Is it possible to unblur a photo when the pixel is a mixture

A pixel that mixed two colors cannot be unmixed with certainty. Unblur the photo and treat the result as a guess you can reject.

The note on the print said is it possible to unblur a photo, and the person who wrote it wanted a yes. I can answer with a chalkboard faster than I can answer with a product. A blurred pixel is not a shy version of a sharp pixel. It is a mixture. Several places in the scene poured light into one bucket while the shutter was open, or while the lens was spreading a point into a disk. You cannot unmix a long mixture back to one true scene, because other scenes would have filled the bucket the same way.

TL;DR. During the exposure, each pixel adds up light from a neighborhood. A point that moves leaves a streak. A streak of the same shape can be made by a dimmer object that was already spread out. The file records the sum, not the diary of the sum. Short streaks are sometimes guessable, because most real scenes are not arbitrary. Long streaks are compatible with too many scenes, including a flat gray wall. Sharpening outlines the sum. It does not invert it. A model that offers a sharp picture is offering one preferred scene, not the scene that necessarily was.

A pixel is a bucket

I draw this for myself whenever I am about to promise a repair. The sensor is a grid of buckets. While the shutter is open, photons land in buckets. The camera writes down how full each bucket got, and then the shutter story is over. The file is a list of totals.

If nothing moved and the lens was focused, a small bright point in the scene fills mostly one bucket, with a little spill into neighbors because lenses are not perfect. If the camera moves during the exposure, that point walks across several buckets and leaves a share of its light in each. If the point was a porch bulb and the camera drifted downward, the bulb becomes a vertical streak. Every bucket in the streak holds a total. None of them holds a label that says "I was the bulb, and the others were the smear."

That missing label is the whole problem. Unblurring, if the word means recovering the scene that existed at one instant, asks for the label. The file has the totals. Arithmetic on totals cannot grow labels that were never written, except by assuming something about the world. The assumption can be decent. It is still an assumption.

The chalkboard, first three rows

I used a real board in a kitchen and five cells, because five is enough and a spreadsheet makes this look more certain than it is. Call the cells A, B, C, D, and E. The numbers are made up and round, so the addition is obvious. Brightness runs from 0 to 100.

Row one is a sharp scene. A single bulb sits on cell C. The cells read 0, 0, 100, 0, 0. If I had photographed that with a fast shutter and a still camera, the file would be that row, near enough.

Row two is the same bulb, and the camera moved one cell to the right while the shutter was open. The bulb spends half the time on C and half on D. I am ignoring the continuous slide and keeping the fractions kind. The buckets record 0, 0, 50, 50, 0. The 100 did not vanish. It was shared. Cell C does not know it used to be the whole bulb. It knows it is half full.

Row three is a different sharp scene, shot with no motion at all. Two bulbs, each half as bright, sitting still on C and on D. The buckets record 0, 0, 50, 50, 0.

Row two and row three are the same file. I can stare at the chalk as long as I like. There is no mark in row two that says "motion," and no mark in row three that says "two bulbs." Is it possible to unblur a photo that looks like this row? Only if I declare which world I prefer. The declaration is not in the buckets.

I erased row three and a friend, who had not seen me draw it, reconstructed the "sharp" picture as a single bulb centered between C and D, value 100, with the argument that bulbs are points. That is a prior. It is often a good prior. It is exactly as compatible with the chalk as the two dim bulbs, and also with a bulb that took a path I did not draw, for example a dimmer bulb that lingered, or a bright bulb that visited three cells unevenly and I rounded. One row of totals, many histories.

Why the gray row is not one picture

I drew a longer exposure underneath, four cells of travel. The bulb spends a quarter of the time on B, C, D, and E. The file reads 0, 25, 25, 25, 25. I then drew a blank scene that is simply a gray wall at 25 across those four cells, shot in focus, no motion. Same numbers, if A is dark in both and I am only looking at the streak.

A gray wall is a boring counterexample and it is the correct one. Long blur moves the recording toward a local average. Averages throw away the differences that made the scene a scene. Once the differences are inside the average, every pattern whose average matches is a candidate. The candidates include the true pattern. They also include patterns that would have been a different photograph, including a photograph of nothing much.

