# Why image deblurring still is an open problem?

This question is not written in order to criticize people efforts in image deblurring, to be honest, I just don't know the difference between techniques.

My question is in title, why deblurring image is still open problem and lots of techniques and methods published for that everyday?

When we have a blurred image, we can deblur that with a simple deconvolution, or using gradient in order to deblur, we can use sharpening filters, or using unsharp mask.

Why everyday we see new techniques for deblurring images? what is the difference between them?

What metric defines that a technique better than the other? Or what things that make a technique weaker than the other? what trade-off do we have in image deblurring?

What is the thing in image deblurring, that none of current deblurring algorithms couldn't solve yet and all of the scientism is struggling to solve that?

Also, I'm looking for any article or book that talks about that (image blurring, deblurring and different method) but I couldn't find

• Image Deblurring is an extremely important technique that finds critical applications in many fields; such as science and consumer. It basically tries to undo the effects of the nonideal optics (or motion) on the captured images, and correct them scientifically, or make them more pleasent to the eye commercially. In typical applications, however, the blur function is not known, and must be estimated from noisy and distorted data. These two facts result in the development of "new" and "improved" algorithms everyday and so forth... Mar 20, 2022 at 1:39
• @Fat32 thanks, and how about when the blur kernel is known? I read this in an article In the non-blind case, the observed blurry image does not uniquely and stably determine the sharp image due to the ill-conditioned nature of the blur operator what does ill-conditioned nature of the blur operator exactly mean? Mar 20, 2022 at 16:47
• roughly speaking when a matrix is ill conditioned, typically its inverse is not stable (its determinant is close to zero, and its inervese will have very large elements in it). Similarly if a forward blur kernel is ill conditioned, then the inverse de-blur kernel, used in deconvolution, will tend to be unstable and useless (it will amplify signal noise)... Mar 20, 2022 at 17:15
• @Fat32 Thanks, suppose that I have a blurred image that I know it has convolved using a gaussian 3x3 matrix or a box filter, then can I deblur that completely? (I mean can I find exactly sharp image?) Mar 20, 2022 at 18:11
• this requires a complete new question... Mar 20, 2022 at 19:25

The linear convolution problem stated as:

$$y = x\ast h+n$$

with $$y$$ the observation, $$x$$ the original data, $$h$$ the blur kernel and $$n$$ the noise is a prototype problem in DSP, for signals, images or other data. Many solutions try to minimize some metric of a data fidelity term $$(x-\hat{x})$$ combined with a penalty, to ensure the solution is somehow proper. Even in the simplest case, where $$h$$ is known and no noise is present ($$n=0$$), the smoothing kernel $$h$$ might remove data from $$x$$ that cannot be recovered. Namely, if its Fourier transform $$H$$ vanishes in the frequency domain, from $$Y=XH$$, you get that whenever $$H(\omega)=0$$, you have indeterminacy in $$X$$ in recovering it from $$X(\omega)H(\omega)=0$$. That is part of what ill-conditioned means. This happens a lot when the blur kernel is of finite support like your $$3\times 3$$ Gaussian example.

Here are a couple of other issues:

• the model is not linear: noise is non additive, measurements undergoes nonlinear effects (saturation) or baseline/background,
• the model is not shift variant: the blur kernel varies with location, data is downsampled, leading to super-resolution questions,
• the blur kernel is unknown, and should be derived from the observations as Fat32 wrote,
• the quality metrics are not appropriate for the purpose,
• the algorithmic metrics (data fidelity, penalty) are not consistent with the purpose.

Each of them is a whole problem per se. My opinion is that we need to work a lot more on sound quality metrics (beside SNR, SSIM, etc.) but research can be conservative, and use the same metrics as others before.

Basically, at least $$x$$ is unknown, so there are a lot of potential estimators for it, and getting the ONE is difficult. However, novel deblurring can be useful because it is a prototype problem. A good algorithm in the theoretical context may turn out to work in practical cases, or may even be re-used in another DSP context.

• Thanks a lot Laurent! almost all of my doubts resolved, so, I find out from your answer that during blur process, some of information in picture vanishes, and these deblurring algorithms are trying to kinda estimate these missing information and competing with each other to have a lower error, by maximizing some arbitrary metrics like SSIM or PSNR. am I right? Mar 21, 2022 at 9:01
• Mostly in line. On the last part : a lot of algorithms are written as a minimization of a cost function to restore missing details or distortions. Adhoc methods like unsharp can be used too. And at the end, restored images are compared wtih more traditional metrics Mar 22, 2022 at 1:35
• And I think it is better to solve this problem using machine learning, can you tell me some good papers or books about solving inverse problems using machine learning? Mar 25, 2022 at 20:28
• What would you include in the machine learning you are interested in? Mar 25, 2022 at 23:24
• something like this: arxiv.org/abs/1803.04189 Mar 26, 2022 at 17:52