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A Matrix-Based Approach to the Image Moment Problem

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A Matrix-Based Approach to the Image Moment Problem A Matrix-Based Approach to the Image Moment Problem JUDIT MARTINEZ, JOSEP M. PORTA and FEDERICO THOMAS Insitut de Rob`otica i Inform`atica Industrial (CSIC-UPC), Llorens Artigas 4-6, 08028 Barcelona, Spain {porta,thomas}@iri.upc.edu Abstract. An image can be seen as an element of a vector space and hence it can be expressed in as a linear combination of the elements of any non necessarily orthogonal basis of this space. After giving a matrix formulation of this well-known fact, this paper presents a reconstruction method of an image from its moments that sheds new light on this inverse problem. Two main contributions are presented: (a) the results using the standard approach based on the least squares approximation of the result using orthogonal polynomials can also be obtained using matrix pseudoinverses, which implies higher control on the numerical stability of the problem; and (b) it is possible to use basis functions in the reconstruction different from orthogonal polynomials, such as Fourier or Haar basis, allowing to introduce constraints relative to the bandwidth or the spatial resolution on the image to be reconstructed. Keywords: The moment problem, image moments, moment-invariant image approximations, basis selec- tion. 1. Introduction reconstructing an image from estimates of its mo- ments. In [13], the moment problem has also arisen A very common problem in physics and engineer- when approximating an image to simplify it. In ing is known under the general title of “the mo- this work, a finite number of moments are used to ment problem” [18]. Corresponding to some fi- reconstruct an approximation of the Fourier coeffi- nite number of observations, we are given a set cients of the corresponding image. Unfortunately, of moments –the integrals of various given func- images are not treated as 2D discrete functions. tions with respect to the measure. Since these Instead, using a zigzag scan, they are converted moments will not determine the measure uniquely, into a linear form. the problem consists in deciding which is the best The reconstruction of an image from a set of its estimate. In pure mathematics, this problem dates moments is not necessarily unique. In other words, back to Theodor Stieltjes who proposed it in a pa- it is an ill-posed problem. Therefore, all possible per published in 1894. His work on the moment methods to solve it must impose extra constraints problem was continued and extended primarily by so that the solution becomes unique. Hausdorff and Hamburger. For classical overviews The standard reconstruction method of an image of the subject, including comprehensive historical from some of its moments is based on the least- attributions of classical results, see [1] or [16]. A squares approximation of the image using orthog- more recent survey of the wide range of approaches onal polynomials [17, 15, 12]. Polynomials are the to the problem, including applications, is [5]. most straightforward choice among all possible or- This paper is concerned with the moment prob- thogonal basis functions because they can be eas- lem for images or, more precisely, with the prob- ily related to the multinomial functions that are lem of reconstructing an image from a set of its used to obtain the geometric moments. Legendre geometric moments. and Zernike polynomials were first used in [17]. The moment problem for images arises in sev- They are orthogonal polynomials for continuous eral applications. In [11], the problem of inverting variables in rectangular and polar coordinates, re- the Radon transform is reformulated into that of spectively. However, they are not orthogonal for 1 2 Submitted to the International Journal of Mathematical Imaging and Vision discrete variables, contrary to what is assumed by matrices are always embraced by parenthesis when some authors [15, 6]. Tchebichef polynomials were superscripts refer to power or transpose. used in [8] and [12] which are orthogonal polynomi- Any discrete image of size a b, say Iab, can als in the discrete domain. Independently of the be seen as a vector in a×b or, alternatively,× as a chosen set of polynomials, the standard method bidimensional functionℜ that maps all the points of assumes null projection coefficients onto the cho- the uniform lattice 1, 2,...,a 1, 2,...,b onto { }×{ } sen polynomial set of order higher than the maxi- real values. Then, Iab can be uniquely expressed mum order of available moments. This solves the as a linear combination of the functions of a basis ill-posesness and the solution becomes unique. In set, i.e., a set containing ab linearly