The Equivalence / Correspondence Problem is a fundamental problem of perspective concerning the relationship between a three-dimensional object or spatial scene and its two-dimensional perspective image.
A single monocular 2-D image of a 3-D object does not normally contain enough information to determine unambiguously the original object’s shape, size, position, depth, orientation or complete spatial geometry. Different three-dimensional objects or scenes can potentially produce the same, or a sufficiently similar, two-dimensional projected image.
The central problem can therefore be expressed simply as:
Different 3-D Objects / Scenes → Same or Equivalent 2-D Perspective Image.
Consequently, the reverse relationship is uncertain:
Single 2-D Perspective Image → More Than One Possible 3-D Spatial Reality.
The problem lies at the heart of Visual Perspective, Optical Perspective, Linear Perspective, Photography, Computer Vision, 3D Reconstruction and the wider theory of perspective. It explains why interpreting an image of spatial reality requires more than simply tracing projection rays backwards from the image.
What Is the Equivalence / Correspondence Problem?
The Dictionary of Perspective defines the Equivalence / Correspondence Problem as a problem of monocular perspective.
When a three-dimensional object is projected onto a two-dimensional image surface, some information about the original spatial structure is necessarily transformed, compressed, hidden or lost.
The resulting image may accurately record a particular projection of the object, but that projection does not necessarily contain sufficient information to determine uniquely:
- the object’s true three-dimensional shape;
- its physical size;
- its distance from the observer or camera;
- its lateral and vertical position;
- its orientation;
- the depth of its individual parts;
- the dimensions hidden from the viewpoint; or
- the complete geometry of the surrounding spatial scene.
The correspondence between object and image is therefore not automatically one-to-one.
Object–Image Equivalence
Volume 1 describes the problem specifically in terms of object/image equivalence.
A single image form may correspond to several possible three-dimensional arrangements.
Thus:
Object A → Image X
Object B → Image X
Object C → Image X
If only Image X is available, the original three-dimensional source cannot necessarily be identified uniquely.
Correspondence therefore concerns the relationship between:
Object Space ↔ Perspective Image Space.
The projection from object to image may be well defined, while reconstruction from image back to object remains ambiguous.
Why a 2-D Image Does Not Uniquely Determine 3-D Space
A two-dimensional perspective image records selected spatial relationships as they appear from a particular viewpoint.
But three-dimensional object space contains an additional depth dimension.
The transformation can be represented as:
3-D Object Space → Perspective Projection → 2-D Image Space.
When the image is formed, different combinations of physical size, distance, orientation and shape may produce equivalent projected relationships.
The observer therefore faces an inverse problem:
How can the three-dimensional source be reconstructed from a two-dimensional projection when several different spatial arrangements may explain the same image?
This is the central Equivalence / Correspondence Problem.
Projection Is Not the Same as Reconstruction
The distinction between projection and reconstruction is fundamental.
If the object geometry, viewpoint and projection system are known, a perspective image can be generated from them.
The forward process is therefore:
Known 3-D Geometry + Known Projection → 2-D Image.
The reverse process is more difficult:
2-D Image → ? → Original 3-D Geometry.
Without additional information, the image does not normally specify one unique solution.
Perspective reconstruction therefore requires constraints, contextual information, recognised structures, known geometrical relationships, additional views or other spatial evidence.
The Problem of Shape
The projected shape visible in an image is not necessarily the object’s absolute or geometrical shape in object space.
Apparent shape depends upon factors including:
- viewpoint;
- viewing direction;
- object orientation;
- projection method;
- foreshortening;
- occlusion;
- projection scale; and
- projection-scale resolution.
A three-dimensional form may therefore present many different two-dimensional apparent shapes when viewed from different positions.
Conversely, one particular two-dimensional outline may be compatible with more than one three-dimensional object.
This makes shape recognition a central part of the correspondence problem.
The Problem of Size
Projected image size does not by itself reveal physical object size.
A small nearby object and a much larger distant object can subtend the same visual angle and produce the same projected image size.
