22.3 Measuring lens quality⧉
After cataloging lens defects, the next step might seem to be correction. But correction and optimization both need measurements first. The optimizer of the next chapter minimizes a number; the deconvolution of Aberrations correction inverts a measured blur; the manufacturer who sells you the lens quotes a spec; and the reviewer who pans it backs the verdict with a chart. Every one of those is a measurement. So this chapter is the assessment step that the rest of the part stands on: how do you turn "this lens is a bit soft in the corners and fringes in blue" into numbers you can optimize against, invert, advertise, or argue about? You cannot improve, target, or correct what you have not quantified: that is the whole reason this chapter comes before Lens optimization.
One measurement dominates, and we have already met the underlying machinery. Locally, a linear incoherent imaging system blurs approximately by convolution with its point-spread function; convolution is multiplication in the Fourier domain. The frequency-domain face of the lens's blur is its transfer function, and its magnitude, which tells you how much contrast survives at each spatial frequency, is the Modulation Transfer Function, the main metric of this chapter. Everything else (spot size, distortion, vignetting, color shift) is a companion measurement that the MTF either contains or sits beside.
22.3.1 The modulation transfer function (MTF)⧉
Imagine photographing not a sharp edge or a point but a smooth sinusoidal intensity pattern: bright and dark stripes whose brightness varies as a sine wave at a single spatial frequency $\nu$, measured in line-pairs per millimeter (lp/mm) at the image plane. A perfect lens would reproduce that sinusoid with its full contrast intact. A real lens, because it spreads each point into a small blob, smears adjacent bright and dark stripes into each other and the reproduced sinusoid comes out with less contrast: the peaks lower, the troughs higher. Define the modulation of a pattern as $(I_\text{max}-I_\text{min})/(I_\text{max}+I_\text{min})$, and the MTF at frequency $\nu$ is simply the ratio of output modulation to input modulation:
Plot that ratio against $\nu$ and you get the curve familiar from lens reviews (Figure 22.3.1). It starts at $1.0$ at zero frequency (the transfer is normalized by its zero-frequency response; a uniform field itself has no sinusoidal contrast) and generally falls toward $0$ at fine frequencies, though lobes and zeros can interrupt a monotonic decline, because the finest stripes are blurred into a flat gray. Coarse detail (low $\nu$) survives well; fine detail (high $\nu$) is attenuated; somewhere the lens simply stops resolving. The MTF is the single most-used summary of what "sharpness" means, and the rest of this chapter is, in one way or another, ways to obtain it and things to plot it against.
The OTF/PSF relationship. The MTF is not separate from the blur studied in the deblurring chapters. It is that blur, viewed in the frequency domain. The lens's full frequency response is the Optical Transfer Function (OTF), and the OTF is exactly the Fourier transform of the point-spread function:
The MTF is the magnitude of the OTF; the discarded phase part is the Phase Transfer Function (PTF), which retains phase shifts lost from the magnitude alone. Coma can affect both magnitude and phase; the MTF alone cannot distinguish a PSF from its mirror image. This is the same convolution-theorem duality the book keeps returning to: blurring an image is convolving by the PSF in space, which is multiplying by the OTF in frequency. The image is $\text{scene} * \text{PSF}$, and Aberrations correction's deconvolution divides by the OTF, so the MTF is literally the thing whose zeros bound what deconvolution can recover (you cannot divide back contrast the lens drove to zero). PSF and complex OTF are equivalent descriptions, while MTF discards phase; this chapter mostly works in the frequency domain because that is where "sharpness vs. detail size" reads off most cleanly.
Scalar summaries: MTF50 and limiting resolution. A whole curve is more than people want when they are ranking lenses, so the curve gets collapsed to a number or two. A common scalar is MTF50, the spatial frequency at which the MTF has fallen to $0.5$, i.e. where the lens still passes half the contrast. It is popular because it correlates well with perceived crispness: a lens with a higher MTF50 usually looks sharper. A second common scalar is the limiting resolution, the frequency at which the MTF drops to some small cutoff (say $0.1$, or the noise floor), the finest line-pairs/mm the lens still resolves at all. The two answer different questions: MTF50 is about how punchy the mid-frequencies look, limiting resolution about the absolute finest detail present.
