Thin Lens Model

The thin lens model is the simplest description of how a lens forms an image. It treats the lens as a plane of zero thickness that bends every ray by an amount proportional to the ray’s height above the optical axis. Under that rule, all rays leaving one point in front of the lens meet again at one point behind it, at a distance given by the thin-lens equation $1/z_o + 1/z_i = 1/f$ . Levoy’s slides call this relation the Gaussian lens formula; Torralba, Isola and Freeman’s Foundations of Computer Vision (hereafter visionbook) calls it the lensmaker’s formula, and ISETCam’s code the Lensmaker’s Equation.

The model connects the pinhole camera used throughout geometric vision to real cameras with glass lenses. It explains why a focused lens camera projects like a pinhole placed at the image distance rather than at the focal length, why refocusing changes the image scale, and why points away from the focused distance spread into blur discs whose size image formation uses to simulate defocus. This article states the model and its approximations, sets out the sign conventions, derives the focusing property and the blur circle from the ray-bending rule, and checks each result with an exact ray trace.

Definition

A thin lens is an ideal lens whose thickness is negligible compared with the other distances along the optical axis. It is characterized by one number, its focal length $f$ . The model makes two approximations, which visionbook states together (§6.2):

the paraxial approximation: rays make small angles with the optical axis, so that $\sin\theta \approx \theta$ . Levoy (slides 17–18) also takes $\cos\theta \approx 1$ and $\tan\theta \approx \sin\theta \approx \theta$ , and calls this first-order optics, because it keeps only the first term of the Taylor series of sine and cosine;
the thin-lens approximation: the ray is bent once, in the plane of the lens.

Two points on opposite sides of the lens, at distances $z_o$ and $z_i$ that satisfy the thin-lens equation, are conjugate points: every ray from one that passes through the lens goes through the other (visionbook §6.3). The lens images a plane at distance $z_o$ onto a plane at distance $z_i$ , both perpendicular to the axis.

Intuition

A pinhole forms a sharp image because only one ray from each scene point gets through. A lens lets in a whole cone of rays from each point, and its job is to send all of them back to one point. Visionbook (§6.2) shows what shape does this under the two approximations: the angle of the lens surface must grow linearly with the distance from the centre, which both spherical and parabolic surfaces satisfy for small angles. A ray through the centre meets two parallel surfaces and is not deviated; a ray near the rim meets the most steeply tilted glass and is bent the most.

That linear rule has three consequences that the rest of the article works out. The ray through the centre (the chief ray) travels straight, so where it lands is exactly where a pinhole at the lens centre would put the point. All the other rays land on the same spot when the sensor sits at the right distance, so the sharp image is a pinhole image. And when the sensor is at the wrong distance, the cone of rays from the point is cut before or after its apex, leaving a disc whose shape is the aperture and whose centre is still the chief ray’s hit.

Formal Definition

Conventions

This article puts the lens in the plane $z = 0$ , with light travelling towards $+z$ . Distances are positive when real: the object distance $z_o > 0$ in front of the lens and the image distance $z_i > 0$ behind it. Transverse coordinates are $(x, y)$ , with positive $y$ upward. This is the convention of Levoy’s slide 25 (“positive $s_i$ is rightward, positive $s_o$ is leftward”), whose $s_o$ and $s_i$ are this article’s $z_o$ and $z_i$ ; visionbook writes $a$ and $b$ . The aperture diameter is $d$ and the f-number $N = f/d$ , as in exposure and image formation (visionbook writes $A$ for the diameter). All lengths below are in millimetres.

The ray-bending rule

In the paraxial thin-lens model, a ray that reaches the lens at height $h$ with slope $u$ (its angle to the axis, in radians) leaves it with slope

u' = u - \frac{h}{f}.

The rule acts on $x$ and $y$ separately. A ray parallel to the axis ( $u = 0$ ) leaves with slope $-h/f$ and crosses the axis at distance $f$ behind the lens, which is the definition of the focal length; a ray through the centre ( $h = 0$ ) is not deviated. Both are visionbook’s ray-tracing rules (§6.3). The rule is our first-order summary of visionbook’s derivation, in which the surface angle at height $c$ is proportional to $c$ (its Eq. 6.1).