Therefore:
Same Image Size ≠ Necessarily Same Physical Size.
To infer physical size, some additional information about distance, scale, recognised object dimensions or spatial context is required.
The correspondence problem consequently overlaps with the wider problems of size, distance, projection scale and spatial measurement.
The Problem of Distance
A monocular perspective image does not normally contain an explicit numerical label specifying the absolute distance of every object from the observer.
Distance must instead be estimated or reconstructed from spatial relationships and other available information.
Such information may include:
- diminution of familiar objects;
- overlap and occlusion;
- height in the visual field;
- texture gradients;
- perspective convergence;
- known metric structures;
- atmospheric effects;
- shadows and illumination;
- movement and optic flow; and
- binocular or multi-view information where available.
These factors help constrain the possible spatial interpretation but do not change the fundamental fact that a single monocular projection is underdetermined.
The Problem of Orientation
An object’s orientation strongly affects its projected shape.
A square facing the observer directly may appear square, while the same square rotated in depth can appear compressed, trapezoidal or otherwise foreshortened.
The image records the apparent form produced by the particular projection angle, rather than automatically revealing the object’s original orientation.
The observer must therefore distinguish between:
- change in true geometrical shape; and
- change in apparent shape caused by viewpoint and orientation.
This is one of the central tasks involved in decoding a perspective image.
Perspective of Form
The Equivalence / Correspondence Problem is closely related to the Perspective of Form.
Perspective projection can transform apparent:
- shape;
- size;
- angle;
- position;
- proportion;
- orientation; and
- visible surface structure.
To interpret the image correctly, these projected transformations must somehow be related back to the object’s possible true or original geometry.
This can be represented as:
True / Absolute Form → Perspective Transformation → Apparent Form → Interpretation → Estimated True Form.
The final step is not automatically determined by the image alone.
Perspective as Object–Image Correspondence
The wider theory of perspective can itself be understood partly through the principle of object–image correspondence.
A perspective image bears some relationship to a spatial object, scene, model or imagined reality.
However, the Dictionary of Perspective notes that this relationship can take several forms, including:
- theoretical correspondence;
- assumed correspondence;
- possible correspondence;
- transposed correspondence; and
- no direct correspondence.
Perspective should therefore not always be understood as a simple one-to-one copying relationship between an image and a physical object.
The nature and degree of correspondence depend upon the particular perspective method, process, system and outcome.
The Problem of Space
The Dictionary also relates the Correspondence / Equivalence Problem to the wider Problem of Space.
Space itself is not directly visible as an independent object. We see objects, surfaces, boundaries, colours, textures, light, shadows and changes distributed within spatial reality.
To understand spatial organisation, these visible elements must be structured and interpreted.
Perspective therefore employs known:
- points;
- lines;
- planes;
- axes;
- horizons;
- vanishing relationships;
- grids;
- scales; and
- geometrical frameworks.
These structures help us segment, order, index, measure and gauge physical space and thereby reduce the ambiguity inherent in the perspective image.
Metric Grids and Perspective Frameworks
One of the principal methods identified in Volume 1 for addressing the Correspondence Problem is the use of a Perspective Framework.
A framework may contain regular physical or geometrical structures such as:
- a known ground plane;
- a metric grid;
- sets of parallel lines;
- orthogonal lines;
- upright planes;
- recognisable geometrical solids;
- known dimensions; and
- established horizon and vanishing relationships.
These known structures provide reference information against which projected changes of shape, size and position can be interpreted.
Thus:
Ambiguous Perspective Image + Known Geometrical Framework → More Constrained Spatial Interpretation.
The framework does not abolish perspective transformation. It provides information that helps the observer decode it.
Linear Perspective as a System for Encoding and Decoding Space
Volume 1 gives Linear Perspective an important role in relation to the correspondence problem.
Linear Perspective does more than create the appearance of depth. Its regular geometrical structure can provide a means for both encoding and decoding spatial relationships.
