Four cameras and a field they cannot see

One view is enough, and the axis pays for it rebuilt an axisymmetric jet from a single photograph. Abel’s integral inverts exactly, because a field with an axis looks the same from every direction: one view is every view, and the only price is that noise collected on the way in from the edge arrives at the axis multiplied.

A flow with no axis gives nothing away. A plume bent by a crosswind, a jet in a swirling combustor, the wake behind a car — each looks different from each direction, and each picture is a set of line integrals through the field along one family of parallel rays. Putting the field back together from several such pictures is tomography, the same mathematics as a medical scanner’s, and the question this essay asks of it is not the one the axisymmetric case asked. That case asked how much noise the inversion amplifies. This one asks whether a few pictures determine the field at all.

They do not, and the way they fail is specific enough to draw.

A picture is a set of sums #

The field here is a grid of cells: 16 across and 16 down, 256 unknown values of density or refractive index. A view is a set of parallel rays crossing the square at one angle, 24 of them spanning its diagonal, and each ray records the sum of the field along its path — computed by sampling every quarter of a cell and interpolating between cell centres, which makes each ray a weighted sum of a handful of cells. Four views equally spaced over half a turn make 96 such sums.

Everything a set of views can say is then a statement about one matrix, with a row per ray and a column per cell. How many independent things the views say is its rank. The fields the views cannot see at all are its null space: combinations of cells whose weighted sums along every ray are zero. How faintly the views see the rest is in its singular values, which were found here from the symmetric matrix the projection makes with itself, by Householder reduction and implicit QL iteration. The decomposition reproduces its own matrix to $7\times10^{-16}$, and a single ray through a Gaussian bump on a finer grid matches the Gaussian’s closed-form projection to 0.6 per cent.

What four cameras miss #

The left-hand pattern is not noise and not an artefact. It is the part of the actual field that lies in the null space of four views: subtract it, and every one of the 96 rays reads the same number as before. No reconstruction from those four views, however clever, can tell the middle field from the right-hand one, because they produce the same data. The pattern is built by subtracting from the field everything the four views can see of it, and what remains is invisible by construction: the null space’s own basis patterns, found separately from the decomposition, project to less than $2\times10^{-14}$ of their own size.

This pattern is one combination of many. Four views of this grid have a rank of 79, so 79 independent patterns of density are seen and 177 of the 256 are not seen at all. The field’s invisible part can be any combination of those 177, and nothing in the pictures constrains which.

The shape of the invisible part tells what the cameras could not know. It is streaked along the four viewing directions and hollowed between them, because a pattern that alternates in sign across a ray integrates to nothing along it — and with only four ray directions there are many ways to alternate so as to cancel along all four at once.

Each view adds one ray’s worth, until the field is pinned #

The rank does not grow in a surprising way, and that is the useful part. Each view adds independent information at roughly the rate its rays cross the field — a little more than the grid’s width — with a small loss to rays that overlap what earlier views already fixed. So the rank rises nearly linearly, and the views needed to pin a field grow in proportion to the resolution wanted: 10 views for a 12-cell grid, 14 for 16, 20 for 24. A grid twice as fine needs about twice as many directions.

That proportionality is the discrete form of a result from structural biology. In 1970 Anthony Crowther, David DeRosier and Aaron Klug showed that reconstructing an object of diameter D to a resolution d needs about πD/d projections spread over half a turn, which is the same statement for a continuous object: the number of views is set by how many resolution elements fit round the object’s circumference. This grid pins itself with fewer views than that estimate would ask, because interpolating between cells smooths the field and reduces what there is to determine; the scaling is the same.

A null space is a cliff, not a slope #

The spectrum makes the distinction that the misconception blurs. A reconstruction’s error has two sources, and they are different in kind.

The patterns to the left of the cliff are seen, some faintly. Their singular values fall smoothly by an order of magnitude or two, and noise in the pictures is amplified in inverse proportion to how faintly a pattern is seen. That is the axisymmetric essay’s subject, and more careful pictures reduce it.

