Spatial Correspondence is the degree of similarity between spatial relationships in one system and those represented, perceived, projected or reconstructed in another. It may concern similarity of spatial position and/or similarity of shape, Form and spatial structure.
In perspective, Spatial Correspondence is especially important when relating Object Space to Image Space: how closely does the position, geometry and organisation represented in a perspective image correspond to the spatial reality, model or scene with which it is associated?
Correspondence need not imply exact equality. A perspective image can preserve some spatial relationships accurately while transforming, reducing, concealing or making ambiguous others.
Correspondence Between Spatial Systems
Perspective continually establishes relationships between different spatial domains.
These may include:
- Physical Space and Object Space;
- Object Space and Image Space;
- Physical Form and Apparent Form;
- Physical Form and Modelled Form;
- represented space and perceived space;
- real and simulated spatial environments; and
- one perspective representation and another.
Spatial Correspondence concerns how closely the relevant spatial properties in one domain relate to those in another.
Correspondence of Position
One aspect of Spatial Correspondence concerns position.
An object possesses a location within a spatial scene. Its perspective image also possesses a corresponding location within an image or represented spatial structure.
Perspective establishes a mapping between these positions, but the coordinates or dimensions involved are not generally identical.
Important positional relationships can include:
- depth or distance from the viewpoint;
- vertical position or height;
- lateral position;
- angular direction;
- orientation; and
- position relative to other objects.
The Dictionary identifies distance, height and lateral location as three primary spatial factors involved in interpreting a perspective image.
Correspondence of Shape and Form
Spatial Correspondence can also concern the relationship between the shape, Form or structure of an object and its representation.
A Physical Form need not appear in an image with an identical shape. Perspective can produce foreshortening, apparent diminution, directional convergence and other changes of Apparent Form.
A circular physical surface may appear elliptical. A rectangular plane may appear foreshortened. Equal spatial lengths may appear unequal when positioned at different distances.
The perspective image can nevertheless correspond systematically to the Physical Form even though the two Forms do not geometrically coincide.
Correspondence Does Not Mean Identity
A central point is that correspondence is not necessarily identity.
A two-dimensional perspective image cannot simply reproduce every three-dimensional property of the spatial reality from which it was formed. Projection transforms spatial relationships as information passes from one spatial domain to another.
Some properties may be strongly preserved, while others may be transformed or lost.
Spatial Correspondence therefore asks not whether two spatial structures are literally identical, but which properties correspond, how closely they correspond and under what perspective conditions.
Object Space and Image Space
The distinction between Object Space and Image Space provides one of the clearest examples of Spatial Correspondence.
An object occupies a three-dimensional position and possesses a particular Physical Form in Object Space. Through a perspective process, information about that object is mapped into an image or view.
The resulting Image-Space structure may preserve important relationships concerning direction, outline, relative position and spatial organisation while transforming depth, apparent size, shape and other properties.
Thus:
Object-Space structure → Perspective Process → corresponding Image-Space structure
The two spatial structures are related, but they are not the same space.
Physical, Modelled and Apparent Form
Spatial Correspondence is closely connected with the relationship among the three Forms used in PRC analysis:
- Physical Form — the actual structure existing in spatial reality;
- Modelled Form — the geometrical approximation used to analyse that structure;
- Apparent Form — the perspective appearance produced under particular conditions.
The degree of correspondence among these Forms can vary.
A Modelled Form may approximate Physical Form very closely over one range of scale but less adequately over another. Apparent Form may resemble Physical Form closely under one viewing condition while differing substantially under another.
Spatial Correspondence therefore provides a useful way of describing these relationships without assuming exact equivalence.
The Correspondence or Equivalence Problem
Spatial Correspondence leads directly to the wider Correspondence or Equivalence Problem.
A single two-dimensional monocular image of a three-dimensional scene does not necessarily contain enough information to identify uniquely the Physical Form, depth, size, position or orientation of the spatial structure that produced it.
Different three-dimensional arrangements can sometimes produce the same or very similar Apparent Form.
