One focus stopped down is as sharp as six in a deep room

A refocused panorama is six lenses shot a room as six frames sixty degrees apart, focusing each on what was in front of it — a table at half a metre, a chair, a doorway, the far wall twice, a shelf. Every frame then had its entrance pupil in its own place, carried forward by its own focusing extension, and the best fixed panorama head left 12.9 arcminutes along the worst seam. A head geared to slide the camera back as the lens extends left 2.82, the floor set by the pupil’s walk with field angle. That essay ended with the comparison a photographer actually faces: whether to refocus at all.

The alternative is to focus once and stop the lens down until the table and the wall are both acceptably sharp. Every frame then has its pupil in the same place, and a plain head aligned on it leaves only the walk. What it costs is blur — defocus at the ends of the room’s depth, and past some aperture, diffraction everywhere — and light.

The measurement below finds the best that one focus can do, and then finds that the question it answers was set up unfairly to it. Refocusing each frame does not make each frame sharp, because the frames are not shallow.

The best one focus can do #

The lens is the fifty-millimetre design of the earlier essays, and blur is measured the way seams were, as an angle in arcminutes, so that the two can be put side by side. At the scale those essays used, sixty pixels to the degree, an arcminute is a pixel.

Two blurs set the answer. A point that is not at the focused distance images as a disc, and seen from the lens the disc subtends an angle equal to the diameter of the entrance pupil times the difference between the reciprocals of the two distances. That is the whole of defocus for small angles, and it needs no focal length once the pupil is fixed: a lens whose pupil is a centimetre across blurs a point at half a metre, focused at one metre, by a centimetre times one per metre, a hundredth of a radian, whatever lens it is. Diffraction goes the other way. A pupil of diameter $A$ spreads every point into an Airy disc whose first dark ring subtends $2.44\lambda/A$, which grows as the pupil shrinks. The figure takes green light, 550 nanometres, and adds the two blurs as a root sum of squares — an assumption about how two blurs of different shape combine, and the one the result is most sensitive to.

The focus that treats the room’s two ends alike is the one that puts them the same distance away in reciprocal terms: the harmonic mean of half a metre and four metres, 0.889 metres. At f/4 the table and the wall then blur by 36 arcminutes each. At f/16, by 9.5. The blur falls as the lens is stopped down until diffraction catches up, and together they are least at f/40, where both ends blur by 5.27 arcminutes. Beyond that, stopping down makes everything worse; at f/128 the room is 11.9 arcminutes soft everywhere.

That is the whole answer to the question as it was posed. One focus beats six focuses on a fixed head, whose worst seam is 12.9 arcminutes, from about f/12 onwards. It never reaches the tracking head, whose worst seam is 2.82. So the essay that proposed the comparison had the ordering right: a geared head and six focuses are the sharp way, a plain head and one focus the cheap way, a plain head and six focuses the worst of both.

Why the floor has no lens in it #

The floor of 5.27 arcminutes has a closed form, and it is worth writing down because of what is missing from it. With $\Delta$ the difference of reciprocal distances the focus has to span, the worst defocus at the best focus is $A\Delta/2$ and the diffraction is $2.44\lambda/A$. Their root sum of squares is least when the pupil is $A = \sqrt{4.88\lambda/\Delta}$, and there it equals

$$\beta_{\min} = \sqrt{2.44\,\lambda\,\Delta}\ \text{radians}.$$

For the room, $\Delta$ is two per metre less a quarter per metre, and $\beta_{\min}$ is 5.27 arcminutes. The focal length does not appear. Neither does the sensor, nor the pixel pitch. The sharpest angle a depth of room can be held to at one focus is a fact about the room and the colour of the light, and every lens reaches it at the f-number that gives it the right pupil: a pupil 1.24 millimetres across, which is f/40 on a fifty and f/19 on a twenty-four. The sharp band is a decision made the point that depth of field is set by how much blur a reader will ignore rather than by the optics alone; here the optics set a hard lower limit on that tolerance, and it is independent of which optics.

Drawn against distance, the blur at a single focus is a V on a scale of reciprocal distance, with its point at the focus. The table at half a metre and the wall at four sit on its two arms at the same height, which is what the harmonic-mean focus is for. The V’s point is not zero once the lens is stopped far down: at f/40 the focused distance itself blurs by 3.73 arcminutes, all of it diffraction, so at the aperture that holds the room best, nothing in the room is sharper than that. Drag the focus off the harmonic mean and the V tips: set at 0.6 metres, the table sharpens to 3.99 arcminutes and the wall coarsens to 7.09; set at 1.5 metres, the table is the end that pays, at 6.79. Either way the room is only as sharp as its worse end, which is why the focus where the two ends are equal is the one worth having.