The patterns to the right of the cliff are not seen at all. Their singular values are zero, the pictures contain nothing about them, and a reconstruction must supply their values from somewhere other than the data — usually by setting them to zero, which is what a least-squares reconstruction does by default. No improvement in the cameras, the exposure or the noise changes this. Only another view does.

What a reconstruction from few views looks like #

The reconstructions are the visible part of the field with the invisible part set to zero, and they have a characteristic look. Each real structure is smeared along the viewing directions, because the views cannot say where along a ray the density sits; and density appears where there is none, in the corners and between the blobs, because the same ambiguity puts some of each structure’s sum in places its rays also cross.

The two-view reconstruction is the extreme case and the most instructive. Two perpendicular pictures of two blobs cannot distinguish them from the two blobs mirrored into the other two corners of the same rectangle — a pair of real features and a pair of phantoms give identical pictures. The smear across the middle is the reconstruction hedging between the possibilities.

Ghosts, by their proper name #

That hedging has a name in experimental fluid mechanics, and it is worth knowing because the most widely used volumetric velocity technique lives with it.

Tomographic particle image velocimetry, introduced by Gerrit Elsinga, Fulvio Scarano and colleagues in 2006, records a volume of seeded flow with typically four cameras and reconstructs the particles in three dimensions before correlating two reconstructions to get velocity. With four views of a volume, the reconstruction contains ghost particles: bright spots where lines of sight from several cameras happen to cross at real particles’ images, but where there is no particle. They are the three-dimensional form of the phantoms above — density the views cannot place, put where every ray is satisfied.

The technique works anyway, and the reason is instructive about the null space rather than an escape from it. Tomographic PIV does not use plain least squares; it uses a multiplicative algorithm that keeps the reconstruction positive and sparse, which is extra information — a prior belief that the field is mostly empty with a few bright points — and it is that prior, not the pictures, that pushes most of the ghosts towards zero. The ghosts that survive are a known source of bias in the measured velocity, and their number rises steeply with seeding density, because more particles mean more chance intersections. A few views of a sparse field can be made to work by assuming sparsity; a few views of a continuous field such as a density or temperature distribution have no such assumption available, and the null space is simply the answer.

Noise, on top of ignorance #

Adding noise separates the two kinds of error cleanly. At two and four views the errors are 69 and 44 per cent, within a point of the noise-free values: the error there is ignorance, and noise barely adds to it. At sixteen views the noise-free error was zero and the noisy one is 6 per cent: the error there is noise, amplified by the faintly-seen patterns at the end of the spectrum, and it is the axisymmetric essay’s problem again.

The two curves say what a measurement campaign should spend its effort on. Below the threshold, better cameras are wasted: the error is set by the views that were not taken. Above it, more views help only slowly — from 6 per cent at 16 views to 4 at 32 — and better pictures help directly. The threshold itself is the number worth knowing before a camera is mounted, and for this grid it is 14.

When the cameras cannot go round #

The views above are spread evenly over half a turn, which is the arrangement that sees most. A wind tunnel rarely allows it. Windows are on the sides, the model is in the middle, and the cameras end up crowded into an arc of ninety degrees or less.

Crowding the same fourteen views into a narrower arc does not reopen an exact null space on this grid — the rank stays at 256 — but it changes how faintly the views see what they see. Counting the patterns whose singular value is below a hundredth of the strongest: 55 with the views spread over 180 degrees, 73 over 90 degrees, and 98 over 60. Those are the patterns noise swamps, and the reconstruction error with one per cent of noise follows: 6 per cent over half a turn, 35 per cent over a right angle, 45 per cent over sixty degrees, with the same number of cameras and the same quality of picture in each case.

So the arc matters as much as the count. Fourteen views are enough to pin this field and not enough to measure it once they are squeezed into the directions a test section allows, and a camera moved to widen the arc buys more than a camera added inside it.