The thin-lens equation

Take an object point at transverse position $\mathbf{p}$ and distance $z_o$ . The ray that meets the lens at $\mathbf{a}$ has slope $(\mathbf{a} - \mathbf{p})/z_o$ before the lens, and by the rule $(\mathbf{a} - \mathbf{p})/z_o - \mathbf{a}/f$ after it. At distance $z$ behind the lens it is at

\mathbf{x}(z) = \mathbf{a}\left(1 + \frac{z}{z_o} - \frac{z}{f}\right) - \mathbf{p}\,\frac{z}{z_o}.

(This and the next two results are our derivation from the rule; the example below checks them.) The dependence on $\mathbf{a}$ , which ray of the cone it is, disappears when the bracket vanishes, that is when

\frac{1}{z_o} + \frac{1}{z_i} = \frac{1}{f}, \qquad z_i = \frac{f\,z_o}{z_o - f}.

This is the thin-lens equation. Levoy (slides 20–21) uses the same argument for a single refracting surface: the ray height cancels, so any ray from a point focuses to its conjugate. Visionbook (Eq. 6.3) obtains it from the lens shape, and Levoy (slide 27) also obtains it from similar triangles in Gauss’s ray construction.

Magnification and the pinhole equivalence

At $z = z_i$ every ray is at $\mathbf{x} = -\mathbf{p}\,z_i/z_o$ . The transverse magnification is therefore

m = \frac{y_i}{y_o} = -\frac{z_i}{z_o},

which is Levoy’s $M_T \triangleq y_i/y_o = -s_i/s_o$ (slide 41). The minus sign says the image is inverted. Visionbook (§6.3) writes the magnification of a lens as $a/b$ , object distance over image distance, which is the reciprocal of $|m|$ : it measures how much larger the object is than its image, so the two sources agree up to that inversion and the sign.

The focused image $-\mathbf{p}\,z_i/z_o$ is also where the chief ray, the straight line from $\mathbf{p}$ through the lens centre, meets the plane $z_i$ . A focused thin-lens camera is therefore exactly a pinhole camera whose centre is the lens centre and whose image plane is at $z_i$ , not at $f$ . Visionbook (§6.3) states the consequence, that lenses, like pinholes, render the world in perspective projection. Levoy (slide 44) notes that in a thin lens the chief ray passes through the optical centre without changing direction, and that in a thick lens the first of the two points where it crosses the axis, the first nodal point, is the centre of perspective.

The lens maker’s formula

The focal length follows from the glass. Visionbook derives, for a lens of refractive index $n$ in air with surface radii $R_1$ and $R_2$ ,

\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} + \frac{1}{R_2}\right),

with both radii counted positive for the convex lens it designs, and notes that some texts, such as Hecht’s Optics, give the back surface a negative curvature. Levoy’s slides 23–24, which follow Hecht, write $(n_l - 1)(1/R_1 - 1/R_2)$ in that convention. The two forms describe the same lens. The naming differs between sources: Levoy calls $1/s_o + 1/s_i = (n_l - 1)(1/R_1 - 1/R_2)$ the lensmaker’s formula and $1/s_o + 1/s_i = 1/f_i$ the Gaussian lens formula (Hecht’s eqn 5.17, by the slide’s attribution), while visionbook gives the name “Lensmaker’s Formula” to $1/a + 1/b = 1/f$ itself.

Properties

Focusing moves the sensor

A fixed lens focuses on different distances by changing $z_i$ , that is, by moving the sensor relative to the lens (Levoy, slide 28). As $z_o \to \infty$ , $z_i \to f$ ; as $z_o$ falls towards $f$ , $z_i$ grows without bound. At $z_o = z_i = 2f$ the magnification is $-1$ , which Levoy (slide 29) calls 1:1 imaging: a 36 mm wide object fills a 36 mm wide sensor. An object at $z_o = f$ sends parallel rays out of the lens, so $z_o = f$ is the minimum object distance a lens can focus (Levoy, slides 30–31; visionbook Fig. 6.9). Solving the equation for $z_o < f$ gives a negative $z_i$ , a virtual image on the object’s side of the lens that cannot be projected onto a sensor (our reading of the sign convention).

