One view is enough, and the axis pays for it

An instrument that takes a derivative establishes what the optical techniques record: an interferometer the density, a schlieren system its first derivative across the rays, a shadowgraph its second. It then names a problem and declines it. Every one of those records is an integral along the line of sight, so a point in the image is a sum over everything the ray passed through, and recovering the field itself from such images is an inversion — ill-posed, that essay says, and how ill-posed is a computation with a number in it.

For a flow with no symmetry the inversion needs many views from many angles. For a flow of revolution it needs one, and it can be done exactly. The number turns out to depend less on the flow than on the instrument, and in the direction the history of the instruments would not suggest.

What one view records #

The test field is a Gaussian of refractive-index excess across a jet, $f(r) = e^{-r^2/\sigma^2}$ with $\sigma$ three tenths of the radius at which the field has vanished — a stand-in for the density defect of a hot plume or the density excess of a cold jet. Since a gas’s refractive index is linear in its density, the field is a density field up to a constant — the kind of field a cold round jet carries until its column breaks up.

A beam of parallel rays passes through it across the axis. A ray at height $y$ above the axis crosses every radius from $y$ outwards, twice, and an interferometer records the phase it accumulated, which is the integral of the field along the ray:

$$P(y) = 2\int_y^R \frac{f(r)\,r\,dr}{\sqrt{r^2 - y^2}}.$$

For the Gaussian that projection is again a Gaussian, $\sqrt\pi\,\sigma\,e^{-y^2/\sigma^2}$, peaking at 0.532 on the axis. A schlieren system records its slope across the rays, $dP/dy$, which is zero on the axis — where every ray is symmetric — and peaks at $y = \sigma/\sqrt2 = 0.212$.

Neither record is the field. The projection is broader than the field in every sense but its shape, because the ray at height $y$ has collected from every ring outside $y$ as well as its own. The deflection is zero at exactly the point where the field is largest. What is wanted is $f(r)$, and what is recorded is a sum over it or a derivative of a sum over it.

Abel’s integral, and why it can be undone #

The relation between the projection and the field is Abel’s integral, and it is older than the instruments. Abel posed it in the 1820s for a mechanical question — given how long a bead takes to slide down a curve from each starting height, what is the curve — and its inversion is exact:

$$f(r) = -\frac{1}{\pi}\int_r^R \frac{P'(y)\,dy}{\sqrt{y^2 - r^2}}.$$

The field at radius $r$ is determined by the slope of the projection at every ray outside $r$. That is the whole reason one view suffices: axial symmetry means every ring is crossed by rays at every height below its own radius, so the rings can be recovered one at a time from the outside in.

The formula also says what the inversion does to its data. It takes a derivative of the projection and then an integral with a kernel that is singular at $y = r$. Together those amount to half a derivative — an operation between doing nothing and differentiating — and any derivative applied to noisy data amplifies the noise. The question is by how much, and for which data.

The convention every figure below uses is worth naming because each part of it is an assumption about the experiment: parallel rays, so that a ray’s height is the same all the way through; perfect axial symmetry about a known axis; a field that vanishes at the outer radius, so that the outermost ray records zero; and refraction weak enough that the rays go straight, which is what makes the record an integral along a straight line.

Peeling from the outside in #

The simplest discrete inversion is onion peeling. The field is taken as constant within each of $N$ concentric rings, and one ray is taken through the middle height of each. The ray through ring $i$ crosses ring $i$ and every ring outside it, and the length of each crossing is the difference of two chords, so each recorded integral is a sum of the ring values weighted by known lengths. The outermost ray meets only the outermost ring and gives its value directly; the next ray meets two rings, one now known; and so on to the axis, which is the last ring solved.

The weights are computed exactly and the construction is checked against the closed form. On 200 rings, the chord sums reproduce the Gaussian’s analytic projection to 1.7 parts in ten thousand of its peak, and inverting the analytic projection recovers the field to four parts in ten thousand everywhere. With clean data, one view is enough, and onion peeling is a perfectly good way of using it.

The construction also shows where trouble will concentrate. Every ring’s value is found by subtracting the known contributions of all the rings outside it from its ray’s integral. An error in any outer ring is subtracted along with its value, into every ring inside it, and the axis — the last ring solved — receives the accumulated residue of all of them.