A ledger of the checks #

The checks are chosen so that each would fail for a different mistake. A ray integral that misweighted its samples would miss the Gaussian’s projection, which has a closed form. An eigen-solver that had lost orthogonality would leave a large residual. A null space computed wrongly would project to something other than zero. And a rank counted with a threshold too loose would report invisible patterns at fourteen views, where the spectrum shows none.

Why this matters more for a flow than for a patient #

Medical scanners take hundreds or thousands of views and never meet the null space, because the patient lies still in a ring that can rotate as long as it takes. A flow does not lie still. To capture an unsteady field at one instant, every view must be taken at once, which means one camera for each direction, and cameras, windows and optical access are what a wind tunnel is short of. The instruments that photograph flows — schlieren, shadowgraph, background-oriented schlieren, interferometry, chemiluminescence — are routinely set up with between two and a dozen views, which is exactly the range the threshold falls in.

The comparison with the scanner is worth one number. A grid 16 cells across needs 14 evenly spread views before it has no invisible part; a clinical scan resolves several hundred elements across the body and takes on the order of a thousand views to do it, comfortably above its own threshold. A flow experiment that wants even a 64-cell grid, which is coarse, needs something like sixty directions by the same scaling, and there is no test section in which sixty cameras look through the flow at one instant. That is why optical flow tomography lives with priors, with symmetry, or with the null space.

So the question for a flow measurement is not the medical one of how finely a large number of views resolves the field. It is how many independent things the views that could physically be fitted around the test section can say, and what fraction of the field that leaves invisible. The axisymmetric case escaped it by symmetry. Every other case has to count.

What the discrete calculation cannot show #

Real optics. Rays here are straight, thin and parallel. Real beams in a density gradient bend, have finite width, and diverge from a lens, and a fan-beam or cone-beam geometry changes the operator — not the existence of a null space, but its shape.

Continuous limited angles. On this grid, views crowded into a narrow arc keep full rank and pay in faint patterns rather than invisible ones. For a continuous field the limited-angle problem is worse than that: every direction outside the arc carries information no view inside it has, and the discrete grid’s full rank is partly the interpolation’s smoothing standing in for it.

Prior information. Least squares assumes nothing about the field. Positivity, sparsity, smoothness, or a known flow structure all reduce the effective null space by excluding fields that could not be the answer, and the right prior can make a few views sufficient. The price is that the reconstruction then reports the prior wherever the data are silent, and nothing in the result says where that was.

Three dimensions and time. The grid is a single plane. A volume needs views that cross it in more than one plane, and a time-resolved flow adds the further constraint that the field must be continuous in time, which is itself a prior.

Who worked it out #

Johann Radon showed in 1917 that a function in the plane is determined by its integrals along all lines, which is the continuous statement that enough views suffice. Allan Cormack worked out a practical inversion in 1963 and 1964, and Godfrey Hounsfield built the first clinical scanner in 1971; they shared the Nobel prize in 1979. The counting argument for how many views an object needs is Crowther, DeRosier and Klug’s of 1970, made for electron micrographs of viruses. The existence of an exact null space for any finite set of views — that a finite number of projections never determines a general function — is a theorem of Kennan Smith, Donald Solmon and Sheldon Wagner from 1977. In fluid mechanics, optical tomography of flames and jets followed from the 1980s, and tomographic PIV’s ghost particles have been the subject of their own literature since 2006.

Still open: how much a known flow structure buys #

The null space is exact only for a reconstruction that assumes nothing. Most flows are not arbitrary: a jet is smooth, a flame front is thin, a vortex has a known core structure. Each of these is a prior that excludes most of the invisible patterns, and the question worth computing is how many views each prior saves.

The next calculation takes the same field and the same views and reconstructs with three priors in turn — smoothness, positivity, and a representation in a small number of known flow structures — and asks for each how many views bring the error below a stated level, and how the reconstruction behaves when the prior is wrong. The last case is the one that matters, because a reconstruction built on a wrong prior is sharp, confident and incorrect, and the pictures cannot say so.