A prior the pictures can refute is the one worth having

Four cameras and a field they cannot see looked at a flow with no symmetry through a few parallel-ray views — the arrangement of a set of schlieren or background-oriented cameras, or a tomographic PIV system — and counted what the views determine. On a field of 16 × 16 cells, four views leave 177 of the 256 patterns of density unseen: combinations that add up to nothing along every ray, invisible by construction. The reconstruction that assumes nothing sets that invisible part to zero, and gets a field smeared along the viewing directions with density where there is none. Pinning the field completely takes views in proportion to the resolution wanted, fourteen for this grid.

The essay ended on the way practice escapes that arithmetic. Flows are not arbitrary. A jet is smooth, a density is never negative, a flow is often a sum of structures of a known kind, and each of these assumptions — a prior — excludes most of the invisible patterns, so that a reconstruction can choose the invisible part rather than zero it. The question was how many views each prior saves, and what a reconstruction does when its prior is wrong. The second question turns out to decide the first.

Four ways to fill in what the views missed #

The operator is the earlier essay’s: a 16-cell grid on a square, each view a family of parallel rays spanning the square’s diagonal, each ray a weighted sum of the cells it crosses. The field is the same too: two Gaussian puffs of different widths and an oblique streak. With noise-free rays, any field that reproduces them differs from the truth only by an invisible pattern, and a prior is a rule for choosing which invisible pattern to add.

The first reconstruction assumes nothing: the minimum-norm field, with the invisible part set to zero, which is the earlier essay’s reconstruction. The second takes, of all the fields that reproduce the rays, the one whose curvature is smallest. It is found by minimising the rays’ misfit plus a curvature penalty so weak that the rays are met and the penalty only chooses among the fields that meet them. The third asks for a field that reproduces the rays as closely as it can while never going below zero, found by projected gradient descent from zero; where a non-negative field meeting the rays exists, this finds one. The fourth is a dictionary: a sum of 64 Gaussian puffs of one width on a regular grid of centres, with the 64 amplitudes fitted to the rays. That is 64 unknowns in place of 256, and a prior that the field is made of structures of a known shape.

The four are not four settings of one knob. The first two differ only in which of the fields meeting the data they prefer; the last two refuse some of those fields altogether, one by a sign and one by a shape. That difference will turn out to be the whole story.

What four views give #

From the same four views, the four reconstructions have errors of 42.8, 39.8, 24.7 and 37.6 per cent. The minimum-norm field is the earlier essay’s streaked cross. The smoothest field is almost the same picture, rounded off: smoothness excludes the jagged invisible patterns, but the ones that matter here — broad hollows and ridges between the viewing directions — are themselves smooth, and the prior lets them through. The non-negative field is visibly better. The empty corners and the space between the puffs, where the minimum-norm field put spurious negative and positive density, are now nearly empty, because any invisible pattern that would put negative density in an empty region is ruled out. The dictionary field is made of puffs, which is its only way of being; it approximates the streak with a row of them.

Views saved #

The figure follows the error as views are added. Every prior’s error falls, and the order of the curves is the order of the four pictures. Non-negativity falls fastest and reaches ten per cent at seven views; assuming nothing takes ten. The smoothest field reaches it at nine. The dictionary, given the regularisation that serves it best, falls to about a fifth and stops, because the streak is not a sum of its puffs and no number of views can make it one.

Counted in views, the saving depends on the field. For the earlier essay’s field non-negativity saves three views of ten. For the same field without its streak — two puffs in an otherwise empty square — it saves five of nine, and four views are enough. That is the prior’s mechanism made quantitative: non-negativity is worth the most where the field is mostly empty, because emptiness is where it excludes the most. It is the same reason sparse images are the ones compressed sensing recovers from few measurements, and dense tomographic PIV of a seeded volume, whose voxels are mostly empty of particles, relies on exactly this constraint. Set against the earlier essay’s count — fourteen views to remove the null space of this grid altogether — non-negativity reaches ten per cent on the puffs with under a third of them. Smoothness saves one view on either field. The dictionary never reaches ten per cent on either, because neither is exactly a sum of its puffs: the two real puffs have widths the dictionary does not contain.