The pupil in the formula is the entrance pupil, not the stop. The hole a scene actually sees found that for a lens of this kind the opening the scene looks through is the stop’s image, larger than the stop and in a different place, and it is that image’s diameter the f-number divides into the focal length. So the floor is set by the size of the hole as seen from the room — the same hole whose position decides where the panorama must turn. A lens whose entrance pupil is far from its stop can reach f/40 with a stop that is physically smaller than 1.24 millimetres, which matters to the lens’s maker and not at all to the picture.

The formula also says what does not help. A longer lens reaches the floor at a larger f-number, and a shorter one at a smaller, but the floor is the same angle. A finer sensor records the same blur across more pixels; a coarser one across fewer. Nothing about the camera moves the sharpest angle a room of this depth can be held to at one focus, and the only way below it is to stop asking one focus to hold the room — which is what refocusing was supposed to do.

Each frame already holds the room’s depth #

The comparison above set one focus’s worst blur against six focuses’ worst seam, as the question proposed, and in doing so assumed that a refocused frame is sharp. It is sharp on its subject. But a frame pointed at a table half a metre away does not hold only the table. The wall is behind it, four metres off, in the same frame.

The figure gives each frame the depth it actually holds: the table’s frame from half a metre to the wall at four; the chair’s from seventy centimetres to four; the doorway’s from a metre and a half through the door to a corridor at six; the two frames on the far wall nothing but the wall; the shelf’s from 1.2 metres to four. Each frame, refocused and stopped down on its own, can be held no sharper than the closed form allows for its own depth. The table’s frame is limited to 5.27 arcminutes, exactly what one focus managed for the room from half a metre to four, because the table’s frame is the room from half a metre to four. The whole room at one focus, now including the corridor at six metres, can be held to 5.39.

So refocusing frame by frame buys 2.3 per cent on the frame that decides the panorama’s sharpness. It buys a great deal on the frames of the far wall, which can be held perfectly sharp — but those were never the problem. The frame that sets the panorama’s worst blur is the one with the nearest subject, and the nearest subject almost always has the room behind it. Focusing is a zoom found that a refocused frame is a different lens; this is the other half of the same observation, that a refocused frame is still a frame with depth in it, and focusing moves the V without narrowing it.

Three ways to shoot the room, counted fairly #

With each frame’s own depth counted, the three strategies can be compared on both of the things they cost: the worst seam and the worst blur.

Six focuses on a fixed head are the worst on both counts: the seam is 12.9 arcminutes, and the blur is still 5.27, because refocusing did not relieve the table’s frame of its depth. Six focuses on the geared head bring the seam down to 2.82 and leave the blur at 5.27. One focus for every frame, stopped to f/41, on a plain head aligned at the pivot that suits that focus, leaves a seam of 2.83 arcminutes — every frame’s pupil in one place, so only the walk with field angle remains, exactly as for the geared head — and a blur of 5.39.

In every column the taller bar is the blur. Once depth within a frame is counted, what limits a room panorama at its best is not where the camera turns but how much of the room each frame has to hold, and that limit is the same whether the frames are refocused or not. One focus on a plain head comes within 2.4 per cent of six focuses on a geared head, and needs no gearing, no unlinked stitcher and no per-frame focal lengths — the last of which the earlier essay found to be the largest error of all, 270 arcminutes, in a refocused panorama read by a stitcher that assumes one lens.

Where the answer depends on the room #

The room here is a hard case: a table half a metre from the lens, a wall at four. The two quantities that compete scale differently with how near the nearest subject is, so the verdict moves with it.

The floor on blur, $\sqrt{2.44\lambda\Delta}$, grows as the square root of the reciprocal distance: from 1.15 arcminutes when the nearest subject is three metres away to 6.99 when it is thirty centimetres. The seam that a head leaves when every frame’s pupil is on its pivot — the tracking head, or one focus on a plain head — grows as the reciprocal itself, from 0.47 to 4.70 arcminutes, because it is a fixed offset seen from a distance. Across the whole range the blur is the larger, by a factor of about two at half a metre and two and a half at three metres. So wherever a frame holds a subject and a wall behind it, blur is the limit and the seam is not.

The seam a fixed head leaves when one frame is refocused on a near subject and its neighbour is not grows much faster, from 1.3 arcminutes at three metres to 80 at thirty centimetres, because the focusing extension itself grows as the square of the reciprocal distance. That is the cost that one focus avoids, and it is the cost that makes refocusing on a plain head the worst choice at every distance drawn. At every distance drawn the refocused seam is larger than the blur the room’s depth already imposes — by about a sixth at two metres, twice at one metre, five times at half a metre and eleven times at thirty centimetres.