The diffraction ceiling. For a fixed pupil amplitude and support in incoherent imaging, removing wavefront aberration gives the diffraction-limited MTF bound, because diffraction is a floor on the PSF that no amount of design removes (FUNDAMENTALS). For a circular pupil the diffraction-limited MTF falls to zero at the cutoff frequency
with $\lambda$ the wavelength and $N$ the image-side f-number in the paraxial air model (use the effective value at finite magnification): beyond $\nu_c$ a perfect lens passes nothing. Stopping down (raising $N$) lowers $\nu_c$, which is the frequency-domain statement of the familiar fact that small apertures soften everything by diffraction. So the diffraction curve is the ceiling in Figure 22.3.1, and a real lens's MTF lives somewhere underneath it, pushed down further by every aberration the previous chapter named.
MTFs multiply. One last fact makes the MTF the right representation for a photographic system, not just a lens. The sensor blurs (finite pixels, the anti-aliasing filter), the demosaic blurs, motion blurs, and the lens blurs, and for linear, shift-invariant stages their transfer functions multiply:
This is why a superb lens on a coarse sensor still looks soft (the sensor's MTF caps the product), and why "system MTF" (lens and sensor and pipeline together) is what photography actually cares about. Adaptive denoising, nonlinear demosaicking, sharpening, and sampling can violate this simple factorization. A measured system response then depends on the target and settings, rather than being one universal product.
22.3.2 Measuring MTF in practice⧉
So far the MTF is a definition. How do you actually obtain it from a lens on a bench? There are two families of method, and the second one is the workhorse of modern testing.
Direct: sinusoidal and Siemens-star targets. The literal approach is to photograph the very thing the definition describes (a chart of sinusoidal patterns at a ladder of known frequencies) and read the output modulation off each, frequency by frequency. It is conceptually pure but fussy: you need an accurate sinusoidal target, careful exposure, and one patch per frequency. A neat refinement is the Siemens star, a radial pattern, sinusoidally modulated for direct MTF measurement or formed from binary wedges for a simpler resolution check, whose spatial frequency rises smoothly toward the center; a single star sweeps a whole range of frequencies in one shot, and the radius at which its spokes blur into gray reads off the limiting resolution. These methods give trustworthy MTF numbers and are still used for careful lab work.
The slanted-edge (SFR) method: ISO 12233. The method that took over, because it needs only one frame of one edge, is the slanted-edge or spatial-frequency-response (SFR) technique standardized as ISO 12233. Photograph a sharp black-to-white edge tilted a few degrees off vertical. The slant provides subpixel sampling: because the edge is slanted, each successive row of pixels crosses it at a slightly different sub-pixel position, so the rows together sample the edge profile at a much finer spacing than the pixel pitch. Pooling all the rows (projecting each pixel onto the axis perpendicular to the edge and binning) yields a super-resolved profile of the edge: the edge-spread function (ESF), what the lens-plus-sensor does to an ideal step. The slant yields finer sampling than the pixel pitch alone.
From the ESF the rest is mechanical (Figure 22.3.2). Differentiate the ESF and you get the line-spread function (LSF), the response to a thin line rather than an edge, which is the two-dimensional PSF integrated along the edge direction. Fourier-transform the LSF and take the magnitude, normalized so the zero-frequency value is $1$, and you have the MTF (in this context usually called the SFR). The compact chain below omits the DC normalization; implementations also linearize pixel values, window the LSF, and compensate the numerical derivative. Avoid clipped edges and document processing. The chain is the whole method in one line:
This method is cheap, repeatable, and automatable. Open implementations (Burns's sfrmat, the open-source MTF Mapper) read the chart and spit out curves, which is one reason camera reviews and QA lines rely on it. Crucially, it measures the whole system: lens, sensor, anti-aliasing filter, demosaic, and any ISP sharpening, all folded into one curve (the product of MTFs from the previous section). For photography that is useful: it is the sharpness you will actually get, though it means you must hold the sensor and processing fixed if you want to compare lenses.