When the prior is wrong #

The earlier essay’s closing worry was the case that matters: a reconstruction built on a wrong prior is sharp, confident and incorrect, and the pictures cannot say so. The figure applies each prior where it is wrong, from eight views, and reports two numbers for each: the error, which needs the truth to compute, and the misfit — how badly the reconstruction fails to reproduce the rays it was given — which needs only the data.

Non-negativity on a signed field. Schlieren and interferometry measure density relative to ambient, and a hot plume beside a dense jet gives a field that is negative in places. Here the second puff is a trough. Non-negativity cannot represent it, and the reconstruction is 72 per cent wrong — but it also misses its own rays by 77 per cent. There is no non-negative field that meets the data, and the data say so at once.

The dictionary on a streak. The puffs cannot make a streak, and the reconstruction is 21 per cent wrong and misses its rays by 4.5 per cent. The data say so, more quietly: a misfit of a few per cent is visible against noise-free or low-noise rays, and hidden in noisy ones.

Smoothness on a sharp front. A front — a flame sheet, a shock seen edge on — is where a density field is least smooth. The smoothest reconstruction is 13 per cent wrong, blurring the front across several cells, and it misses its rays by a part in a million. Nothing in the data objects. This is the essay’s worry exactly, and it is exact for this prior.

The priors the data can catch #

Putting every prior on every field together shows the distinction as a geometry. The minimum-norm and smoothest reconstructions always sit on the misfit floor: they choose only among fields that meet the data, so whatever their error, the rays cannot object. Non-negativity sits on the floor when it is right and far off it when it is wrong. The dictionary is never on the floor, because none of the four fields is exactly a sum of its puffs.

The two kinds of prior differ in a way that can be stated generally. A prior that only chooses among the fields the data allow — a penalty, however clever — is unfalsifiable by those data: the reconstruction always fits, whatever the truth. A prior that excludes fields the data allow — a constraint, or a representation with fewer unknowns than the data have independent facts — can fail to fit, and then the data refute it. And the result of this calculation is that the second kind is the one that saves views. Non-negativity helps because it excludes most of the null space; smoothness helps little because it excludes nothing, only ranks.

There is a caveat, and it is the one that keeps this from being a guarantee. Non-negativity is refuted only when no non-negative field meets the data. If the negative part of a field lies in the null space, some non-negative field may meet the rays, and the reconstruction will fit and be wrong. A constraint can be caught, but only when the error it makes is visible to the views.

A dictionary nearly right #

The dictionary has a failure the other priors do not, and it is worth a figure. Fitted without regularisation, from three to six views, its error runs to hundreds and thousands of per cent: 157 times the field at three views. The dictionary is not wrong by much — puffs approximate the streak tolerably — but the puffs overlap, and some combinations of them are almost invisible to the views. Fitting amplitudes to data that no combination of puffs can match exactly, the fit pours the mismatch into those nearly invisible combinations with enormous coefficients. The reconstruction is then confident in the sense that it is decisive, and wrong by two orders of magnitude.

Regularisation cures it — a small penalty on the amplitudes, which is itself a second prior, that the amplitudes are modest — and the regularised error stays near the no-prior error for few views and stalls near a fifth. A dictionary is exact when it is right: a field built from three of its puffs comes back from six views to four parts in a hundred thousand. When it is nearly right, it needs either many views or a second prior to keep it from inventing structure.

What an experimenter already assumes #

None of this is hypothetical for the instruments in use. Tomographic PIV reconstructs the light intensity scattered by seeding particles in a volume, and its standard algorithm, the multiplicative algebraic reconstruction technique, cannot produce a negative intensity: it is a non-negativity prior built into the iteration. It works with four to six cameras, far fewer than the resolution would demand, for exactly the reason drawn here — a seeded volume is mostly dark, so non-negativity excludes nearly everything the views cannot see. It also fails in the way drawn here. When the seeding is too dense, the volume is no longer mostly empty, and the reconstruction fills with ghost particles: bright spots that meet every camera’s rays and are not there. Ghost particles are the null space of a non-negativity reconstruction becoming visible once the field stops being sparse enough for the constraint to pin it.

