Perspective Calculation

A Perspective Calculation is a numerical, mathematical or geometrical result derived through a perspective process. As a Perspective Product, it expresses a spatial or image relationship in calculated form rather than only as a visible view, image or representation.

Perspective calculations may concern position, distance, direction, scale, size, proportion, projection, viewpoint, image coordinates or other relationships between Object Space and Image or Perspective Space.

Calculation as a Perspective Product

The Product analytical level asks what a perspective process has produced. In mathematical, graphical, computational and technical systems, one possible result is a calculation.

The calculation may be an intermediate result used to construct an image, or it may itself be the required final product—for example, a calculated spatial position, distance, scale, projection coordinate or geometrical relationship.

From Spatial Reality to Calculated Result

A perspective calculation establishes or reconstructs relationships between a spatial object or scene and its corresponding perspective representation.

The input may consist of known geometry, measured image quantities, viewpoint information, camera or optical parameters, scale information or coordinates. Mathematical or geometrical operations are then used to derive another quantity or spatial relationship.

The calculated result can subsequently be used to construct, compare, measure, model or interpret a perspective view or image.

Geometrical Perspective Calculation

Many classical perspective constructions are geometrical calculations expressed graphically. Lines, intersections, ratios and projected positions are used to determine where spatial features should appear within an image.

In central perspective, selected object points may be related to a fixed centre of projection and picture plane. Intersections between projectors and the picture plane determine corresponding image positions.

Vanishing points, measuring points, distance points and proportional constructions can similarly be used to establish calculated spatial relationships within a graphical perspective construction.

Mathematical Perspective Calculation

Mathematical Perspective expresses perspective relationships through numbers, equations, coordinates, ratios or transformations.

A mathematical perspective calculation may determine:

the projected position of a spatial point;
the image size associated with a specified depth;
scale or magnification;
viewing or projection angles;
the position of a vanishing point or line;
the relationship between Image-Space and Object-Space coordinates;
camera or viewpoint parameters;
distances or dimensions reconstructed from image information.

Perspective Projection Calculations

Projection is one of the most important areas of perspective calculation. A projection calculation determines how spatial points, lines or surfaces correspond to positions within an image or projected representation.

In a central projection system, the calculation must consistently relate the spatial geometry, centre of projection and image plane. Parallel spatial directions should then produce the vanishing behaviour required by the chosen projection geometry.

Computer graphics can perform the same general operation numerically, transforming three-dimensional coordinates into image coordinates according to defined viewing and projection parameters.

Calculation and Measurement

Calculation and Measurement are closely connected but are not identical.

Measurement obtains or assigns a quantified value to a spatial or image property.
Calculation derives a further result by applying mathematical or geometrical relationships to known, measured or assumed quantities.

For example, a length may first be measured within a calibrated image; that measurement can then enter a perspective calculation used to estimate an Object-Space distance or size.

Forward and Inverse Calculation

Perspective calculations can operate in two broad directions.

Forward calculation begins with spatial geometry and viewing or projection conditions and determines the resulting image or perspective relationships.
Inverse calculation begins with an image, view or set of measurements and attempts to infer properties of the spatial reality or perspective system that produced them.

Inverse calculation is generally less straightforward because a single perspective image may not uniquely determine every property of the original three-dimensional arrangement. Additional measurements, assumptions, calibration or multiple views may be required.

Graphical Calculation

A perspective calculation does not always need to appear as an equation. Graphical construction can itself perform a calculation through geometrically defined lines, intersections and proportional relationships.

Classical perspective drawing contains many such procedures. A constructed intersection can determine a projected position just as a numerical equation could determine the corresponding coordinate.

This helps explain why Graphical Perspective and Mathematical Perspective can overlap: one may express geometrical relationships through construction while the other expresses comparable relationships numerically or algebraically.

Computational Perspective

Digital perspective systems can perform very large numbers of perspective calculations rapidly. Computer graphics, computer vision, photogrammetry, CAD and other computational systems may calculate transformations between three-dimensional spatial models and two-dimensional or three-dimensional Image Spaces.

These calculations can support image generation, spatial reconstruction, measurement, modelling, matching, tracking and the extraction of spatial information from images.

Calculation Depends on the Perspective System

A calculation is meaningful only in relation to the assumptions and conditions of the perspective system being used. Different projection methods can produce different relationships between spatial dimensions and image dimensions.

Central, parallel, orthographic, cylindrical, spherical and other perspective systems therefore require different geometrical or mathematical relationships. A calculation appropriate to one projection system cannot automatically be transferred unchanged to another.

Examples of Perspective Calculations

calculating the projected position of an object point;
calculating diminution with distance;
determining image scale or magnification;
calculating a vanishing point from a spatial direction;
calculating a perspective projection matrix;
deriving object dimensions from calibrated image measurements;
calculating camera or viewpoint position;
transforming three-dimensional coordinates into image coordinates;
reconstructing spatial position from multiple perspective views;
calculating scale, distance, direction or angle from perspective data.

Calculation, Data and Information

A calculation frequently forms a bridge between a measured perspective relationship and the information derived from it.

A Perspective Image may provide measurable quantities; calculations transform or combine those quantities; and the resulting values can become Data / Information about the object, scene, image or perspective system.

This sequence is not compulsory. A perspective process may produce a calculation directly, and one process can produce several different Products at the same time.

Calculation Within the Analytical System

Within the PRC analytical framework, Calculation is treated here at the Product level because a calculated result can be produced by a perspective process.

The method used to obtain it may belong to a mathematical, graphical, optical, instrument or New Media Category; the process may operate through a particular Class and Type; and the calculated result may support Functions such as matching, measuring, modelling, representing or interpreting spatial reality.

The Product question is: what has the perspective process produced? In this case, a calculation.