People hear this and say the rest of the picture will disambiguate it. Sometimes the rest of the picture helps, and I will get to that. It does not repeal the row. If the entire frame is the gray row, because the motion was long and the scene was small, there is no "rest of the picture." A tiny cropped face that smeared for the whole exposure is this row, dressed up as a portrait.

A letter is a stricter board than a face

Faces are how this discussion goes soft. A sharp face and a guessed face can both look like someone. Letters have a correct shape, so I drew an E.

A sharp capital E, crude, five rows by five columns, is a vertical spine on the left and three horizontal bars. In chalk I filled the spine and the bars with 100 and left the gaps at 0. You can read the E from across the kitchen.

I then smeared every column downward by more than the gap between the bars. Each new pixel is the average of a bar, a gap, and the next bar. The three bars and the two gaps become one tall gray block. The spine, which ran along the smear, is still a darker vertical, because averaging a spine with itself leaves a spine. What is left on the board is a gray rectangle with a darker left edge.

Which letter was that? An E, an F, a ruined B, a bold I with a thick side, a barcode, a shadow. The file fits all of them. A tool that returns a crisp E has chosen the E. It has not read the E off a label inside the gray. If the true letter was an F, the crisp E is a confident error. I care about this more than about the bulb, because I have watched repairs "restore" signage into the wrong word and look successful at arm's length.

If the smear is shorter than the gap, the bars do not fully pool. A dip remains where the gap was. That dip is information. It is not the sharp bar, but it is a clue the gap existed. Short blur is a damaged measurement. Long blur is a measurement of the average only. The difference is the length of the smear compared with the detail you hoped to keep, not a moral difference between tools.

Horizontal smear of the same E is a different damage. The bars run with the motion, so they stay bars, thinned or doubled at their ends. The spine, which crosses the motion, fattens and can merge with the start of the bars. You might still read E. You might read a set of three lines and a blob. Direction decides which part of the letter died. A repair that assumes the wrong direction will "fix" the bars that were fine and leave the spine wrong. I have a practical case of that kind of wrong direction on a shop sign shot from a bus. The physics is the same chalkboard. The sign is only the place I could check the spelling.

The weights are rarely equal

Real blur is not always "half and half." A shaky hand slows down at the turnaround. The bulb spends more time at the ends of the streak than in the middle. The buckets then read something like 0, 40, 20, 40, 0 instead of a flat 50, 50. That peaked-at-the-ends streak is still a mixture. It is a better fingerprint of the path than a flat average is, and a careful estimate of the path uses it.

It is still not unique. A sharp scene that was already brighter at two ends and dimmer in the middle, shot without motion, fills the buckets the same way. So does a shorter path with a different brightness. If I know the path, from a highlight that should have been a point, I can write the weights down and attempt an inverse. If I only have the buckets, I am guessing the weights and the scene at the same time. Two guesses that explain one file are not a measurement. They are a pair of stories.

There is a further nuisance, and it is the reason inverses get loud. Suppose the weights are even, and suppose some tiny pattern of plus and minus, bright and dark, averages to almost nothing under those weights. A fine checkerboard, smeared by an even streak, can nearly vanish. The inverse, trying to be helpful, decides the checkerboard might have been huge, because a huge checkerboard smeared hard can leave a tiny residue. Noise is a tiny residue. The inverse treats noise as a hidden checkerboard and paints worms. This is not a software bug in the moral sense. It is what happens when you divide by a very small number. The smear destroyed the checkerboard's evidence, and the inverse invents a large checkerboard to explain a speck.

That is why a damped inverse, the kind that refuses to amplify the vanished patterns, looks softer and less wormy. It is admitting the chalkboard. Some frequencies are gone. Leaving them gone is more honest than amplifying the noise that sits in their place. A sharp wormy file is the inverse pretending the evidence survived.

Defocus fills a disk, not a line

I have been drawing lines because motion is a line, or an arc I straighten for the board. Missed focus is a different mixture. A point becomes a disk. Every bucket inside the disk gets a share, in a ring if the lens is a typical lens with its aperture wide open. The file of a small light is a circle, not a streak.

The ambiguity is the same species. A disk can be a sharp point seen out of focus, or a dull circle that was painted on the wall and shot in focus, or a smaller point with a larger disk. The rest of a photograph often breaks the tie, because not everything is a disk of the same size. Edges of a building, if they all share the same disk radius, support "the lens was wide, the focus was off" rather than "the building was made of circles." That support is statistical. It is not a receipt.