Refocusing changes the image size

Because the pinhole-equivalent image plane is at $z_i$ , the image scale is $z_i/z_o$ , not $f/z_o$ . Refocusing from infinity to a near object moves the sensor back and enlarges the whole image slightly; Levoy (slide 45) puts it as “as you refocus a camera, the image changes size”. In terms of the pinhole model the relative difference between the true image height and a pinhole at $f$ is $(z_i - f)/f = f/(z_o - f)$ (our derivation): about 5% for a 50 mm lens focused at 1 m, 1% at 5 m. In our reading, this is why the intrinsic matrix of a calibrated camera is valid only at the focus setting used for calibration (camera parameters), and why a design that sizes a lens from a working distance must use the thin-lens equation rather than the pinhole shortcut (vision system design).

A lens maps 3D space to 3D space

Every object distance has its own image distance, so a lens maps the volume in front of it onto a volume behind it, and the sensor takes one slice of that image volume (Levoy, slides 45–46). The mapping is not proportional in depth: as an object moves linearly along the axis, its image moves non-proportionately, and a linear motion of the lens relative to the sensor moves the in-focus plane non-linearly (slide 45). Differentiating $z_i = f z_o/(z_o - f)$ gives $dz_i/dz_o = -f^2/(z_o - f)^2 = -m^2$ (our derivation): depth is compressed by the square of the transverse magnification, so for distant objects a large range of object depths maps into a thin slab around $f$ .

Converging and diverging lenses

A lens with $f > 0$ (convex) makes parallel rays converge to a real focus behind it; with $f < 0$ (concave) parallel rays diverge as if from a virtual focus at distance $|f|$ in front of the lens (visionbook §6.3.2; Levoy, slides 32–34). The same equations hold with the sign of $f$ . Lens power is $P = 1/f$ , in dioptres when $f$ is in metres, and the powers of thin lenses in contact add (Levoy, slides 35–36). Levoy’s example is a +2 dioptre close-up filter on a 200 mm lens, which shortens the combined focal length to 143 mm and, at the same image distance, brings the in-focus object distance from 1000 mm to 334 mm (slides 38–39).

Defocus and the circle of confusion

Put the sensor at $s_f = z_i(z_f)$ , the image distance of a chosen focus distance $z_f$ , and consider a point at another distance $z$ , whose rays converge at $s = z_i(z)$ . Substituting $z_o = z$ into the ray position above, and using the thin-lens equation to write the bracket as $1 - s_f/s$ , gives the position on the sensor (our derivation):

\mathbf{x}(s_f) = \mathbf{a}\left(1 - \frac{s_f}{s}\right) + \frac{s_f}{s}\left(-\mathbf{p}\,\frac{s}{z}\right).

The second term is the chief ray’s hit. The first is the aperture point $\mathbf{a}$ scaled by $|1 - s_f/s|$ . The defocused image of a point is therefore a copy of the aperture, shrunk by that factor and centred on the chief ray: a uniform disc for a circular stop, of diameter

c = d\,\frac{|s - s_f|}{s} = \frac{f^2\,|z - z_f|}{N\,z\,(z_f - f)},

where the second form uses $d = f/N$ . This circle of confusion is the formula image formation gives as a derivation, and the first form, aperture times $|s - s_f|/s$ , is the similar-triangles construction in ISETCam’s opticsCoC.m header comments. Visionbook (§6.3.1) describes the circle of confusion as this blur circle at the sensor plane, and Levoy (slides 52–56) derives depth of field from its largest acceptable diameter $C$ . Three facts follow directly. The blur is proportional to the aperture diameter, that is, inversely proportional to $N$ , as exposure notes, so stopping down by one stop ( $N$ up by $\sqrt{2}$ ) shrinks it by $\sqrt{2}$ . Off-axis points blur by the same amount as on-axis points at the same depth, because only the scale factor depends on depth. And since the disc is a scaled aperture, a non-circular diaphragm gives a non-circular blur, which image deblurring describes as the defocus point spread function.

The blur is not symmetric about the focus plane. For points the same distance $\Delta$ in front of and behind $z_f$ , the formula gives

\frac{c(z_f - \Delta)}{c(z_f + \Delta)} = \frac{z_f + \Delta}{z_f - \Delta} > 1

(our derivation): a point in front of the focus plane is always blurred more than a point equally far behind it. Levoy (slide 53) states the consequence for depth of field, that it “is asymmetrical around the in-focus object plane”; the sharp range, its near and far limits and the hyperfocal distance follow from choosing a largest acceptable $c$ . Levoy (slide 53) also distinguishes this object-side range, the depth of field, from the image-side range of sensor positions that keep a point acceptably sharp, the depth of focus.