What one focus costs in light #

None of this is free. A lens at f/41 admits a hundredth of the light it admits at f/4, and the price rises steeply near the floor, because once diffraction is half the blur, a little more stopping down buys almost nothing.

Holding the room to within 2 per cent of its floor takes f/36, 6.3 stops below f/4. Relaxing to six arcminutes saves half a stop, to eight saves one and a half more, and matching the fixed head’s refocused seam of 12.9 arcminutes takes only f/12.3, 3.2 stops, nine times the exposure. For a still room on a tripod — which is where a panorama head is used at all — six stops is a longer exposure and nothing more. For anything that moves, or for a hand-held panorama, it is the whole cost, and the geared head that lets each frame be shot wide open on its own subject is the only route to a sharp foreground.

Even then, as the section on each frame’s depth showed, wide open on its own subject is not sharp everywhere in the frame: the table’s frame shot at f/4 blurs the wall behind the table by far more than any seam. Refocusing buys light, not depth. The two strategies spend different currencies — one focus spends exposure time to buy depth, six focuses spend mechanical complexity and per-frame calibration to buy aperture — and in a still room with depth in every frame, only the first buys the thing the panorama needs.

What the stopped-down panorama keeps #

The earlier essays in this sequence traced the centre of a panorama from a point, to a pupil, to a pupil that walks with field angle, to a pupil that moves with focus. The eye is a place, not a point started it, and focusing moves the pivot past its best place found that focus carries the pupil further than the walk does. A single focus undoes that last step. Every frame has the same lens, the same pupil and the same principal distance, so the panorama is once again one instrument turned about one place, with only the walk left along its seams — the walk the entrance pupil walks with the angle measured, 5.53 millimetres across the field for this front element — and the pivot that is not the eye already priced what such an offset costs. And the harmonic mean that sets the best single focus is the same harmonic midpoint that a rig is right on one surface found for the depth a multi-camera stitch should be computed at: in both, depth enters as its reciprocal, and the fair middle of a range of depths is the middle of their reciprocals.

What it gives up is sharpness at the focus itself. At f/40, nothing in the room is sharper than 3.73 arcminutes, because diffraction applies everywhere. A refocused frame at a wide aperture is much sharper than that on its subject and much softer everywhere else. A reader who cares most about one subject — a face at the table — should refocus for it; a reader who wants the room evenly held should not. The centre has an area showed that a pupil of finite size keeps every theorem of perspective for the centre of each blur disc; the stopped-down room is that fact used deliberately, trading the disc’s size for the certainty of its centre.

What was assumed #

Defocus and diffraction add as a root sum of squares. Adding them outright would raise the floor by exactly 41 per cent, at the same best aperture, since at that aperture the two are equal; a detailed optical transfer function would put it somewhere between. The ordering of the strategies does not depend on it, since every strategy pays the same blur for the same depth.

The lens is perfect at every aperture. A real fifty-millimetre lens has aberrations that fall as it is stopped down, so wide open it is softer than drawn, and its sharpest aperture is usually a few stops down from wide open. That favours stopping down.

Each frame’s depth is its subject to the wall behind. A frame looking into a corner has less depth; a frame with its nearest subject in front of a window has more. The table’s frame decides the verdict, and a room with nothing near the camera has a far lower floor and a much smaller seam, which is the regime where refocusing is also unnecessary.

Blur and misregistration are the same kind of error. A seam blended over a width becomes a smear of about the misregistration’s size, which is why the two are drawn on one scale; a seam cut hard instead shows as a step, which a viewer notices more than a blur of the same size.

Still open: whether a stopped-down room is sharper stacked than at one focus #

One more strategy sits between the two measured here, and it may beat both. A photographer can shoot each frame several times at different focuses and wide apertures, then merge the sharp parts of each — focus stacking. Each exposure has its pupil in its own place, so the stacked frame is not one camera; but the merge keeps the table from the exposure focused on the table and the wall from the one focused on the wall, and at a wide aperture neither suffers diffraction.

The measurement that settles it takes the table’s frame, from half a metre to four metres, splits its reciprocal depth into $n$ equal slices, and for each slice finds the aperture that holds the slice to its own floor. The stacked frame’s worst blur is then the slice’s floor, $\sqrt{2.44\lambda\Delta/n}$, which falls as the square root of the number of exposures. Against that, each exposure’s pupil sits at its own extension, so neighbouring slices disagree about where the table’s edge is by the refocused seam of the section above — and the question is how many slices make the stacked frame sharper than one stopped-down focus before the disagreement between slices, which a stacker must hide at every boundary, becomes the larger error.