One per cent of noise #

Real images carry noise, and the test gives each ray an independent error of one per cent of the largest value that instrument records — the projection’s peak for the interferometer, the deflection’s peak for the schlieren system — so that the two are equally good photographs by their own standards. The noise is seeded, so the figure is reproducible.

From the interferogram, fifty rings, the axis comes out at 1.431 where the truth is 1: a one per cent error in the image has become a forty-three per cent error at the centre of the jet. Near the edge the recovered field is close to the truth. The error gathers towards the axis exactly as the construction predicted, and it gathers where the field is largest and where the flow — the core of the jet, the centre of the flame — is usually most interesting.

From the schlieren deflection, the axis comes out at 0.876, and the error over the whole radius is a third of the interferogram’s. That is the first sign that the two instruments are not equally suited to the job, and it is the opposite of what their reputations would suggest: interferometry is the quantitative technique and schlieren the qualitative one.

Where the noise ends up #

A single noisy reconstruction is one sample. The noise gain of each ring — the standard deviation of its recovered value per unit of relative noise on the data — can be computed exactly from the inverse of the chord matrix, without sampling at all.

For an interferogram on fifty rings the gain is 19.78 on the axis and 1.88 at the edge; a seeded set of four hundred noisy reconstructions gives 20.05 on the axis, within the sampling error of the exact figure. On 25 rings the axis gain is 9.89 and on 100 rings it is 39.56. Doubling the resolution doubles the noise on the axis.

The reason is in the last step of the peeling. The innermost ring’s value is what remains of the central ray’s integral after every outer ring’s contribution has been subtracted, divided by the length of that ray’s chord through the innermost ring. That chord is proportional to the ring’s width. Halve the ring width, and the residue — noise and all — is divided by half the length, while the number of noisy subtractions feeding it has doubled. The finer the rings, the thinner the slice each ray has to be divided by, and nothing in a sharper photograph compensates.

Finer is noisier, unless the optics took the derivative #

Across nine resolutions from 12 rings to 200 the interferogram’s axis gain grows as the number of rings to the power 1.000, from 4.75 to 79.11. The schlieren deflection’s gain does not grow at all: it is 0.94 at 12 rings and 0.94 at 200, a power of 0.002.

This is the surprising connection, and it runs back to what the previous essay found each instrument is. A schlieren system records the derivative of the projection. To use it, the projection is recovered by integrating the deflection inwards from the edge, where the projection is zero, and the integration turns each ray’s independent noise into a step whose size is that noise multiplied by the ray spacing. Onion peeling then takes differences of neighbouring rays and divides by a length proportional to the ray spacing — and the multiplication and the division cancel. The noise that reaches the axis no longer depends on the resolution.

The derivative the schlieren optics took is exactly the derivative the inversion needed. Abel’s inversion formula begins by differentiating the projection; a schlieren image arrives already differentiated, with its noise on the derivative rather than on the integral; and what remains to be done to it is the half of the inversion that smooths. The instrument praised for being qualitative is, for this question, the quantitatively better one — and its blindness to a uniform stream, which the previous essay found exact, costs nothing here, since the one constant the integration needs is supplied by the field vanishing at the edge.

What a better photograph actually buys #

The result invites a conclusion that is half right: that an interferogram should be inverted at low resolution. The half that is right is that the rings of an onion-peeling inversion should be no finer than the noise allows, and the arithmetic says exactly how fine that is. The gain on the axis is about 0.4 times the number of rings, so for noise of one per cent of the projection’s peak the axis of a fifty-ring reconstruction carries a twenty per cent uncertainty and a two-hundred-ring reconstruction nearly eighty. To hold the axis to ten per cent at one per cent of image noise, the reconstruction can use about twenty-five rings — whatever the camera resolves.

The half that is wrong is that the extra pixels are wasted. They are not wasted on the rays: averaging neighbouring pixels into one ray per ring reduces each ray’s noise by the square root of the number averaged, which buys back part of what finer rings cost. A camera with sixteen pixels to a ring and rings sized to the noise is better than one pixel to a ring, and the trade is the same one between resolution and noise that the PIV window forces on a velocity field — a filter chosen deliberately instead of one imposed.