Where, and in which direction. A single curve hides the fact that the MTF is not one number even for one lens at one aperture: it varies across the field and with the orientation of the detail, so a real measurement is a small matrix of curves. Across the field, a lens is nearly always sharpest on the optical axis and softer toward the corners (aberrations grow with field angle), so the bench measures the MTF at several image heights (center, mid-frame, and corner) not just on-axis. Orientation matters because at any off-axis point two special directions behave differently: sagittal structures, oriented radially (spoke-like, pointing at the frame center), and meridional (also called tangential) structures, oriented across the radius (concentric, rim-like). Off-axis astigmatism (Chapter 22.1, Aberrations and optical challenges) brings these two orientations to focus at slightly different distances, so the lens resolves one better than the other and their MTF curves separate; on-axis, by symmetry, the two coincide. This is exactly what a manufacturer's MTF chart shows (Figure 22.3.3): contrast plotted against image height, with often a solid curve for sagittal and a dashed curve for meridional (check the legend and whether the curves are measured or calculated), usually at two frequencies: a low one (around 10 lp/mm) that reads as contrast and "character," and a higher one (around 30 lp/mm) that reads as fine-detail sharpness. Do not confuse the orientation of test lines with the perpendicular direction in which their intensity varies. Manufacturers and measurement programs can use different sagittal/tangential labeling conventions. Once you know the conventions, the chart is easy to read: curves that stay high and flat mean even sharpness to the edge; a large sagittal–meridional gap indicates directional differences that can arise from astigmatism, coma, and other effects; both falling together means plain corner softness; and a low-frequency curve that sags signals veiling flare or low micro-contrast rather than lost resolution.
22.3.3 Spot diagrams and the PSF⧉
The MTF lives in the frequency domain; its space-domain twin is the point-spread function and, for a designer working from a ray model, the spot diagram. As Lens optimization develops in detail, you can estimate the PSF geometrically by tracing a fan of rays from one field point through the lens and scattering their landing points on the image plane (Figure 22.4.1). A tight cluster means the lens brings that point nearly to a point (sharp; a smeared blob means it does not) and, as the previous chapter showed, the shape of the blob diagnoses which aberration dominates (a round haze is spherical, a one-sided comet is coma, an ellipse is astigmatism).
For measurement purposes the spot diagram hands us a scalar (the RMS spot radius, the root-mean-square spread of the landing points) which is the classical sharpness metric the merit function of the next chapter shrinks. And experimentally, rather than tracing rays, you can simply image a point source (a pinhole or a distant star) through the real lens and photograph its PSF directly. That measured PSF is precisely what Aberrations correction feeds to deconvolution. So the same lens quality appears in three forms: as a measured PSF (image of a point), as a ray-traced spot diagram (the design's prediction), and as an MTF (the magnitude of the Fourier transform of a normalized PSF). A ray spot diagram omits diffraction and needs appropriate ray weights to approximate an intensity PSF. It is useful when geometric blur dominates; it is not interchangeable with a measured diffraction-inclusive PSF near the diffraction limit.
22.3.4 MTF through focus and depth of focus⧉
Everything above measured the MTF at the best focus. But focus is itself an adjustment, and asking how the MTF behaves as you change focus answers a different and very practical question. Fix a single spatial frequency, then slide the sensor (or refocus the lens) and re-measure the MTF at each position. The contrast rises to a peak at best focus and falls away on either side: the through-focus MTF (Figure 22.3.4). The position of the peak is best focus; the width of the peak is the depth of focus: the range of sensor positions over which that frequency stays above a usable contrast. (Depth of focus is the image-side cousin of the FUNDAMENTALS depth of field on the subject side; they are linked by magnification.)
Two features make this measurement diagnostic. First, higher spatial frequencies give narrower through-focus peaks: fine detail demands more precise focus than coarse detail, so the depth of focus shrinks with the detail you ask for: a low-frequency target tolerates sloppy focus, a high-frequency one does not. Second, if the through-focus peaks for different frequencies (or for different field points, or different colors) sit at different focus positions, that offset is a signature of residual aberration: spherical aberration shifts best focus with frequency, field curvature shifts it with field, axial chromatic aberration shifts it with color (the next section). A perfectly corrected lens would have all its peaks aligned; the offsets reveal the residual aberrations.
22.3.5 Field-dependent MTF: sagittal vs tangential⧉
A lens is sharp in the center and softer toward the corner, so a single MTF number for "the lens" hides where it is good and where it gives up. The standard published chart therefore plots the MTF against image height (distance from the optical center, in millimeters, from $0$ at the center out to the corner) at a coarse frequency (say $10$ lp/mm, which tracks contrast and "punch") and a fine frequency (say $30$ lp/mm, which tracks fine resolution). And each of those is split again into two curves, because off-axis a lens treats two orientations of detail differently: sagittal detail (lines running radially, toward the center) and tangential detail (lines running across the radius, azimuthally) focus differently. So the manufacturer chart is four curves (sagittal and tangential at each of two frequencies) plotted center to corner (Figure 22.3.5).