Examples

An exact paraxial ray trace

The script traces rays through an ideal thin lens with $f = 50$ mm at f/2 (aperture 25 mm). The aperture is sampled by 225 fixed points: a grid strictly inside the stop and 32 points exactly on its rim, built from rational coordinates, with the lens centre (the chief ray) as sample 0. There is no randomness. Each ray is bent by the rule $u' = u - h/f$ and nothing else, so the script checks the derivations above against a direct computation; the parameters were chosen to show the derived effects, and the example illustrates the derivation rather than testing an open question. It checks four statements, each with a tolerance of $10^{-12}$ relative error, far above double-precision rounding:

every ray from $z_o$ crosses the plane $z_i = f z_o/(z_o - f)$ at one point (spread relative to the aperture diameter);
that point is the pinhole projection with image plane at $z_i$ ; the magnification is $-z_i/z_o$ ; and the image differs from a pinhole at $f$ by the relative amount $f/(z_o - f)$ ;
with the sensor fixed at the focus for $z_f = 1000$ mm, the footprint of a point at another depth has the diameter $c$ given above, and its centroid lies on the chief ray’s hit;
points 100, 250 and 500 mm in front of the focus plane blur more than points equally far behind it, in the ratio $(z_f + \Delta)/(z_f - \Delta)$ .

The object points are on the axis, at $(0, 100)$ mm and at $(-80, 60)$ mm. The footprint diameter is twice the largest distance of a ray’s hit from the hits’ centroid, which is exact here because the rim samples lie exactly on the stop.

import os
os.environ["OPENBLAS_NUM_THREADS"] = os.environ["OMP_NUM_THREADS"] = "1"
import numpy as np

# Units: mm. Thin lens in the plane z = 0, light travels towards +z.
# Distances are positive when real (object in front, image behind the lens).
f, N = 50.0, 2.0
d = f / N                                   # aperture diameter, 25 mm
R = d / 2

# Aperture samples: a grid strictly inside the stop plus rational points on its rim
# (no randomness); index 0 is the lens centre, i.e. the chief ray.
pts = {(0, 0)}
pts |= {(i / 8, j / 8) for i in range(-7, 8) for j in range(-7, 8) if i * i + j * j < 64}
for k in range(-8, 9):
    t = k / 8
    x, y = (1 - t * t) / (1 + t * t), 2 * t / (1 + t * t)
    pts |= {(x, y), (-x, y)}
A = R * np.array(sorted(pts, key=lambda p: (p != (0, 0), p)))
r = np.hypot(A[:, 0], A[:, 1])
print(f"aperture samples {len(A)}, on the rim {np.sum(np.isclose(r, R))}, "
      f"max r/R {r.max() / R:.6f}, |sum x|/R {abs(A[:, 0].sum()) / R:.1e}, "
      f"|sum y|/R {abs(A[:, 1].sum()) / R:.1e}")

def trace(p, z_o, z):
    """Hits at plane z of the rays from object point p = (x, y) at distance z_o,
    one ray per aperture sample a: slope (a - p)/z_o, bent by -a/f at the lens."""
    u = (A - np.asarray(p)) / z_o
    return A + (u - A / f) * z

z_img = lambda z_o: f * z_o / (z_o - f)     # thin-lens equation solved for z_i
TOL = 1e-12
points = [(0.0, 0.0), (0.0, 100.0), (-80.0, 60.0)]

print("\n1-2. Focus plane z_i, against pinholes at z_i and at f")
print("  z_o    z_i     point          spread/d  err(z_i)/d  m        m err    "
      "(y-y_f)/y_f  f/(z_o-f)  mismatch")
ok1 = ok2 = ok2b = ok2c = True
for z_o in (100.0, 150.0, 1000.0, 5000.0):
    z_i = z_img(z_o)
    for p in points:
        H = trace(p, z_o, z_i)
        spread = np.abs(H - H[0]).max() / d          # every ray against the chief ray
        pin = -z_i / z_o * np.array(p)               # pinhole with image plane at z_i
        err = np.abs(H.mean(0) - pin).max() / d
        ok1 &= spread <= TOL; ok2 &= err <= TOL
        line = f"  {z_o:<6.0f} {z_i:<7.3f} {str(p):<14} {spread:.1e}   {err:.1e}"
        if p[1] != 0:
            m = H[0, 1] / p[1]
            m_err = abs(m + z_i / z_o) / (z_i / z_o)
            y_f = -f * p[1] / z_o                    # pinhole with image plane at f
            rel = (H[0, 1] - y_f) / y_f
            mis = abs(rel - f / (z_o - f)) / (f / (z_o - f))
            ok2b &= m_err <= TOL; ok2c &= (rel >= 1e-3) and (mis <= TOL)
            line += f"     {m:<8.5f} {m_err:.1e}  {rel:<11.6f}  {f / (z_o - f):<9.6f}  {mis:.1e}"
        print(line)
print(f"  H1 spread <= {TOL:g}: {ok1};  H2a pinhole at z_i <= {TOL:g}: {ok2};  "
      f"H2b magnification: {ok2b};  H2c not f: {ok2c}")

