o(t. l] aj) = j-- aj). ~(~,~j) -- Aj ~(). j-- j-- Because ~() = g(,...,) = c~ by Axiom, the conclusion follows immediately. [3 the weak cascade axiom and the affine transformation axiom. Further work is underway to reveal relationship among resolution measures, and generalize the theory into the real and complex domains. Acknowledgments This work was supported in part by the ational Institutes of Health (IDCD R0 DC0058). Ming Jiang's work was supported in part by an KBRSF grant. References [] Tom L. Williams, The Optical Transfer Function of Imaging Systems, Institute of Physics Publishing, Bristol and Philadelphia, 999. [2] G. Wang and Y. Li, "Axiomatic approach for quantification of image resolution," IEEE Signal Procesing Letters, vol. 6, pp , 999. [3] J. A. O'Sullivan, Ming Jiang, Xiao ming Ma, and Ge Wang, "Axiomatic quantification of multidimensional image resolution," IEEE Trans. Information Theory, 999, accepted. [4] L. HSmander, The Analysis of Linear Partial Differential Operators, vol. I, Springer-Verlag, Berlin, second edition, Discussion and Conclusion In [2, 3], it was proved that the function F in Axiom S is F(x, y) = v/x 2 + y2. (26) and that the square of f defined in (6) is linear. Therefore Axiom S restricts the possible resolution measures at the arithmetic aspect, by the arithmetic property of the combination function F. That is one reason why we call it the arithmetic measure. The affine axiom Axiom A is an axiom about the resolution change under affine geometric transformations. This is also one reason why we call it geometric measure. Another reason for adopting those terms is that fortunately they are consistent with their mathematical constructions as the arithmetic and geometric means of its constituent elements. We presented a general axiomatic system for quantification of image resolution and demonstrated that any resolution measure consistent with the general axiomatic system must be a homogeneous symmetric function of order ½ of eigenvalues of a PSF. With additional axioms, resolution measures can be uniquely determined in the family of resolution measures permissible in the general axiomatic system. Specifically, the squared root of the geometric means of PSF covariance matrix eigenvalues been singled out with 70