# What Is the Relation Between Deblurring and Deconvolution in Computer Vision and Image Processing?

The deblurring problem can be modelled as follows

$$f = \phi u + \epsilon, \; \epsilon \sim N(0, \sigma)$$

where $$\phi$$ is a filter (e.g. a low-pass filter) and $$\epsilon$$ is a Gaussian noise.

In computer vision, what is the relation between deblurring and deconvolution?

• Possible duplicate of Why can convolution only be applied to compute the output of a linear filter? Commented May 19, 2019 at 16:50
• As you've asked (and got an explanation in the above question), filtering is convolution. So, if you want to invert a blur, and you define blur to be a filter, then the question "what is the relationship of deblurring and deconvolution" becomes "what is the relationship between reversing a convolution and deconvolution" and the answer to that is: it's the same. It's literally the definition of deconvolution. Commented May 19, 2019 at 16:52
• Marcus, in general deblurring does not only concern lti filters. It is only the most primitive approaches which do. Commented May 19, 2019 at 19:03
• @mathreadler you'll notice OP defined blurring to be a filter in his own question: "… is a filter (e.g. a low-pass filter)". So, not my idea! Commented May 19, 2019 at 19:40
• It says nothing about the filter being linear nor time / spatially invariant. Just that it is low pass. Commented May 19, 2019 at 19:42

Let me present the following Diagram:

So, both Deblurring and Deconvolution are operations within the family of Image Restoration (Which is a subset of Inverse Problem set).

Basically we build the Image Restoration set by different Degradation Models.
The one related to the question are:

2. Spatially Invariant Model
A Model where the degradation is the same for any place in the image.
We created it as a subset of Linear Model though it doesn't have to be. But for clarity.
Any operator which is both Linear and Spatially Invariant can be defined by a Convolution Operation. Hence it can be reversed by Deconvolution.
3. Low Pass Operator
A set of images degraded by a Low Pass Filter. Namely by a convolution with a Low Pass Filter.

Now, there is a set of degradation which basically create a blurry image.
Reversing this operation is called Deblurring.
In case the blurring is made by a Low Pass Filter applied by Convolution the Deconvolution of this operation is also a Deblurring process.

In the context of image processing (and machine vision as well), blurring is an operation that reduces the sharpness of an image by some lowpass filtering applied on it.

There are different causes of blurring such as lens blur, motion blur, or just LSI (linear shift invariant) lowpass filtering.

Deblurring refers to any restoration performed on the image that try to remove the effect of a previous blurring, by outputting a sharper (similar to original) version of the image.

When blurring can be mathematically defined as an LSI convolution operation (aka LSI filtering), then the operation of deblurring can be defined as a deconvolution (i.e; inverse of convolution), and that's the sole relation between deblurring and deconvolution.

• In image processing, LTI becomes LSI - Linear Spatially Invariant.
– Royi
Commented May 20, 2019 at 4:27
• Ah thank you! so easy to ignore. Although LSI stands for Linear Shift Invariant to my knowledge..? But spatial invariance is also quite fitting... ;-) Commented May 20, 2019 at 18:20
• I think shift invariant would be generalization of both :-). Just like we assume in 1D the coordinates are time and we call it Time Invariant, I think in 2D the assumption is the coordinates are spatial -> Spatial Invariant.
– Royi
Commented May 21, 2019 at 2:11