z_f = 1000.0
s_f = z_img(z_f)                            # sensor fixed at the focus for z_f
coc = lambda z: f**2 * abs(z - z_f) / (N * z * (z_f - f))   # Image Formation's c
print(f"\n3. Sensor fixed at s_f = {s_f:.4f} mm (focus for z_f = {z_f:.0f} mm)")
print("  z      point          diameter um  formula um   rel err  centroid-chief/d")
ok3 = ok3b = True
diam = {}
for z in (500.0, 750.0, 900.0, 1100.0, 1250.0, 1500.0):
    for p in (points[0], points[2]):
        H = trace(p, z, s_f)
        cen = H.mean(0)
        D = 2 * np.hypot(*(H - cen).T).max()
        rel = abs(D - coc(z)) / coc(z)
        off = np.hypot(*(cen - H[0])) / d
        ok3 &= rel <= TOL; ok3b &= off <= TOL
        diam[z, p] = D
        print(f"  {z:<6.0f} {str(p):<14} {D * 1e3:<12.3f} {coc(z) * 1e3:<12.3f} "
              f"{rel:.1e}  {off:.1e}")
print(f"  H3 diameter rel err <= {TOL:g}: {ok3};  H3b centroid on chief ray: {ok3b}")

print("\n4. Same distance in front of and behind the focus plane")
ok4 = True
for D_ in (100.0, 250.0, 500.0):
    for p in (points[0], points[2]):
        ratio = diam[z_f - D_, p] / diam[z_f + D_, p]
        pred = (z_f + D_) / (z_f - D_)
        rel = abs(ratio - pred) / pred
        ok4 &= (ratio > 1) and (rel <= TOL)
        print(f"  Delta {D_:<5.0f} {str(p):<14} c(front)/c(behind) {ratio:.6f}  "
              f"(z_f+Delta)/(z_f-Delta) {pred:.6f}  rel err {rel:.1e}")
print(f"  H4 front blur larger, ratio as derived: {ok4}")

Output (NumPy 2.3.5; the script runs in well under a second):

aperture samples 225, on the rim 32, max r/R 1.000000, |sum x|/R 0.0e+00, |sum y|/R 0.0e+00

1-2. Focus plane z_i, against pinholes at z_i and at f
  z_o    z_i     point          spread/d  err(z_i)/d  m        m err    (y-y_f)/y_f  f/(z_o-f)  mismatch
  100    100.000 (0.0, 0.0)     0.0e+00   0.0e+00
  100    100.000 (0.0, 100.0)   5.7e-16   0.0e+00     -1.00000 0.0e+00  1.000000     1.000000   0.0e+00
  100    100.000 (-80.0, 60.0)  5.7e-16   0.0e+00     -1.00000 0.0e+00  1.000000     1.000000   0.0e+00
  150    75.000  (0.0, 0.0)     7.1e-17   0.0e+00
  150    75.000  (0.0, 100.0)   2.8e-16   0.0e+00     -0.50000 0.0e+00  0.500000     0.500000   2.2e-16
  150    75.000  (-80.0, 60.0)  2.8e-16   0.0e+00     -0.50000 0.0e+00  0.500000     0.500000   0.0e+00
  1000   52.632  (0.0, 0.0)     7.1e-17   0.0e+00
  1000   52.632  (0.0, 100.0)   1.4e-16   6.8e-16     -0.05263 1.3e-16  0.052632     0.052632   1.5e-15
  1000   52.632  (-80.0, 60.0)  1.1e-16   5.5e-16     -0.05263 1.3e-16  0.052632     0.052632   2.5e-15
  5000   50.505  (0.0, 0.0)     7.1e-17   0.0e+00
  5000   50.505  (0.0, 100.0)   7.1e-17   0.0e+00     -0.01010 0.0e+00  0.010101     0.010101   6.4e-15
  5000   50.505  (-80.0, 60.0)  4.9e-17   1.2e-16     -0.01010 0.0e+00  0.010101     0.010101   6.4e-15
  H1 spread <= 1e-12: True;  H2a pinhole at z_i <= 1e-12: True;  H2b magnification: True;  H2c not f: True

