Volume estimation from two silhouettes: method and limits

Two photographs can support a useful volume estimate, but they do not uniquely determine a three-dimensional shape. The practical approach is to calibrate each silhouette, align both views to shared physical heights, and integrate an assumed cross-section at every height. This is useful for opaque parts, shape studies and controlled measurement experiments.

For human subjects, posture, clothing, movement and hidden body contours add substantial uncertainty. You will need to separate calibration error from the larger question: does an ellipse actually represent the material inside each measured outline? Start with a stationary reference object before interpreting measurements of a person.

1. Decide whether the cross-section model fits

The simplest model assigns an ellipse to each horizontal slice. Its width comes from the front silhouette, while its depth comes from the side silhouette. This assumes those dimensions describe the same cross-section along perpendicular, aligned axes.

That assumption works best for a single, approximately elliptical section. It becomes unreliable when an outline contains several disconnected regions, deep indentations or overlapping components.

A person's arms and torso illustrate the problem. Measuring the outermost front edges can include empty space between the arms and trunk. Multiplying that width by the overall side depth fills space that contains no body material.

You can model separate regions only when their correspondence between views is known. Two segmented intervals in one image do not automatically identify the same physical components in the other image.

Define the measurement before building the algorithm:

• Is the result an external envelope or an estimate of material volume?
• Which regions are included, and how are overlapping regions handled?
• Is the cross-section assumed elliptical everywhere or only within selected regions?
• What reference measurement will establish acceptable error?

Treat the ellipse assumption as a model requiring validation, not as a consequence of having two images.

2. Establish a common pose and height coordinate

Mount the cameras with approximately perpendicular viewing directions and a shared vertical reference. Their optical axes should be approximately horizontal if image rows will represent horizontal slices.

For each view, establish a physical height origin and vertical direction. A calibrated target with known height marks can define these references. The article on calibration target coverage and margins explains why references must cover the measurement region rather than occupy one small central patch.

Matching image row numbers is not sufficient. Different resolutions, cropping and camera heights mean that row 500 can represent different physical heights in the two images.

Instead, define a shared height grid, such as increments of 5 mm. Convert each height into the corresponding image location in each calibrated view. Interpolate the contour intersections at that location.

For moving subjects, capture both views as close together as the verified acquisition timing permits. Separate sessions introduce changes in stance, breathing and orientation. Image registration cannot recover an outline hidden by a different pose.

A planar transformation can align a reference plane, but it cannot remove perspective differences throughout a thick object. For coordinate-frame checks, see extrinsic calibration accuracy beyond reprojection error.

3. Convert pixels into physical dimensions

First correct lens distortion using calibration appropriate to the actual focus, resolution and image processing settings. Then establish horizontal and vertical scale for each view.

A simple scale conversion is valid only when perspective variation is negligible or bounded within your error budget. The reference must be at a representative measurement plane, not against a distant background.

Suppose a vertical reference spans 400 mm and measures 800 pixels in the front image. Its vertical scale is:

If the same reference measures 1,000 pixels in the side image, that view has a vertical scale of 0.40 mm/pixel. Both images can still be sampled at identical physical heights.

Measure horizontal scale separately, or establish from calibration that the applicable horizontal and vertical scales agree. Resizing an image differently along its axes invalidates a previously shared scale.

The image-marker calibration guide covers the distinction between image distances and physical lengths. Use that calibration discipline before adding volume integration.

4. Segment the silhouette before refining its edges

For an opaque bench target, controlled backlighting usually gives a more useful starting image than an elaborate edge detector. The objective is a stable foreground mask, not every visible texture boundary.

A practical processing sequence is:

1. Capture a background reference without the target.
2. Check illumination uniformity and avoid saturation near the silhouette boundary.
3. Segment foreground using a validated intensity threshold or background-difference method.
4. Identify connected regions and retain those belonging to the measurement target.
5. Locate left and right boundary intersections at each shared height.
6. Flag ambiguous or missing sections rather than silently interpolating through them.

Inspect the mask overlaid on the original image. Small gaps, reflections and translucent edges can otherwise remain invisible in the final volume number.

Morphological closing may repair small segmentation defects, but it can also fill real gaps between components. Set its scale in physical units and verify its effect on known features.

For example, at 0.50 mm/pixel, a 4-pixel boundary displacement represents 2 mm. That may be tolerable for one dimension and unacceptable for another.

Clothing and hair become part of an optical outline. A silhouette algorithm cannot infer the underlying body contour simply by producing a smoother edge.

5. Integrate shared slices with an explicit formula

For an axis-aligned elliptical cross-section, width and depth are full diameters. The area is therefore:

Using full diameters without dividing by four overestimates the area by a factor of four. Keep the distinction between radii and diameters explicit in variable names.

For equally spaced samples, trapezoidal integration gives:

Consider a synthetic segment extending from 0 to 400 mm. These dimensions describe only that segment, not an entire body or a catalog product.