What the schlieren route buys is the freedom not to make that trade. Its axis gain does not depend on the number of rings, so its reconstruction can be as fine as the camera allows without paying for it at the centre. The cost moves elsewhere. The deflection has to be integrated from an edge where the field is known to vanish, and any error in that assumption — a slow skirt of warm gas outside the imaged region — is carried inwards to every ring, as a smooth bias rather than as noise. The interferometer’s weakness is noise and the schlieren system’s is the boundary, and which matters more depends on whether the flow ends inside the picture.

That trade has a familiar shape. It is the difference between telling an equation too much at its boundary and too little: the interferogram’s inversion needs no boundary value and pays in amplification, and the deflection’s needs one and pays in its sensitivity to it.

What is assumed rather than measured #

The inversion is exact under its conventions, and every one of them is a claim about the flow the photograph cannot check.

Axial symmetry. A flow that is almost symmetric is reconstructed as a symmetric flow, and whatever asymmetry it has is folded into the rings — a slightly bent jet appears as a jet with a spurious ring structure. The image cannot say how symmetric the flow was.

The axis. The centre must be located before inverting, from the image itself. A centre misplaced by a fraction of a ring width puts the central ray through the wrong ring, and the axis value, which is the most sensitive, is also the most sensitive to that.

The edge. The field must have vanished at the outer radius so that the outermost ray reads zero and the schlieren integration starts from a known value. A field that extends beyond the imaged region, like a plume’s slow-decaying skirt, contaminates every ring from the outside in.

What the construction leaves out #

Refraction. Strong density gradients bend the rays they are being measured with, so a ray’s path is not the straight line the integral assumes. Near a shock inside a supersonic jet, where the density jumps, the assumption fails outright.

Better inversions. Onion peeling is the simplest scheme and among the noisiest. Schemes that fit smooth basis functions, or that add a penalty on roughness, trade a little bias for much less noise, and they do reduce the interferogram’s gain. None of them removes the half-derivative, which is a property of the problem rather than of the scheme, and none of them makes the axis anything other than the worst place in the reconstruction.

Noise that is not independent. Real camera noise is correlated between neighbouring pixels, and a correlated error is amplified differently from an independent one — less at the finest scales and more at the coarse ones.

And more than one view. A flow without symmetry needs views from many directions, and the inversion is then the tomographic problem in full, with a conditioning far worse than this one’s and a strong dependence on how many directions are available. The axisymmetric case is the best case of optical tomography, and it is already the case in which the centre of the flow is the least certain part of it.

Where the inversion came from #

Abel’s integral equation is from 1823 and 1826, and it entered physics through spectroscopy rather than fluid mechanics: the light from an electric arc or a flame is an integral of the local emission along the line of sight, and recovering the emission as a function of radius from a side-on photograph is the same inversion. Onion peeling, the three-point method and filtered back-projection were compared systematically in the early 1990s, and the comparison found, as the arithmetic here does, that the noise behaviour is dominated by how the derivative is handled.

The instrument half of the story is the older one. Schlieren photography was a working technique for nearly a century before anybody inverted an axisymmetric schlieren image, and interferometry displaced it for quantitative work because an interferogram’s fringes are a number and a schlieren grey level is a slope. Which of them is better for recovering a field from one view is not a question of which is more quantitative. It is a question of which of Abel’s two operations the optics has already performed.

Still open: a flow without an axis #

The case this leaves open is the general one: a flow with no symmetry at all, seen from a handful of directions, where the field must be recovered from projections that do not individually determine any part of it. That is optical tomography, and its conditioning depends on the number of views in a way that makes the axis’s cost here look mild.

Beside it is a question about the other instruments in this family asked in the same terms. The window every vector is averaged over is a filter and this is an inverse, and a PIV system viewing a flow through a curved window, or a pressure-sensitive paint on a surface seen obliquely, each combine the two. What an instrument does to noise is as much a part of its operator as what it does to the signal — the lesson the paint and the shutter each reached separately, and one that a photograph of a flow is always silently subject to.