What are the best methods for deconvolving images with unknown kernels?

1 What is deconvolution?

Deconvolution is the inverse operation of convolution, which is a mathematical way of describing how an image is transformed by a filter or a point spread function (PSF). For example, when an image is blurred by a camera lens, the lens can be modeled as a PSF that convolves with the original image to produce the observed image. Deconvolution aims to reverse this process and recover the original image from the observed image and the PSF. However, in many situations, the PSF is unknown or only partially known, which makes deconvolution more challenging and prone to errors.

2 Why is deconvolution hard?

Deconvolution with unknown kernels is hard for several reasons. First, it is an ill-posed problem, meaning that there are infinitely many solutions that can fit the observed image, and some of them may not make sense or be desirable. For example, a sharp image can be deconvolved with a very noisy kernel, or a noisy image can be deconvolved with a very smooth kernel, but neither of these results would be useful or realistic. Second, it is a nonlinear problem, meaning that small changes in the input or the parameters can lead to large changes in the output, making it sensitive to noise and instability. Third, it is a high-dimensional problem, meaning that the unknown kernel can have many pixels or coefficients that need to be estimated, which increases the computational complexity and the risk of overfitting.

3 What are some methods for deconvolution?

There are many methods for deconvolving images with unknown kernels, but they can be broadly classified into two categories: blind and non-blind. Blind methods try to estimate both the original image and the kernel simultaneously, without any prior information or constraints. Non-blind methods try to estimate only the original image, using some prior information or constraints on the kernel, such as its size, shape, or sparsity. Both categories have advantages and disadvantages, depending on the application and the data.

4 How do blind methods work?

Blind methods work by optimizing an objective function that measures the discrepancy between the observed image and the convolution of the estimated image and kernel, as well as some regularization terms that impose smoothness or sparsity on the image and kernel. The objective function can be linear or nonlinear, depending on the assumptions and the model used for the convolution. The optimization can be done by gradient-based methods, such as gradient descent or conjugate gradient, or by non-gradient-based methods, such as alternating minimization or expectation-maximization. Blind methods can achieve good results when the kernel is large or complex, but they can also be slow, unstable, or prone to local minima.

5 How do non-blind methods work?

Non-blind methods work by using some prior information or constraints on the kernel to simplify the deconvolution problem. For example, some methods assume that the kernel is known up to a scaling factor, or that it belongs to a certain family of functions, such as Gaussian or Laplacian. Other methods assume that the kernel is sparse, meaning that most of its pixels or coefficients are zero or close to zero, which reduces the number of parameters to estimate. Non-blind methods can be faster, more stable, or more robust than blind methods, but they can also be less accurate or more biased, depending on the quality and validity of the prior information or constraints.