At the middle slice, the area calculation is:

Keeping the unrounded areas, the integrated segment volume is:

The nonzero end areas mean this is a truncated segment. Any volume beyond its end planes is excluded.

Reduce slice spacing until further reduction changes the result negligibly relative to your tolerance. Numerical convergence only validates the integration spacing; it does not validate the assumed cross-section shape.

6. Budget perspective, boundary and model errors separately

For small, systematic dimensional errors, a useful first-order estimate is:

If each dimension is biased high by 1%, the resulting volume bias is approximately 3%. Do not assume averaging many slices removes calibration or segmentation bias.

Perspective produces another systematic effect. Under a pinhole approximation, the magnitude of relative scale variation is approximately the depth displacement divided by camera distance:

At a nominal distance of 2,000 mm, a 30 mm displacement changes scale by roughly 1.5%. If both measured diameters carry that bias, the cross-section area error is approximately 3%, before other contributions.

Moving the camera farther away can reduce relative perspective variation, but requires suitable optics, space and adequate image sampling. A calibrated perspective model helps describe the error; it does not make unknown surface depths disappear.

Model error can be larger than either effect. A rectangle and an ellipse can have identical front and side extents. For the same width and depth:

The rectangular area is about 27% larger, despite matching those two measured widths. General cross-sections permit still more ambiguity.

Report separate contributions for scale, pose repeatability, segmentation, integration spacing and shape assumptions. A single precise-looking decimal hides these distinctions.

7. Distinguish silhouette integration from stereo reconstruction

Having two cameras does not by itself make a measurement stereo triangulation. The distinguishing feature is how the images constrain shape.

The ellipse method imposes a rounded cross-section rather than reconstructing surface points. Silhouette intersection uses a different geometric assumption and need not produce the same volume.

If your task requires surface depth rather than an outline-based approximation, start with paired-image stereo calibration. That workflow addresses a different measurement problem, not a more accurate implementation of the ellipse formula.

8. Select equipment for a stationary validation bench

Choose area scan cameras by required contour sampling, exposure behavior and acquisition compatibility. The camera finder helps narrow candidates, but interface compatibility alone does not establish synchronized measurement.

The LS-CMR-0053 and LS-CMR-0166 are candidates to evaluate for a stationary bench. Verify acquisition controls and timing in the supplied documentation before planning simultaneous capture.

All 2 share: Sensor size 1/1.7". Values as printed in each product's data sheet; open a part for its drawing and full specifications.

Both candidates use rolling shutters. Keep the reference target stationary unless motion and row-dependent exposure timing have been characterized.

For a hypothetical 600 mm field width, assigning 0.50 mm to each pixel requires:

That is a sampling requirement, not a guarantee of dimensional accuracy. The guide to resolution for dimensional inspection explains the additional optical and edge-location constraints.

Select machine vision lenses for sensor coverage, working distance and edge definition across the complete field. Use the lens finder to narrow the geometry. The LS-FFL-0567 is a candidate for optical evaluation, not a confirmed match to an unspecified installation.

For small opaque references, compare backlights using the lighting finder. The LS-BKL-0750 and LS-BKL-0830 offer different illumination approaches for bench testing.

The comparison of collimated and diffuse backlighting explains why illumination direction can change measured edges. These examples do not establish full-body coverage or suitability for exposing people. Assess optical exposure, electrical safety and mounting risks separately.

9. Do not treat external volume as measured body mass

Mass requires density as well as volume:

Two silhouettes provide no direct measurement of tissue composition or internal air spaces. Clothing can further separate the external optical envelope from the intended body volume.

For illustration only, an estimated volume of 70 L multiplied by an assumed density of 1.00 kg/L gives 70 kg. A different assumption of 0.95 kg/L gives 66.5 kg from exactly the same images. Neither density value is established by the silhouettes.

Use a suitable weighing instrument when mass is the required quantity. If studying a population-based estimator, validate it against independent measurements and report prediction uncertainty. Do not present this geometric method as a medical measurement or use its output alone for health decisions.

10. Verify the complete method before interpreting results

Start with independently characterized reference objects, including shapes that challenge the ellipse assumption. A cylinder alone cannot validate a method intended for irregular, separated or concave sections.

Use this acceptance checklist:

• Scale: verify dimensions at the center and edges of both measurement fields.
• Depth sensitivity: move a reference through the expected depth range and record dimensional bias.
• Height matching: confirm that a shared physical mark is sampled at the same height in both views.
• Segmentation: vary the threshold within a justified range and record the volume change.
• Repeatability: remove and replace the target at least 10 times, reporting spread and mean error separately.
• Slice spacing: halve the interval and compare the integrated result.
• Shape validity: include non-elliptical references to expose model bias.
• Traceability: retain raw images, masks, calibration settings, rejected slices and calculation units.

Set acceptance limits before reviewing the final results. If pose or cross-section assumptions dominate, additional pixels are unlikely to solve the problem.

For equipment selection, bring your field dimensions, depth range, smallest required contour change and validation tolerance to the camera configuration step, or request application support before ordering.