3. Sensor fixed at s_f = 52.6316 mm (focus for z_f = 1000 mm)
  z      point          diameter um  formula um   rel err  centroid-chief/d
  500    (0.0, 0.0)     1315.789     1315.789     1.9e-15  0.0e+00
  500    (-80.0, 60.0)  1315.789     1315.789     7.9e-15  1.8e-16
  750    (0.0, 0.0)     438.596      438.596      1.1e-14  1.6e-19
  750    (-80.0, 60.0)  438.596      438.596      5.1e-14  3.6e-16
  900    (0.0, 0.0)     146.199      146.199      2.7e-14  7.9e-20
  900    (-80.0, 60.0)  146.199      146.199      1.1e-13  2.2e-16
  1100   (0.0, 0.0)     119.617      119.617      6.3e-15  7.9e-20
  1100   (-80.0, 60.0)  119.617      119.617      7.0e-14  1.4e-16
  1250   (0.0, 0.0)     263.158      263.158      2.5e-15  1.6e-19
  1250   (-80.0, 60.0)  263.158      263.158      3.9e-14  2.7e-16
  1500   (0.0, 0.0)     438.596      438.596      3.4e-15  3.2e-19
  1500   (-80.0, 60.0)  438.596      438.596      3.4e-14  3.1e-16
  H3 diameter rel err <= 1e-12: True;  H3b centroid on chief ray: True

4. Same distance in front of and behind the focus plane
  Delta 100   (0.0, 0.0)     c(front)/c(behind) 1.222222  (z_f+Delta)/(z_f-Delta) 1.222222  rel err 2.0e-14
  Delta 100   (-80.0, 60.0)  c(front)/c(behind) 1.222222  (z_f+Delta)/(z_f-Delta) 1.222222  rel err 4.1e-14
  Delta 250   (0.0, 0.0)     c(front)/c(behind) 1.666667  (z_f+Delta)/(z_f-Delta) 1.666667  rel err 8.7e-15
  Delta 250   (-80.0, 60.0)  c(front)/c(behind) 1.666667  (z_f+Delta)/(z_f-Delta) 1.666667  rel err 1.2e-14
  Delta 500   (0.0, 0.0)     c(front)/c(behind) 3.000000  (z_f+Delta)/(z_f-Delta) 3.000000  rel err 1.5e-15
  Delta 500   (-80.0, 60.0)  c(front)/c(behind) 3.000000  (z_f+Delta)/(z_f-Delta) 3.000000  rel err 2.6e-14
  H4 front blur larger, ratio as derived: True

The data check on the first line held: all rim points lie on the stop (largest radius exactly $R$ ) and the samples are symmetric about the centre.

Every check held. The single-point focus held: at $z_i$ all 225 rays from each of the twelve object positions coincide to within about $6 \times 10^{-16}$ of the aperture diameter, the size of double-precision rounding, so the ray height has indeed dropped out. The pinhole equivalence held: the focused point is the pinhole projection with image plane at $z_i$ , and the magnification is $-z_i/z_o$ in all eight off-axis cases. The “not $f$ ” comparison held as well: a pinhole at $f$ misplaces the image by exactly $f/(z_o - f)$ , which is 100% at $z_o = 2f$ (where $z_i = 2f$ , the 1:1 case), 5.3% at 1 m and 1.0% at 5 m.

With the sensor fixed for 1 m, the footprint check held: every footprint has the closed-form diameter, with the largest relative error $1.1 \times 10^{-13}$ , at 900 mm off axis. The centroid check held too: the off-axis point at $(-80, 60)$ mm is blurred by exactly as much as the on-axis point at the same depth, and its disc is centred on the chief ray. In our reading, this is what lets a defocused thin-lens image be modelled as the pinhole image convolved with a disc, valid wherever depth is locally constant. The asymmetry check held for both points: 100, 250 and 500 mm in front of the focus plane give 1.22, 1.67 and 3 times the blur of the same offsets behind it. The table also shows the same 438.596 µm disc at 750 mm and at 1500 mm, so in this set-up a point 250 mm in front of the focus plane is as blurred as one 500 mm behind it.