If sufficient information is known about:
- the viewpoint;
- picture-plane geometry;
- ground plane;
- parallel directions;
- vanishing points;
- metric grid;
- object dimensions; and
- projection scale;
then considerably more can be inferred about the three-dimensional space represented by the two-dimensional image.
In this sense, Linear Perspective can operate as a quantitative construction and analytical framework for spatial reality.
Regular Perspective
Regular Perspective provides recognisable structures that make image interpretation easier.
A spatial scene containing regular forms such as:
- squares;
- rectangles;
- cubes;
- parallel walls;
- regular floors;
- repeated architectural elements; and
- metric grids
provides geometrical information that can help reveal the direction, orientation and spatial organisation of the scene.
Much architecture and technical representation contains precisely these kinds of framework structures.
This helps explain why a regular architectural scene can often be easier to interpret geometrically than an irregular natural object.
Irregular Perspective
Irregular Perspective makes the Equivalence / Correspondence Problem especially apparent.
Natural forms such as:
- flowers;
- plants;
- trees;
- rocks;
- clouds;
- irregular terrain; and
- other complex organic forms
may contain few obvious metric grids, parallel-line systems or familiar geometrical frameworks.
A flower petal, for example, may possess almost any apparent projected shape depending upon its actual form and orientation.
The viewer cannot simply assume that an unfamiliar curved outline corresponds to one particular three-dimensional shape.
This is why irregular forms can be particularly difficult both to draw convincingly and to reconstruct mentally from a 2-D image.
Recognised Objects and Prior Knowledge
Another way of constraining image ambiguity is through knowledge of the objects being viewed.
If an observer recognises an object and already possesses information about its normal:
- shape;
- size;
- proportions;
- orientation;
- function; or
- relationship to surrounding objects,
then the projected image becomes easier to interpret.
Recognition therefore introduces information that is not contained solely in the isolated geometry of the image itself.
The interpretation becomes a combination of projected image information and contextual knowledge.
Depth Cues and the Correspondence Problem
The human visual and perceptual system uses numerous Depth Cues to help overcome the Equivalence / Correspondence Problem.
These can provide information about apparent spatial organisation, including:
- relative size;
- overlap and occlusion;
- texture gradients;
- height within the visual field;
- aerial or atmospheric perspective;
- light and shade;
- known image form;
- motion parallax;
- optic flow;
- focus-related information;
- binocular disparity; and
- other contextual and environmental information.
No one cue necessarily provides a complete solution. Spatial perception emerges from the combination and interpretation of available information.
Visual Perspective Type 2
The Correspondence Problem is therefore directly relevant to Visual Perspective Type 2.
The retinal image is itself a perspective projection of spatial reality.
Human perception must interpret changing retinal information in order to estimate:
- object identity;
- size;
- shape;
- position;
- distance;
- orientation;
- depth; and
- spatial relationships.
The simple geometrical projection onto the retina therefore does not by itself explain the full capabilities of visual perception.
Perception involves the interpretation of the available image together with other geometrical, optical, physiological and psychological information.
Apparent Shape and Shape Recognition
The Dictionary of Perspective introduces Shape Grammars as one means of addressing the wider problem of interpreting viewpoint-dependent shapes.
The aim is to understand systematically how apparent shape changes with:
- viewing position;
- object aspect;
- true scale;
- projection scale; and
- other perspective transformations.
The basic problem is that the same object produces different apparent shapes under different projection conditions.
A sufficiently developed system of shape recognition should therefore be capable of relating:
Changing Apparent Forms → Possible Underlying Object Form.
This is relevant not only to human vision but also to robotic vision and artificial intelligence.
A Single View Is Partial
A single perspective view provides only a partial description of a three-dimensional object or scene.
Some surfaces may be visible while others are hidden.
Some dimensions may lie largely across the image plane while others extend strongly in depth.
Some geometrical relationships may be apparent while others remain ambiguous.
Thus:
One Viewpoint → One Partial Projection of 3-D Reality.
This is one reason why changing viewpoint can reveal additional structural information.
Multi-View Perspective as a Solution
Volume 1 states that certain forms of Multi-View Perspective, including multi-view parallel perspective, can provide a solution to aspects of the Correspondence Problem.
