Blind Estimation of PSF for Out of Focus Video Data

A method has been proposed for blindly estimating the Point Spread Function (PSF) of video data. The PSF’s of the images in a sequence are assumed to be of compact support and hence admit FIR modeling. The zeros (roots) of the Optical Transfer Function (OTF) i.e. the Fourier transform of the PSF are first estimated by locating the singularities of the magnitude of the cross power spectrum. These roots are then used for spectrum factorization leading to complex polynomial approximation of the OTF. It has been shown that a 2 0 spectrum can be factorized under the assumption of local isotropy around the roots. 1. I N T R O D U C T I O N Estimating the PSF blindly is a challenging problem undertaken by numerous researchers in recent years. To overcome lack of sufficient information (i.e. blindness), a priori information need to be introduced. The following approaches can be identified in the literature: 1. Parametric model fitting approach [4][7][8] 2. Optimizationbased approach [ 1][14][ 151 [21]. 3. Adaptive estimation [ 121 [13][18] 4. Using higher order statistics [X][lO][ll] All these methods, in fact, introduce the a priori information by imposing empirical models either on the actual PSF or the statistics of the PSF and the input signal. The use of higher order statistics in recent years has proven to be a very efficient approach, but at the cost of increased complexity and non-Gaussianity assumption of the statistics of the input signal. The approach adopted in this work is based on using cross-statistics rather than auto-statistics, and hence increased efficiency is achieved without increased complexity. Clearly the method would then require at least two observations of the same signal. The proposed method is based on identifying the zeros of the OTF and factorizing the spectrum. The idea of characterizing the PSF by finding and using the zeros of the OTF is, in fact, relatively old [8]. The approach, however, has been abandoned due to the difficulty of detecting the zeros of the OTF in noise. Use of a denoising stage proposed in later literature [5][7] does not improve the estimation This work was partially supported by the ONR/MURI grant N00014/95/1/0521 (ARPA order C635) of the zeros either, as was originally reported in [3], too. Herein, we will demonstrate that the zeros can be readily identified using cross-statistics rather than the spectrum of the observed signal itself. Moreover, it will be shown that the spectrum factorization in 2D has an asymptotic solution although no direct solution is known. The following section will describe the method followed by a section in designing a simple stable inverse filter for testing the method. Results are illustrated in the final section. 2. M O D E L I N G THE PSF In this section, we will define our model for the PSF (or the OTF). We will then develop a method for the blind estimation of the OTF. As we will see in the next section, the proposed method is particularly applicable to video data, since it requires two images with only relative shifts between them. In fact, in video sequences transformations between successive frames can be closely approximated by shifts and a small rotation angle which can be compensated by registration. We will show below how an adaptive PSF model can be built assuming local isotropy. For this purpose we will assume that the OTF of the imaging system is linear shift invariant and of finite duration and hence can be represented by a FIR filter: k(u) = ckezp(-i(u,k)) (1) k e T where ck’s are some constants, (,) is the usual inner product in IR’, and T is a discrete compact subset of Et’. As is well known [16], due to the fundamental theorem of algebra, in the univariate case the transfer functipn k can be factorized as the product of its roots. Clearly h is then specified up to a scale factor, if its roots are known. Unfortunately, due to general inability of factorizing polynomials in higher dimensions, this simple convenient factorization of the spectrum in univariate problems does not extend to higher dimensions. However, we will show that for the 2 D case the problem can have a solution in an asymptotic sense. We start by assuming that the zeros of our OTF occur a t isolated points’. Since the OTF is then either locally convex I I n general, the zeros of an entire function of two or more variables do not have to be isolated. However, in practice they can typically occur at isolated points [19]. 0-8186-8821-1/98 $10.00