Background-oriented schlieren and interferometric tomography of a flame or a plume measure a refractive index relative to ambient, and there a sign constraint is wrong, since a hot region is less dense than its surroundings. What those instruments use instead is smoothness, usually as a regularisation weight chosen by eye, and this calculation says what that buys: little, and without warning when the field has a front. A flame imaged tomographically with few views and a smoothness prior has its flame sheet blurred by an amount the data cannot report. The remedies are the ones the figures point to: more views, or a prior that claims something the data could contradict — a known flame thickness, a known number of structures — so that when it is wrong, the misfit says so.

What a photograph shows made the general point that every flow picture is a measurement with an operator between the flow and the image; the window every vector is averaged over was the same point for a PIV vector. A tomographic reconstruction adds a second operator on top — the prior — and it is the one the experimenter chooses. Choosing a prior the data can refute is the only way to learn afterwards whether it was right.

What was checked #

Three checks. The minimum-norm, smoothest and non-negative reconstructions of a field of puffs reproduce the four views’ rays to two parts in a hundred thousand, which is the statement that they act only on the invisible part. With twenty views the grid has no null space, and the minimum-norm and smoothest fields both return the true field to four parts in ten thousand. A field made of three dictionary atoms is recovered from six views to four parts in a hundred thousand. The tests also refuse a prior the calculation does not know, a field it does not know, and a tolerance of zero.

What the picture cannot show #

Noise. Every ray here is exact. With noise the misfit floor rises to the noise level, a dictionary’s few per cent of misfit can hide under it, and a constraint’s refutation needs a misfit well above the noise. The smoothest field, which was unfalsifiable already, stays so.

Priors chosen with the answer. The dictionary’s regularisation weight was chosen for the smallest error, which only a calculation that knows the truth can do. In practice it would be chosen from the misfit, and would be a little worse.

One grid and one geometry. Sixteen cells and parallel rays over half a turn. Cone-beam cameras, limited optical access — views crowded into a narrow arc, as the earlier essay also drew — and finer grids change every number and none of the distinctions.

An old problem in a new instrument #

The question of what a set of measurements determines and what must be assumed is not special to tomography. The inside a flow does not decide is the same question about the singularities inside a body, of which many give the same outside flow; every instrument examined so far — the paint that measures the wrong field, the shutter that is part of the answer, an instrument that takes a derivative — has an operator that loses something, and a reading that must supply it. One view is enough, and the axis pays for it was the case in which a prior — axial symmetry — is so strong it removes the null space entirely, and its price was the noise it amplified at the axis. What the priors here add is the observation that a prior can be tested by the same data it acts on, but only if it claims something the data could contradict. A prior that only expresses a preference is untestable by construction, and it is also the one that helps least.

Who worked it out #

Algebraic reconstruction and its null space are Gordon, Bender and Herman’s, from 1970, and the counting of views against resolution is Crowther, DeRosier and Klug’s of the same year. Regularisation of ill-posed inverse problems is Tikhonov’s, from the 1960s. Non-negativity as a constraint that recovers sparse objects from few measurements is the core of the compressed-sensing results of Candès, Romberg and Tao and of Donoho, from 2006, and the multiplicative algebraic reconstruction used in tomographic PIV, which enforces positivity, is Elsinga, Scarano, Wieneke and van Oudheusden’s, from 2006.

Still open: the views a noisy measurement needs #

The contrast here was drawn with exact rays, which is why a smooth reconstruction’s perfect fit was damning and a constraint’s misfit decisive. A real camera adds noise, and the question changes shape: a prior now has to be judged by how its error and its misfit compare with the noise, and the regularisation weight has to be chosen from the data alone. The next calculation adds noise at a stated level, chooses every prior’s weight by the discrepancy principle — the misfit set equal to the noise — and asks how many views each prior then needs, and whether a constraint’s refutation survives when its misfit is comparable with the noise it has to be seen against.