Instead of relying upon one image, the object is examined from several viewpoints.
The principle is:
View 1 + View 2 + View 3 + Additional Constraints → Better Estimate of 3-D Object Geometry.
Each viewpoint can reveal:
- previously hidden surfaces;
- new object aspects;
- different projected dimensions;
- changes in overlap;
- parallax;
- additional shape information; and
- new relationships between spatial features.
The additional views reduce the number of plausible three-dimensional interpretations.
However, multiple views do not mean that every reconstruction problem automatically becomes exact or complete.
Multi-View Perspective and 3D Reconstruction
The same principle extends into modern 3D Reconstruction.
Several projected views of an object can be aligned and combined so that a three-dimensional model is inferred from their common and changing features.
The general relationship is:
Multiple 2-D Projections + Viewpoint Information + Constraints → Estimated 3-D Model.
Volume 1 gives electron cryo-tomography as an example in which images obtained from different viewing angles are computationally reconstructed into a three-dimensional model.
A single projection does not normally contain enough information to determine complete 3-D form, whereas several projected aspects provide a more complete basis for reconstruction.
The resulting model nevertheless remains dependent upon factors such as viewing coverage, resolution, image quality and the reconstruction method.
The Problem of Viewpoint
The Correspondence Problem is closely connected with the Problem of Viewpoint.
Each viewpoint produces a different perspective projection and therefore provides different information about the object.
Volume 1 emphasises that each view contains unique but partial information about three-dimensional spatial reality.
To understand an object more completely, perspectives from different positions can therefore be combined.
But a new difficulty then appears:
How should many different partial views be accurately registered and integrated into one coherent model?
Thus Multi-View Perspective reduces one problem while introducing the additional problem of viewpoint integration.
Parallel Perspective Does Not Eliminate the Problem Completely
Parallel projection can remove some of the transformations associated with central perspective, particularly diminution with depth.
Orthographic and other parallel views can therefore be particularly useful for analysing and representing three-dimensional form.
However, Volume 1 points out that even parallel perspective remains dependent upon viewpoint and orientation.
An inclined or hidden feature may still appear foreshortened or may not be visible at all.
Several coordinated orthographic views may therefore be required to define a three-dimensional object adequately.
The important distinction is:
Parallel Perspective can reduce some correspondence ambiguities without making every single view a complete description of 3-D reality.
The Correspondence Problem and Photography
A photograph provides a particularly familiar example of the Correspondence Problem.
A photographic image may appear highly realistic because it was formed optically from a physical scene.
Yet photographic realism does not mean that every three-dimensional property of the original scene can be uniquely recovered from the photograph.
The image still represents the scene:
- from a selected camera position;
- along a selected viewing direction;
- through a particular optical system;
- within a particular field of view;
- at a particular projection scale;
- with limited resolution; and
- with hidden regions excluded by occlusion.
A photograph is therefore evidence of spatial reality, but it is not automatically a complete geometrical specification of that reality.
The Correspondence Problem and Linear Perspective Drawing
A Linear Perspective drawing can reproduce many of the geometrical transformations characteristic of a central view.
But once the drawing exists as a two-dimensional image, an observer may still require additional information to reconstruct its complete object space.
The ambiguity becomes particularly great if the image lacks:
- a recognisable horizon;
- clear vanishing points;
- regular parallel systems;
- a ground-plane grid;
- recognised objects;
- known dimensions; or
- other contextual spatial information.
Linear Perspective is therefore both subject to the Correspondence Problem and, when its geometrical framework is known, one of the principal methods for helping to constrain it.
The Correspondence Problem and Computer Vision
The same fundamental problem applies when a computer attempts to interpret camera images.
A machine receiving a two-dimensional image must identify or infer information about:
- objects;
- surfaces;
- shape;
- scale;
- depth;
- position;
- orientation;
- movement; and
- relationships between image features.
The image itself may be compatible with several possible spatial interpretations.
Computer Vision systems therefore use additional models, constraints, learned information, multiple views and other forms of contextual evidence to interpret perspective images.