A disk and a streak in the same frame are two mixtures. The porch example below has both: the bulb streaked, and the far hedge was already a soft disk because it was out of the plane I focused. One kernel cannot be both shapes. An inverse that assumes a streak will turn the hedge into echoes. An inverse that assumes a disk will turn the bulb into a ring. Looking at the highlight tells me the shape. Skipping that look is how a single "blur" slider becomes a random editor.

Rotation is the other shape I have to mention. If the camera twists instead of sliding, the center of the frame moves little and the corners move a lot. One streak length cannot fit the frame. The center is a short mixture. The corner is a long one. A global inverse uses one set of weights and is wrong somewhere. Local weights exist in careful tools, and they multiply the number of stories that fit, because now each patch has its own chalkboard. More freedom, not less ambiguity.

Noise means the totals are not even the totals

The bucket number is itself noisy. Too little light, a high ISO, a dark coat, and the recorded 50 might be the true 50, or 44, or 57. Two histories that were distinguishable when the numbers were exact become the same history once I have to allow a few counts of slop.

On the five-cell board, row "50, 50" and a neighboring story "48, 52" are not different files in practice. They are the same file twice. Any method proud enough to distinguish them is fitting noise. This is the worm problem again, seen from the exposure rather than from the frequency side. Dark frames are where unblur looks most magical on a screen and least deserved, because the totals were uncertain before the mixture even started.

JPEG comes after. The camera stores approximate totals, in blocks, and throws away more of the small differences. If I blur and then compress, I damage the clues twice. If someone compresses, messages the file, and I try to invert the blur, I am inverting a blur that is no longer the one the lens made. I am inverting a blur plus a blocky approximation. The chalkboard gets an eraser taken to it before I arrive. Starting from the raw file, when it exists, is not a fetish. It is a way to read the totals before a second process rewrites them. I have written practical versions of that choice on specific jobs. The reason lives here.

Sharpening does not unmix anything

Unsharp mask and its cousins take each pixel and push it away from its neighbors. Edges get a bright lip and a dark lip. The operation is easy to see, and it feels like the opposite of blur because blur was a pulling toward the neighbors and this is a push.

It is not the inverse of the mixture I drew. The inverse of "replace C and D with the average" is not "make a halo." The inverse, when it exists, is "put the 100 back in one cell and the 0 back in the other," and we already saw that the file does not say which cell gets the 100. A halo leaves the 50s in both cells and paints a bright line beside them. The two-bulb story and the one-bulb story both grow halos. Neither is selected. The smear is decorated.

On the letter E, sharpening the gray rectangle draws a bright edge around the rectangle. It does not cut the gaps back in between the bars. Anyone who has outlined a motion smear and called it fixed has done this. The picture got crisper. The chalkboard did not gain a row.

Clarity sliders and "structure" sliders are in this family when they boost mid-sized contrast. They can make a short, mild disk look punchier, which is sometimes all a small print needs. They are not evidence that unblur happened. I use them as cosmetics after I have decided the mixture is too short to worry about, or too long to invert, and the print only needs to look awake. Calling the cosmetic an unblur is how the gray rectangle gets delivered as an E.

A prior is a preference between stories

Here is the part where software re-enters, briefly, without being the point. When many scenes explain one file, a model or a person picks one. The pick is a prior: bulbs are points, letters are a known alphabet, walls are flat, faces have two eyes of a familiar size. A learned model is a prior fitted on many photographs. It prefers scenes that look like those photographs.

That preference can be right. Most porch bulbs were points. If I must guess between "one bulb that moved" and "two half bulbs that were always there," I guess one bulb, and on a real porch I am often correct. The guess is still a guess. It will be wrong on the night someone hung two lamps. It will be wrong on the letter F when the prior has seen more E shapes. It will be wrong on a face when the prior has a more average face handy.

Is it possible to unblur a photo, then, in the sense people mean when they ask? It is possible to choose a sharp story that fits the totals and fits a prior. It is not possible to extract, from a long mixture, the single scene that was there, because the single scene is not a function of the totals. Short mixtures plus a strict prior plus a highlight that reveals the path: that is the case where I am willing to try, and where I expect to keep some of the results. Long mixtures, no highlight, a dark noisy file, a letter or a face I cannot check: that is the case where the sharp story is fan fiction with a histogram.