Process note: the plan for this example listed its departures from the batch plan before the pilot: an added check that each footprint’s centroid lies on the chief ray; the asymmetry measured as the blur ratio at equal offsets; the “not $f$ ” comparison as a printed measure; fixed rational sampling with no seed; and a printed data check. The plan did not name the points for the asymmetry check; the pilot computed it for both the on-axis and the off-axis point, and the editor accepted this before the first run. After the pilot, and before the first run, the only change to the code was a print-format repair approved by the editor: the width of the “point” column was increased from 12 to 14 characters so that the rows align. No value, measure or check changed. The tolerance of $10^{-12}$ was fixed before the pilot, as a margin for double-precision rounding over an exact result, using a rational-arithmetic version of the same trace written for planning.

Common Misconceptions

“The image plane of a camera is at the focal length.” Only for objects at infinity. For an object at $z_o$ the sensor sits at $z_i = f z_o/(z_o - f) > f$ , and projection scales with $z_i$ . The example shows a 5% scale error for a 50 mm lens focused at 1 m if $f$ is used.
“Out-of-focus blur is symmetric about the focus plane.” Equal distances in front of and behind the focus plane give different blur, larger in front, by the ratio $(z_f + \Delta)/(z_f - \Delta)$ .
“Defocus moves points.” In the thin-lens model the blur disc is centred on the chief ray, so defocus spreads a point without shifting its centre. Displacement without blur is lens distortion, a departure of real lenses from this model.
“Every source means the same formula by ‘lensmaker’s formula’.” Visionbook (and ISETCam, as Lensmaker’s Equation) uses the name for $1/z_o + 1/z_i = 1/f$ ; Levoy uses it for the relation of the object and image distances to the refractive index and the surface radii, and calls the former the Gaussian lens formula.
“Magnification is $z_o/z_i$ .” That is the factor by which the object is larger than its image, which visionbook calls the magnification of a lens; Levoy’s transverse magnification, used here, is its signed reciprocal $-z_i/z_o$ , and vision system design uses its magnitude $z_i/z_o$ .

Limits of the Model

The thin lens is an idealization at three levels.

Thickness. Real camera lenses are thick assemblies of many elements, which Levoy (slide 43) notes can still be characterized by a few numbers. The chief ray then enters and leaves through two nodal points, which for a lens in air coincide with the principal points, and the first nodal point is the centre of perspective (slide 44). Levoy (slide 31) warns that the object distance in the lens formula may be measured not to the front of the barrel but to a plane inside the assembly that one cannot see, and that many lenses cannot extend far enough to reach the theoretical minimum object distance.
Paraxial rays. Spherical surfaces do not bring all rays to a point; Levoy (slides 13–14) attributes this to spherical aberration and notes that nearly axial (paraxial) rays behave best, while visionbook (§6.2) notes that real lens surfaces are designed by numerical optimization. Spherical, chromatic and other aberrations, and their geometric cousin distortion, are summarized in image formation and modelled for calibration in lens distortion.
Geometric optics. The model ignores the wave nature of light (visionbook §6.1; Levoy, slide 8). A real aperture images a point as an Airy pattern of finite size, whose radius grows with the f-number; image formation gives the arithmetic and notes that closing the aperture, which shrinks the defocus disc, eventually blurs the whole image by diffraction.

Where It Is Used

Geometric vision uses the thin lens through its pinhole equivalence: camera calibration estimates the pinhole parameters of the focused lens, which is why they change with focus (camera parameters). Image formation uses the circle of confusion to simulate defocus and the thin-lens irradiance formula for exposure and vignetting, and exposure relates the f-number to the light reaching the sensor. Vision system design solves the thin-lens equation for the focal length that images a required field of view at a given working distance. Image deblurring inverts the defocus blur whose size and shape the model predicts. Depth from focus and depth from defocus, which depth estimation mentions, turn the dependence of the blur diameter on depth into a range measurement, and computational photography simulates shallow depth of field from estimated depth.