The Correspondence Problem consequently links traditional Perspective Theory directly with modern problems of machine vision and image analysis.
Spatial Perception and Three Primary Factors
The Dictionary of Perspective identifies three primary factors that must be interpreted in a spatial perspective image:
- Distance — depth location of the object;
- Height — its dimension or position in the vertical direction;
- Lateral Location — its position across the lateral dimension.
Many additional factors are also involved, particularly object size, shape, aspect and angular orientation.
From a single viewpoint it may be difficult or impossible to identify all these properties correctly.
This is precisely why human spatial perception employs multiple depth cues and contextual relationships rather than relying upon isolated projected shape alone.
The Equivalence Problem Is Not the Scale–Shape–Size Problem
The Equivalence / Correspondence Problem is closely related to, but different from, the Scale–Shape–Size Problem.
The distinction can be stated clearly:
Equivalence / Correspondence Problem: Can the original three-dimensional object or scene be uniquely identified from its perspective image?
Scale–Shape–Size Problem: How do apparent or measured shape and size depend upon viewing distance, projection scale, resolution and measurement scale?
A multi-view system may substantially improve object/image correspondence while still remaining subject to scale and resolution limitations.
Volume 1 therefore specifically warns that solving aspects of the Correspondence Problem does not eliminate the wider Scale–Shape–Size Problem.
The Equivalence Problem Is Not the Shape-Sufficiency Problem
The Correspondence Problem should also be distinguished from the Shape-Sufficiency Problem.
The two questions are different:
Correspondence: Which three-dimensional reality could have produced this image?
Shape Sufficiency: Is the geometrical model used to describe that reality sufficiently accurate at the chosen scale and for the required purpose?
A reconstructed three-dimensional model may resolve much of the image ambiguity while remaining only an approximation of the physical object’s detailed structure.
The problems therefore interact but should not be collapsed into one concept.
Correspondence, Scale and Resolution
Any claim of object–image correspondence must also be considered at a particular projection scale and resolution.
An image may adequately identify a large-scale object form while failing to contain smaller structural details.
Increasing magnification or using another imaging scale may reveal features that were previously unresolved.
The apparent or measurable form of the object can therefore become more complex as additional structure becomes visible.
Thus object/image correspondence is not independent of the scale at which the image has been captured and interpreted.
Image Ambiguity and Perspective Illusions
The Equivalence / Correspondence Problem also helps explain why perspective illusions are possible.
If more than one spatial reality can produce a similar projected image, then a deliberately constructed false geometry can potentially imitate the image relationships normally produced by another physical arrangement.
This principle underlies techniques such as:
- Forced Perspective;
- accelerated and decelerated perspective;
- Ames-type distorted spaces;
- miniature photography;
- false backgrounds;
- blended scene perspective; and
- other simulated spatial constructions.
The observer interprets the projected image according to one plausible spatial arrangement even though a different physical geometry actually produced it.
A General Model for Solving the Correspondence Problem
The sources identify several kinds of information that can be combined to reduce correspondence ambiguity.
A general model is:
Perspective Image + Known Viewpoint + Projection Geometry + Perspective Framework + Recognised Objects + Depth Cues + Scale / Resolution Information + Additional Views → More Constrained 3-D Interpretation.
No single component is necessarily sufficient in every situation.
The appropriate solution depends upon the particular object, scene, perspective method and information available.
Encoding and Decoding Spatial Reality
The Correspondence Problem can be understood through two complementary operations:
Encoding
A spatial object or scene is transformed into a perspective image.
Spatial Reality → Perspective Process → Image.
Decoding
An observer or analytical system attempts to infer the underlying spatial reality from the image.
Image + Contextual Information → Estimated Spatial Reality.
The first operation can be geometrically exact according to a defined projection method while the second may nevertheless remain ambiguous.
This asymmetry is one of the most important consequences of the Equivalence / Correspondence Problem.
The Perceptual Leap from 2-D to 3-D
Volume 1 describes the interpretation of perspective as requiring a remarkable perceptual leap.
