3.3 Image Deblurring
3.3.8 Semi-Blind Models
Semi-blind deblurring [2, 11] involves recovering a hidden true image with only par- tial knowledge or some assumptions about the blur function, such as the type of blur [123, 32]. Such models perform well and can obtain improved results over blind de- blurring when the blur type may be known or estimated. Such techniques are useful in related areas such as the segmentation of blurred images [11] and super-resolution [130, 217] where the blur is often of Gaussian or out of focus type. Parametric deblur- ring is a particular type of semi-blind deblurring which assumes that we may be able to model the blur degradation term h as a parametric function dependent on only a few parameters. Our aim in this case is to recover the parameters and thus reconstruct
the blur function.
Intensity-constraints in deblurring models [13, 15, 44, 182, 181] have become popular due to their ability to avoid intensity values falling outside of their expected range, which can cause poor results, particularly in astronomical imaging, medical imaging and blind and semi-blind deconvolution. We present a parametric model which incorporates intensity constraints implicitly for the image. This is found to enhance the quality of results when compared to unconstrained models.
Much research involving parametric models either assumes that Gaussian blur is the cause for degradation or models alternative blur functions as piecewise constant functions. Assuming that the blur is of Gaussian type is a limitation which may prevent the accurate restoration of images corrupted by other blur types, so we would like to extend this to accommodate other blurs. We note that modelling a blur function as a piecewise constant prohibits the recovery of the parameters within a variational framework.
Many papers present techniques which may be classed as semi-blind in the sense that their aim is to restore images which have been corrupted by blur of a particular type [32, 123, 165, 179, 214]. [123] presents a method for blind deconvolution assuming that the degradation in the image is caused by motion blur. While not presented as a parametric model, the authors attempt to find the length or blurring distance of the blur function as shown in Equation (3.37) by using image statistics. [32] also presents a non-parametric model which assumes that the degradation is caused by motion blur, and makes use of multiple images to estimate the precise blur function.
Many papers attempt to improve the estimation of point spread functions by con- sidering them as piecewise functions after discretisation. This reduces the estimation of many unknowns to that of only a few variables which determine the entire psf. Semi-blind deblurring using piecewise representations of blur functions In [18], the authors give several equations for piecewise constant representations of kernel functions. Motion blur, caused by relative motion between the subject and the imaging device, is presented in the case of constant horizontal velocity translation [18, 102] as
h(x, y;s, t) =h(x−s) =
( δ(y−t)
V T for 0≤x−s≤V T
0 otherwise,
whereV is the measure of the velocity and [0, T] is the exposure interval. The discrete equivalent uses the blurring distanceL and is thus presented as
h(i, j;k, l) =h(i−k) =
(
1
L+1 for 0≤i−k≤L, j=l
Out of Focus blur, which occurs when some parts of the scene are in focus and others are not, concerns the intensity distribution within the circle of confusion of radius r around a point. The corresponding point spread function is presented [18, 102] as
h(x, y) =
(
1
πr2 forx2+y2≤r2
0 otherwise.
The parameters are estimated from the power spectrum of the blur function by approximating the power spectrum of the true image using a wide variety of images [186].
[78] makes use of semi-blind deconvolution by reducing the number of unknown variables in the PSF to a small number of variables from which it can be given in full. [117] also presents discrete piecewise equations for linear motion blur and out of focus blur, making use of frequency domain zeros to influence the identification of the parameters.
[9] presents discrete equations for several blur types including box blur, also known as uniform 2D blur, as
h(i, j) =
( 1
L2 for −L2 ≤i, j≤ L2
0 otherwise,
where L is assumed to be an odd integer. This can be viewed as a composite motion blur in horizontal and vertical directions.
Recently, [130] and [217] have made use of parametric formulations of blur functions for super-resolution techniques. The parameters are estimated prior to deblurring in cases where it is known or assumed that the blur function may be of Gaussian or out-of-focus type.
Semi-blind deblurring using continuous representations of the blur function Gaussian blur has a well-known formulation given by Equation (3.38). Recently, papers aimed at semi-blind deconvolution assuming Gaussian blur have introduced this for- mulation in the functional. The authors of [11] assume that Gaussian blur is the cause of degradation of the image and so replace the kernel term with a Gaussian equation
hσ(x, y) = 1 2πσ2exp −x 2+y2 2σ2 (3.38) which is dependent on the variable σ. The objective functional for the restoration is similar to total variation regularised deblurring [169, 54] made up of a fitting term for deblurring in the presence of Gaussian noise and a smoothness term given as the L2
norm of the gradient which allows for smooth kernels, in contrast to total variation which permits piecewise constant functions. The equation is modified to allow for the
parametrically defined kernel functionhσ. It is thus presented as fBSK(u, σ) = 1 2 Z Ω (hσ∗u−z)2dA+γ Z Ω|∇ hσ|2dA
whereγ is a regularisation parameter controlling the smoothness of the recovered blur function, u is the restored image andz is the received data. In order to minimise this functional, the authors derive the Euler Lagrange equations for u and σ and proceed with alternate minimisation of the arguments.
[64] also considers the case of semi-blind parametric deblurring in a constrained blind deconvolution problem by assuming that the blur function may be of Gaussian type and so modelled as (3.38), leaving only a vector containing a small number of parameters to be found.
The representation of a blur kernel by a differentiable function which allows it to be incorporated into the minimisation problem is useful but in the above work is limited to Gaussian functions which cannot be used to approximate many blur types. In Chapter 7, we consider an extension to this which allows more classes of blur to be incorporated into an energy functional.