Reading the chart is straightforward once you know what the curves mean. Both curves falling toward the corner is the ordinary off-axis softening, as coma, field curvature, and lateral color pile up where the previous chapter said they would. The more telling feature is the gap between the sagittal and tangential curves: that gap can indicate astigmatism or other directional aberrations. Through-focus measurements can reveal whether radial and tangential detail reach best focus on different planes; when they do, one sensor plane cannot render both at their sharpest. A lens whose four curves stay high and together across the whole frame is "even"; one whose corners collapse, or whose sagittal and tangential curves split wide apart off-axis, is dumping quality into the edges, exactly the compromise Lens optimization will show being negotiated, and exactly the corner softness Aberrations correction may be asked to claw back.
22.3.6 Color, geometry, and illumination measurements⧉
The MTF and its through-focus and field variants quantify blur. But the previous chapter's catalog had defects that are not blur at all (color misregistration, displaced geometry, darkened corners) and each has its own measurement, which (not coincidentally) is exactly the data the lens-profile correction of Aberrations correction needs.
Lateral (transverse) chromatic aberration is a difference in magnification between color channels, so it is measured as a per-channel radial scale: image a high-contrast target, find where the red, green, and blue edges land as a function of field, and fit the tiny scale factor that separates them. It comes out as one magnification offset per channel, which is precisely the per-channel radial rescale the ISP plays back to collapse the colored fringe.
Longitudinal (axial) chromatic aberration is instead a difference in focus depth between colors, so it is measured through focus: find the best-focus sensor position for red, green, and blue separately, and the spread between them is the axial color. It is the through-focus measurement of the previous section, repeated per wavelength, and it is what achromat and apochromat design (and, partially, deconvolution) aim to flatten.
Geometric distortion is measured by imaging a ruled grid and quantifying how far the imaged lines bow from straight, then fitting the Brown–Conrady coefficients $k_1, k_2, \dots$ (or the division-model $\lambda$) defined in the challenges chapter; the result is reported as a percent distortion versus field. Those fitted numbers are exactly the coefficients the radial-warp correction inverts.
Vignetting / relative illumination is measured by photographing a uniform field and recording the radial brightness falloff: corner brightness relative to center, in stops or percent. A single uniform-field image measures their combined effect, including sensor angular response; it does not separate the mechanisms by itself. Aperture-dependent measurements and a model can help isolate optical clipping. The calibrated combined response supplies the gain map used to even the frame.
Field curvature is measured by mapping best-focus depth versus field: the surface on which the image is actually sharp is not the flat sensor but a curved (Petzval) surface, and tracing that surface predicts how soft the corners go when the center is nailed. It is the spatial bookkeeping behind why "flat-field" designs are prized.
Each of these is a small, targeted measurement, and each one's output is the input to a correction in the next chapters: the clean illustration of why measurement has to come first.
22.3.7 Scalar quality summaries⧉
Finally, the field wants a single number to rank lenses and to hand an optimizer, so the rich curves above get collapsed, at a cost. The common scalars are the ones we have already met: MTF50 and limiting resolution (lp/mm) from the MTF curve, RMS spot radius from the spot diagram, and the wavefront-error metrics: RMS wavefront error and its companions the Strehl ratio (peak intensity of the real PSF relative to the diffraction-limited one) and the Maréchal criterion (a lens is "diffraction-limited" if its RMS wavefront error is below about $\lambda/14$). Some summaries go further and weight the MTF by the human contrast-sensitivity function to predict perceived sharpness: the family of "blur index" or acutance scores (SQF and its relatives).
The plain caveat is that every one of these is a lossy summary of a much larger object: a fuller description includes MTF and phase over frequency, field, focus, and color, together with transmission, distortion, flare, and other properties, effectively a hypercube, and any single number throws most of it away. A scalar is genuinely useful for ranking lenses and, especially, as an optimization target the next chapter can drive down; but it hides where in the frame, at what detail size, and in what color the lens is good or bad. That is the right note to end on, because it is the seam to the rest of the part: this chapter produces the scorecard, Lens optimization compresses it into a merit function and minimizes it, and Aberrations correction takes the measured PSF, distortion, vignetting, and color maps and inverts them. Measurement first; then you optimize, and then you correct what optimization left behind.