Humans routinely look at:
- retinal images;
- drawings;
- paintings;
- photographs;
- cinema and television images;
- computer displays; and
- other perspective representations
and infer three-dimensional scenes from them.
Yet a single flat image does not uniquely specify the spatial world that produced it.
The visual system therefore combines image information with additional cues, constraints, contextual information and previous knowledge to construct a probable spatial interpretation.
No Universal Monocular Solution
The sources do not propose one universal method capable of reconstructing every unknown three-dimensional scene perfectly from one monocular image.
If the spatial reality is unspecified and its geometry completely unknown, reliable interpretation requires additional information such as:
- a known viewpoint;
- known projection geometry;
- a standard picture plane;
- recognisable objects;
- a metric grid;
- known depths;
- projection scale and resolution;
- contextual relationships; or
- additional viewpoints.
The correspondence problem is therefore not merely a defect in one particular perspective construction. It is a fundamental limitation of trying to infer three-dimensional spatial reality from limited projected information.
The Equivalence / Correspondence Problem and Perspective Category Theory
Within Perspective Category Theory, the Equivalence / Correspondence Problem is a cross-cutting problem rather than a separate principal Perspective Category.
It can arise wherever a perspective system establishes a relationship between spatial reality and an image, view, measurement, calculation or representation.
It is especially relevant to:
- Natural Perspective — determining physical spatial structure from available appearances;
- Visual Perspective — interpreting projected retinal information;
- Optical Perspective — relating optical images to their target objects and scenes;
- Mathematical Perspective — modelling object/image relationships and possible reconstructions;
- Graphical Perspective — encoding and decoding represented spatial geometry;
- Instrument Perspective — interpreting camera and other instrument images;
- Simulated Perspective — exploiting alternative object/image correspondences to create illusions; and
- New Media Perspective — computer vision, multi-view imaging, 3D reconstruction and computational modelling.
The problem therefore cuts across the entire field because perspective fundamentally concerns relationships between spatial realities and their images, views and representations.
Why the Equivalence / Correspondence Problem Matters
The Equivalence / Correspondence Problem matters because it places a fundamental limit upon what can be known from a perspective image considered in isolation.
It explains why:
- a realistic photograph need not uniquely specify its original 3-D scene;
- the same projected shape can arise from different object forms;
- image size does not by itself determine physical size;
- one viewpoint provides only partial spatial information;
- context and depth cues are essential to visual perception;
- metric grids can make perspective images easier to decode;
- irregular natural forms are particularly difficult to reconstruct;
- multi-view systems provide more complete spatial information;
- perspective illusions can substitute one physical geometry for another; and
- 3D reconstruction is fundamentally an inferential rather than simply reversible operation.
The problem can therefore be summarised as:
A perspective image is evidence about spatial reality, but it is not necessarily a unique specification of spatial reality.
Equivalence / Correspondence Problem — Frequently Asked Questions
What is the Equivalence / Correspondence Problem?
It is the problem that a single 2-D monocular perspective image of a 3-D object or scene does not normally contain sufficient information to determine uniquely the original object’s complete shape, size, position, orientation and spatial geometry.
Why is it called an equivalence problem?
Because several different three-dimensional objects or scenes can potentially produce the same or equivalent two-dimensional projected image.
Why is it called a correspondence problem?
Because the problem concerns the correspondence or relationship between spatial objects in object space and their projected forms in image space.
Can a single photograph uniquely determine the original 3-D scene?
Not normally. A single photograph provides substantial spatial evidence, but several different three-dimensional arrangements can potentially be compatible with the same two-dimensional projection.
What information is lost in a 2-D perspective image?
The image does not automatically provide a unique specification of absolute depth, physical size, complete 3-D shape, orientation or hidden structure. These properties must be inferred from additional information.
Can two different 3-D objects produce the same 2-D image?
Yes. Under suitable viewpoint and projection conditions, differently shaped, sized or positioned three-dimensional objects can produce the same or sufficiently equivalent projected form.
