Machine Vision Lens Calculator

$H_s$ is the size of the sensor along the fitted side of the scene and $R_s$ its number of pixels along that side; the camera orientation decides whether this is the sensor’s long or short side. $\mathrm{FOV}$ is the size of the scene along that side: its height, $\mathrm{FOV}_y$, or its width, $\mathrm{FOV}_x$. $M$ is the magnification, the size of the image on the sensor divided by the size of the scene, and $R_w$ the length of the scene one pixel covers on the plane in focus. Along the other side the image covers $H_o / M$, where $H_o$ is the sensor’s other side.

• Pixels are taken as square.
• Their size is the sensor maker’s pixel size or, where the maker gives the sensor’s size, its width divided by the number of pixels across it; every other dimension follows from those two. For Other sensor, they are the values entered.
• The pixel counts are those a camera delivers by default.
• $R_w$ is the sampling of the scene by the pixels; the detail the image shows also depends on the sharpness of the lens, the focus and the lighting.

The distance $s$ runs from the sensor to the scene, and $F$ comes out in the same unit; $v$ is the distance from the lens to the sensor when the scene is in focus. The inputs must give a focal length between 4 and 600 mm and a magnification of at most 1.

• The calculation uses the thin-lens model: the lens is treated as a single plane, and the scene is assumed to be in focus on the sensor.
• A real lens is thick, with two principal planes some distance apart, so the camera position differs from the calculated one, most at short distances.
• Each lens has a minimum focusing distance.
• Prime lenses come in standard focal lengths; one slightly shorter than $F$ keeps the whole scene in the image.
• The lens must also cover the sensor’s format, and the lens maker’s data decides the final setup.

$p$ is the pixel size, and $d$ the object depth, centred on the plane in focus at $u$ from the lens; $z_n$ and $z_f$ are its near and far edges. $N$ is the f-number set on the lens and $\lambda$ the wavelength of the light. $b_d$, $b_m$ and $b_f$ are the diffraction, motion and defocus blur in pixels, and $B$ is the blur budget: the largest total blur $b$ a measured edge may show. With the focus in the middle, the near edge has more defocus blur, so it decides the f-number.

An f-number is usable when the total blur at the near edge stays within $B$. At $N_{\mathrm{best}}$, defocus and diffraction blur are equal and the total is lowest. When the scene moves, $v_m$ is the speed of the parts across the field of view and $t_e$ the exposure time; $t_{1\,\mathrm{px}}$ is the longest exposure that keeps motion blur within one pixel.

• Focused close, a lens works at a higher f-number than the one set on it, the working f-number $N(1+M)$, and diffraction is calculated with it.
• The best f-number is limited to f/1, since faster lenses are rare.
• The blur budget is a choice.
• Measuring diffraction to the first dark ring and adding the blurs as a root sum of squares are conventions.
• Motion blur lies along the direction of motion, so the sum overstates it for edges along that direction and understates it for edges across it.
• Motion blur is taken at the plane in focus; at the near edge it is larger by $u/z_n$.
• Rolling-shutter sensors also skew a moving part; moving scenes usually call for a global shutter or a short flash.

The depth of field is worked out at $N_{\mathrm{best}}$, or at f/1 when $N_{\mathrm{best}}$ is below it. $c$ is the part of the blur budget left for defocus after diffraction and motion; when nothing is left, the page says so. $H$ is the hyperfocal distance, and $D_n$ and $D_f$ the nearest and farthest distances in focus, measured from the lens; the page adds $v$ to give them from the sensor. Once $u$ reaches $H$, the far limit is infinite.

• $c$ is not the circle of confusion on lens scales, so the results differ from photographic depth-of-field calculators.
• The calculation uses the thin-lens model and works at the centre of the image; toward the corners real lenses blur more.
• Real lenses are thick, and the aperture can look a different size from the front and the back.
• Lens sharpness, focus accuracy and vibration add blur that the calculation does not include.
• The lens maker’s data and a test on the parts decide the final setting.

$\theta_x$ and $\theta_y$ are the angles off the optical axis at which the lens sees the left and right, and the top and bottom edges of the scene, and $\theta_c$ the angle at a corner; $u$ is the distance from the lens to the plane in focus. On a plane square to the camera at distance $z$ from the sensor, one pixel covers $R_w(z)$, which equals $R_w$ at the plane in focus.

The larger the angle, the further an edge that lies nearer or farther than the plane in focus shifts in the image. $\varepsilon$ is the resulting error in a measured length, as a fraction of that length, when the part lies $\Delta z$ farther than the plane in focus, or nearer when $\Delta z$ is negative; a farther part is measured short. Per millimetre of $\Delta z$, the error is $100\,\%/u$, with $u$ in millimetres, for any length in a plane square to the camera. At the edges of the object depth, $\Delta z = \pm d/2$, the error is $\mp 50\,\%\,d/u$.

• The camera is calibrated at the plane in focus.
• The lens is an ordinary one, whose view widens with distance; in a real lens, $u$ and the angles are measured from its entrance pupil.
• The error remains unless each part’s depth is measured and corrected, for example with a second camera or structured light.
• A telecentric lens, whose view does not widen, removes most of it.
• At large viewing angles the sides of three-dimensional parts also show at the edges of the scene.
• Lens distortion is not included; calibration measures and corrects it.