independent order to avoid this assumption, which is difficult bidimensional functions, which will be denoted by kl to interpret in terms of the image properties, a the set Ξab , so that maximum entropy method was proposed in [14]. { } It consists in obtaining the image with maximum a b kl kl entropy with the desired moments. Solving the Iab = α Ξab. k l problem using Lagrange multipliers permits to ob- X=1 X=1 tain an explicit form of the reconstructed image in To avoid in what follows this double summation terms of an exponential function. Alternatively, in the formulation of the problem, we introduce a [10] proposes minimizing the divergence of the im- matrix-based formulation, but first we need some age, instead of maximizing its entropy, using also definitions. a variational approach. Unfortunately, both ap- proaches assume a continuous domain for the im- Definition 1 (Basis matrix). The functions in age. any basis set are assumed to be separable and In this paper, we propose a reconstruction kl equally defined for both coordinates, i.e., Ξab = method that permits introducing constraints that k l t k l φa(φb) , where φa and φb are vectors which can be interpreted in terms of image properties, will be grouped in matrices of the form Φab = such as bandwidth or spatial resolution. We also 1 b (φa,..., φa) called basis matrices. show how the standard least-squares reconstruc- a tion method can be seen as a particular case of Definition 2 (Gram matrix). The matrix Γb = t it. First, we introduce the necessary mathematical (Φab) Φab, containing the inner products between background. Section 3. reformulates the standard the elements of the corresponding basis matrix, is method in terms of the presented formalism. Sec- called a Gram matrix. tion 4. generalizes the result to other orthogonal a k l bases different from polynomials. Finally, section Note that, since Γb [k, l] = φa, φa , the Gram h i 5. contains the conclusions and prospects for fu- matrices are diagonal for orthogonal basis sets and ture research. the identity for orthonormalized basis. Definition 3 (Projection matrix). The matrix 2. Mathematical background containing the projection coefficients of image Iab kl onto the first m n elements of Ξab are called 2.1. Notation and definitions projection matrices× , which can{ be expressed} as m t Ω Φ t I Φ . Let zm denote a column vector, z its trans- mn = ( am) ab bn ∈ ℜ m pose, and zm[k], with k = 1,...,m, each of its el- kl k t l m×n Note that Ωmn[k, l] = Iab, Ξab = (φa) Iab φb. ements. Likewise, let Zmn denote a ma- h i ∈ ℜ Image Iab can be approximated in terms of the trix of size m n and Zmn[k, l], its element (k, l), × first m n elements of Ξkl by where k = 1,...,m and l = 1,...,n. For simplic- × { ab } ity, square matrices will only have one subscript. m n Superscripts are used to denote any parameter on ˆmn kl kl t Iab = λ Ξab = Φam Λmn (Φbn) , which a matrix depends. Two unary matrix opera- k=1 l=1 tions are used: ( )t denotes the transpose of a given X X −·1 kl matrix; and ( ) , its inverse. To avoid confusions, where m a, n b, and Λmn[k, l] = λ . · ≤ ≤ Submitted to the International Journal of Mathematical Imaging and Vision 3 Definition 4 (Expansion matrix). If the image approximation coefficients λkl are chosen so that the truncation error is minimized using the least- Iab squares error criterion, Λmn is called an expan- sion matrix. mn Iab 2.2. A Theorem Lemma 1. The approximation of image Iab, in the least-squares sense, can be expressed in terms ˆImn of the projection matrix Ωmn as ab ˆmn t Iab = Φam Λmn (Φbn) a −1 b −1 t = Φam (Γm) Ωmn (Γn) (Φbn) t −1 = Φam ((Φam) Φam) t −1 t Ωmn ((Φbn) Φbn) (Φbn) Figure 1: Lemma 1 permits to obtain the best approx- mn imation, ˆIab , of image Iab in the least-squares sense, = (Φ )− Ω (Φ )+, am mn bn contained in the subspace represented by the plane in gray, that is, its orthogonal projection onto this sub- where ( )− and ( )+ stand for the left and right space. Theorem 1 is a generalization of this lemma Moore-Penrose· pseudoinverses.· mn that permits to obtain the image Iab contained in kl other subspaces, here represented by a white plane, Proof. Since λ is chosen so that the truncation mn mn that also projects orthogonally onto ˆIab . Both Iab error is minimized according to the least-squares mn and ˆIab preserve the first m × n moments of Iab. error criterion, the subspaces generated by the er- ror and that in which the approximated image is contained are orthogonal. That is, There are infinite images, not only Iab, that lead to the same projection matrix, Ωmn, resulting ij t Ξab, Iab Φam Λmn (Φbn) = 0, from projecting them onto the first m n elements h − i kl × of the basis Ξab . The above lemma permits to for i = 1,...,m and j = 1,...,n.
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