How do humans solve the Correspondence Problem?
Humans use combinations of depth cues, object recognition, perspective frameworks, prior knowledge, movement, binocular information and other contextual evidence to constrain possible spatial interpretations.
What is the role of a metric grid?
A metric grid provides known geometrical structure that helps establish scale, direction, depth, parallel relationships and the spatial organisation of an image. It therefore assists image decoding.
Why are regular geometric scenes easier to interpret?
Regular scenes often contain recognisable parallel lines, right angles, grids and standard shapes. These provide reference structures from which perspective transformations can be interpreted.
Why are flowers and natural forms difficult to draw in perspective?
Flowers, plants and other irregular forms often lack obvious metric frameworks or regular geometrical structures. Their projected shapes can therefore be difficult to relate unambiguously to their three-dimensional forms.
Can Linear Perspective solve the Correspondence Problem?
Linear Perspective can greatly constrain the problem when its viewpoint, picture plane, metric framework, vanishing relationships and other geometrical information are known. It does not make every arbitrary monocular image a unique description of 3-D reality.
Does Orthographic Projection solve the problem?
Orthographic and other parallel projections can remove some central-perspective transformations, but one view still contains only selected information about a three-dimensional object. Multiple coordinated views may be required.
Can Multi-View Perspective solve the problem?
Multi-View Perspective can provide a substantial solution because different viewpoints reveal additional information about shape, depth, orientation and hidden surfaces. The resulting views must nevertheless be accurately related and integrated.
How is the Correspondence Problem related to 3D Reconstruction?
3D Reconstruction attempts to infer three-dimensional geometry from perspective images. Because one image does not normally determine a unique 3-D scene, reconstruction commonly requires multiple views, known geometry, measurements, constraints or other contextual information.
How is it related to Computer Vision?
Computer Vision systems must infer objects, depth, shape, position and orientation from images. They therefore encounter the same fundamental problem of relating projected image information to possible three-dimensional spatial realities.
Is the Correspondence Problem the same as the Scale–Shape–Size Problem?
No. The Correspondence Problem concerns whether the source 3-D object or scene can be uniquely identified from an image. The Scale–Shape–Size Problem concerns how apparent or measured form depends upon scale, resolution, distance and other perspective conditions.
Is it the same as the Shape-Sufficiency Problem?
No. Shape Sufficiency asks whether a geometrical model is sufficiently accurate for a particular scale and purpose. Correspondence asks which spatial reality could have produced a particular perspective image.
Why do depth cues matter?
Depth cues provide additional spatial evidence that constrains possible interpretations of an otherwise ambiguous perspective projection.
Why are perspective illusions possible?
Because different physical arrangements can produce equivalent projected relationships. A deliberately altered scene can therefore generate the image normally associated with a different spatial reality.
Is the Equivalence / Correspondence Problem a Perspective Category?
No. It is a fundamental cross-cutting problem concerning the relationship between spatial reality and perspective images, views and representations across several Perspective Categories.
The Equivalence / Correspondence Problem within the Wider Field of Perspective
The Equivalence / Correspondence Problem exposes one of the deepest limitations—and one of the most important functions—of perspective.
Perspective converts spatial reality into images, views, measurements and representations. But a projection is not simply an interchangeable copy of its source. Viewpoint, orientation, scale, projection geometry and resolution transform the information carried by the resulting image.
As a result, the journey from object to image and the journey from image back to object are not equivalent operations:
3-D Object → Perspective Projection → 2-D Image
does not automatically imply:
2-D Image → Unique 3-D Object.
Additional spatial information is required to constrain the possible solution. Perspective frameworks, depth cues, known geometry, recognised forms, scale information and multiple viewpoints all help establish a more reliable correspondence between image and spatial reality.
Seen in this wider context, the Equivalence / Correspondence Problem is fundamental to vision, representation, photography, drawing, perspective illusion, computer vision and 3D reconstruction. It demonstrates that perspective is concerned not only with making images of spatial reality, but also with the much harder problem of understanding what spatial reality those images can legitimately be said to represent.