What deconvolution can and cannot recover from a photograph

A blurred photograph is not a damaged photograph in the way a scratched one is. It is the result of an operation, and the operation has a name and a shape. During the exposure, the light from each point in the scene fell not on a single sensor cell but across several, in a pattern called the point spread function. Every point contributed its share to every cell it touched, and the recorded image is the sum of all those overlapping contributions. That summing is a convolution, and the point spread function is the kernel.

The useful consequence is that if you know the kernel, the sum can be partly unpicked. In the frequency domain a convolution becomes a multiplication, so undoing it becomes a division — take the transform of the blurred image, divide by the transform of the kernel, transform back. That is deconvolution in its naive form, and it works perfectly on a synthetic image with no noise and a known kernel. It fails catastrophically on a real photograph, for a reason worth understanding before trusting any result.

The transform of a line-segment kernel is a sinc function: it oscillates and passes through zero repeatedly, with the zeros spaced inversely to the length of the segment. At each of those frequencies the kernel recorded nothing at all, so the blurred image contains nothing there except sensor noise. Dividing noise by something arbitrarily close to zero produces something arbitrarily large, and the inverse transform turns that into a violent periodic ripple across the whole frame.

The Wiener estimate adds a constant to the denominator — a stand-in for the ratio of noise power to signal power — so that the division is bounded no matter how small the kernel's response was. That constant is the noise term in the panel, and it is the whole safety of the method. Set it low and more genuine detail comes back, along with ringing. Set it high and the pass approaches doing nothing. There is no setting that gives detail without ringing, because the information needed to distinguish them is not in the file.

The model assumes the camera moved in a straight line at a constant speed. A hand does not. Over a thirtieth of a second a hand translates and rotates, accelerates and slows, and the path traced by any point in the frame is a short irregular curve — often with a hook at one end where the shutter closed. Fitting a single straight segment to that curve gives a kernel that is right about some of the smear and wrong about the rest, so part of the frame sharpens and part of it rings, and no length or angle makes both behave.

Rotation makes it worse in a specific way: the length of the smear then varies across the frame, being near zero at the centre of rotation and largest at the corners. One length cannot describe that. So can distance: a camera that translates blurs a near subject more than a far one, in inverse proportion to depth, which is why a deconvolved landscape often looks right at the horizon and rings in the foreground.

• Short exposures shaken slightly fit the model best. A 1/60 handheld frame with three or four pixels of smear is the case this works on.
• Long exposures fit it worst. Half a second of handheld shake is a scribble, and no straight segment is a useful approximation of a scribble.
• A tripod frame with a moving subject does not fit at all, because the blur is confined to the subject and the operator is global.
• A frame that has been resized since it was shot has had its kernel resampled along with everything else, which widens it and makes the estimate less reliable.

A defocused lens spreads each point into a filled disc rather than a line. The distinction is not cosmetic. A line kernel has a transform with isolated zeros — narrow frequencies where information was lost — and information survives between them. A disc kernel attenuates a broad band of frequencies towards nothing, and what is attenuated below the noise floor is not recoverable by any method, because there is nothing left to divide. No amount of computation distinguishes a signal that is absent from a signal that was never there.

This is why the defocus case is not offered here. It is not difficult to implement — a disc kernel is fewer lines than a line kernel — and a slider marked defocus would produce output that people would be pleased with at fit size. That output would consist of detail the solver generated from noise and from its own assumptions, arranged plausibly. Shipping it would mean telling people that their out-of-focus photograph had been recovered when it had been replaced.

Three tests, all done at 100% and all done against the original rather than against memory. First, look at a flat area — sky, a painted wall, skin. Recovery leaves flat areas flat. Invention fills them with a fine periodic texture, often at a consistent angle, which is the ringing pattern expressed where there was no signal to mask it.

Second, look for detail that could not have been recorded. If small lettering has become legible, or a fabric has acquired a weave, ask whether the original frame — before any shake — had the resolution to hold it. A 4 megapixel phone photograph of a number plate forty metres away did not contain the digits before the camera moved, and a solver cannot supply them.

Third, move the length estimate by a pixel in each direction and watch what changes. A result that is genuinely following the file changes gradually. A result that has settled into a self-consistent pattern of its own will often change abruptly, or barely at all, which means the pattern is coming from the solver rather than from the photograph.

Non-blind deconvolution against a straight-line kernel does useful work on a mildly and evenly shaken frame from a clean file, and the improvement is real, measurable and worth having. It does partial work on a more complicated shake, and the result is a judgement call best made at 100%. It does nothing for a defocused frame, nothing for a moving subject against a still background, and nothing for a photograph that was too small to hold the detail being asked for. Those four sentences are the whole of the position this site takes, and they are the reason the deblur route opens by drawing the distinction rather than by promising a fix.