The inverse relationship is therefore underdetermined:
one Apparent Form → several possible Physical Forms
This means that recovering spatial reality from a single perspective image is not simply the reverse of producing the image.
Why a Single Image Can Be Ambiguous
A perspective image may preserve some spatial information while simultaneously:
- reducing three dimensions to two;
- concealing depth;
- occluding surfaces;
- transforming apparent shape;
- changing apparent size;
- reducing fine detail;
- combining several spatial locations along the same viewing direction; or
- allowing several possible three-dimensional interpretations.
Consequently, Spatial Correspondence between an image and its source scene cannot always be established uniquely from image geometry alone.
Additional Information Improves Correspondence
The ambiguity of a single perspective image can often be reduced by adding further information.
Useful constraints can include:
- additional viewpoints;
- known viewpoint or Station Point;
- known Picture Plane geometry;
- metric grids;
- known dimensions;
- recognised objects;
- known Projection Mode;
- scale and resolution information;
- depth cues;
- scene context; and
- known geometrical constraints.
Such information can narrow the range of possible Physical Forms corresponding to a particular Apparent Form.
Multiple Views and Spatial Correspondence
Multiple views can provide substantially more spatial information than a single monocular image.
If an object is viewed or represented from several known directions, spatial relationships hidden or ambiguous in one view may become clearer in another.
This principle is used in technical drawing, photogrammetry, stereo imaging, computer vision and three-dimensional reconstruction.
Multiple views do not eliminate every possible problem of correspondence, but they can greatly constrain the range of spatial arrangements compatible with the available image information.
Spatial Correspondence and Perspective Accuracy
Spatial Correspondence also provides a way of discussing the accuracy or adequacy of a perspective representation.
A perspective image may correspond closely to the spatial scene in one respect while differing in another.
For example, a representation might preserve:
- directional relationships very accurately;
- relative positions approximately;
- shape only under a particular viewing condition; or
- scale only within a specified range or reference framework.
It is therefore often more meaningful to ask what kind and degree of correspondence is achieved than simply whether an image is correct or incorrect.
Spatial Correspondence and Projection Mode
Different Projection Modes preserve and transform different spatial relationships.
Central, parallel, spherical, cylindrical, anamorphic and other projection systems can map the same spatial scene into different image structures.
The resulting images may therefore possess different kinds of Spatial Correspondence with the same Object-Space reality.
A representation should consequently be evaluated according to the geometrical rules and purpose of its particular perspective system rather than against a single universal image geometry.
Spatial Correspondence and Scale
Correspondence also varies with scale and resolution.
A representation may correspond adequately to the large-scale structure of a physical object while omitting small-scale features. Increasing image scale or resolution may reveal details that were previously absent or visually insignificant.
The degree of correspondence therefore cannot always be evaluated independently of the scale at which the spatial structure is being viewed, represented or analysed.
This connects Spatial Correspondence with the wider Scale–Shape–Size and Shape-Sufficiency problems.
Representation and Reconstruction
Spatial Correspondence operates in two complementary directions.
Representation:
Spatial reality → perspective image or view
Here the problem is how the structure of Object Space is transformed and represented in Image Space.
Reconstruction or interpretation:
Perspective image or view → possible spatial reality
Here the problem is how reliably Image-Space information can be used to infer the Physical Form and spatial organisation that produced it.
Perspective therefore concerns both the production of correspondence and the interpretation of correspondence.
Why Spatial Correspondence Matters
Spatial Correspondence provides a general concept for analysing how faithfully or systematically one spatial structure relates to another.
It is important because perspective does not simply duplicate spatial reality. It maps, transforms and represents selected spatial relationships through particular viewing, optical, geometrical or technical conditions.
The concept therefore helps distinguish between:
- similarity and identity;
- Object Space and Image Space;
- Physical Form and Apparent Form;
- spatial reality and geometrical modelling;
- representation and reconstruction; and
- the information supplied by an image and the complete spatial reality that may have produced it.
Spatial Correspondence is consequently central to understanding both the power and the inherent limitations of perspective representation.