The pages that actually run a prior on a file are unblur and fix blurry photo. They do not change the chalkboard. A slider is a way to look at two stories, the original totals and one preferred sharp relative, and to refuse the relative. Refusal is the correct outcome often enough that I would not want a tool that skipped it. What the model is doing when it prefers one face is the subject of can AI unblur an image. How I sort a stack of family pictures into "try" and "do not try" without drawing this board every time is is there a way to make blurry pictures clear. The third related note, can you unblur a photo, is the practical companion people usually wanted when they asked. I am staying with the sums.

The porch bulb, where I stopped

I kept one file as a personal copy of this board, so the numbers would be pixels and not only chalk. porch-01 is a night frame, a bare bulb over a door, camera moved. The crop I measure is porch-bulb-crop.tif, 16-bit, 800 by 800, 3.8 megabytes, cut from a larger raw conversion with sharpening off.

The bulb near the middle of the crop is a vertical streak. I call the bright core 40 pixels long. A dim tail adds a few more, and I do not pretend to know the tail to a pixel. A second highlight, a wet nail on the door frame toward the corner of the full frame, is a longer streak, about 70 pixels, and it is slightly curved. The center moved less than the corner. The camera rotated a little while it also dropped. One global length is already a lie on this file.

Forty pixels of mixture on a bulb that should be a few pixels wide is a long mixture for the detail "how is the filament shaped," which was never going to survive, and a moderate mixture for the detail "where was the bulb," which is a blob either way. I do not need the filament. I need to know not to invent a second lamp to explain the streak. The prior "one bulb" is right here because I was standing under one bulb and I can see one fitting on the wall. The file alone, cropped tight to the streak and handed to someone who was not there, does not say one fitting. It says a line of light.

I ran an inverse-style pass with a vertical length near 40, on a copy. The streak tightened and grew a dark ring, the worm from the vanished fine detail, and the corner highlight, which needed a longer and slightly curved path, split into two knots. The ring is the checkerboard problem in plain sight. The two knots are the wrong weights. I deleted the output and kept the crop. The sharp story was worse than the totals, because it added structure the totals do not support and got the path wrong in the corner.

A shorter smear on a different night, a few pixels, with a still camera and a slightly soft lens, is the other end. I might let a prior tighten it, and I would still call the result a preference. The porch file is my reminder of the long end. The answer on that file to the question on the print is no. Not because a tool is weak. Because a 40 pixel sum is not a filament, and a second story that looks like a filament is a drawing.

The board is only useful if I keep four rules in view while I draw it:

• The same totals can come from motion or from a still scene that was already spread out.
• A longer smear fits more scenes, including a flat gray patch.
• A prior picks one of those scenes. It does not retrieve a label from the buckets.
• Sharpening outlines the totals. It does not choose among the scenes.

Chalk row	Numbers	One history	Another history that fits
Sharp bulb	0, 0, 100, 0, 0	Still point on C	Almost nothing else, if the numbers are exact
Short smear	0, 0, 50, 50, 0	Bulb moved from C to D	Two half bulbs, still
Long smear	0, 25, 25, 25, 25	Bulb travelled four cells	A gray patch, no motion
Ended path	0, 40, 20, 40, 0	Shaky path, slow at the ends	A sharp scene already bright at the ends
Smeared E	A gray block, darker spine	The letter E	An F, or a blotch with a side

What I will say out loud

I will say that a short, straight, global smear, measured on a highlight, with a prior I can check on letters or on a freckle, is worth an attempt. I will say that the attempt produces a preferred sharp relative, and that I have kept some of those relatives when the checks held.

I will not say that the true scene is sitting inside the file as a hidden sharp copy. It is not sitting there. The shutter summed it and threw the diary away. On a long smear the sum fits a crowd of scenes, and the sharp one a model prefers is a member of the crowd, sometimes the right member, sometimes a tidy stranger.

I will not sharpen a gray block and call the halo an E. I will not treat a disk and a streak as one disease. I will not run one path length across a frame that rotated. Those are not policies. They are the chalkboard, copied onto the files I actually have, including a porch bulb that stayed a streak on purpose.