Image Space

Image Space is the space in which a resulting perspective image, view, model or representation is formed, located, displayed or perceived.

Within the Dictionary of Perspective, Image Space and Perspective Space are used as equivalent general terms. Image Space may be optical, visual, graphical, mathematical, instrument-generated or digital, and may be two-dimensional or three-dimensional depending upon the perspective system involved.

Image Space should be distinguished from Object Space or Target Space, which contains the object, scene or spatial information towards which the perspective process is directed.

The basic relationship can therefore be expressed as:

Object or Target Space → Perspective Process or System → Image or Perspective Space → Visual Interpretation

Image Space and Object Space

Perspective normally involves a relationship between two broad kinds of space.

Object Space or Target Space — contains the physical, artificial, simulated or imagined object, scene or spatial information to be viewed, measured or represented;
Image Space or Perspective Space — contains the resulting image, view, model or representation.

The two spaces are related, but they are not the same.

An object may possess one size, shape, position and orientation in Object Space while possessing a different apparent size, shape, position or orientation in Image Space.

Perspective therefore concerns the relationship between an original spatial structure and the image-space structure produced from it.

Image Space and Perspective Space

In its general sense, Perspective Space is synonymous with Image Space.

It refers to the space in which a perspective image, view, model or representation is formed, located or perceived.

A narrower historical use of the term perspective space refers specifically to the fictive or represented space suggested within a picture. This usage is common in art theory, but it should not be confused with the broader PCT meaning.

For general perspective analysis:

Image Space = Perspective Space

The broader term therefore applies not only to drawings and paintings but also to optical images, retinal images, photographs, mathematical representations, instrument images, digital models and New Media environments.

Image Space as a transformed space

Image Space is produced when a perspective process relates or transforms Object Space into a resulting image or view.

This transformation may preserve some properties of the original spatial reality while altering others.

Properties may be:

preserved;
reduced;
compressed;
concealed;
distorted;
repositioned; or
represented at a different scale.

Perspective therefore involves both correspondence and transformation.

The resulting Image Space is not simply a duplicate of Object Space. It is a structured spatial result produced through the particular optical, geometrical, graphical, instrumental, computational or perceptual processes involved.

The dimensionality of Image Space

Image Space is not necessarily two-dimensional.

Depending upon the perspective process, an image or representation may occupy a:

1-D Image Space;
2-D Image Space;
2.5-D represented space; or
3-D Image or Model Space.

A conventional photograph, drawing or flat display normally employs a two-dimensional Image Space.

A three-dimensional computer model, holographic system, volume display or interactive virtual environment may instead employ a three-dimensional Image Space or model space.

The dimensionality of the original object and the dimensionality of its resulting image therefore need not be identical.

The geometry of Image Space

The geometry of Image Space depends partly upon the perspective process and partly upon the geometry of the surface or spatial structure on which the image is formed.

A flat picture plane produces a planar Image Space.

An image projected onto a curved surface produces a correspondingly curved or curvilinear spatial arrangement.

Image Space may therefore employ geometries including:

planar;
curvilinear;
cylindrical;
spherical;
Euclidean;
non-Euclidean;
Cartesian; and
other coordinate or model spaces.

There is consequently no single geometry that defines every form of Image Space.

Planar Image Space

A conventional drawing, painting, photograph or flat-screen image normally occupies a planar Image Space.

Three-dimensional Object-Space relationships are mapped onto a two-dimensional surface.

In central or Linear Perspective, this can produce characteristic transformations such as:

diminution of apparent size with distance;
convergence of projected parallel lines;
vanishing points;
horizon or vanishing lines;
foreshortening;
overlap and occlusion; and
changes in apparent shape and position.

These features belong to the resulting image geometry. They should not automatically be treated as properties of the original Object Space.

Curved Image Space

Image Space can also be formed on a curved image surface.

When an image is projected onto a cylindrical, spherical or otherwise curved surface, its size, spacing, position and shape are mapped according to that surface and the associated projection process.

Curvilinear and spherical perspective systems provide important examples in which image geometry cannot adequately be described as though the image were confined to an ordinary flat picture plane.

Curved Image Space is especially important when considering wide-field perspective and the relationship between planar graphical projection and natural or optical vision.

Retinal Image Space

Human vision provides a particularly important form of curved Image Space.

Light from spatial objects and scenes is optically projected onto the curved retinal surfaces of the two eyes.

The retinal image therefore differs geometrically from an image projected onto a flat picture plane.

Retinal curvature forms one physical and geometrical contribution to the curvilinear character associated with Visual Perspective Type 2.

The resulting human visual experience, however, should not be equated simply with the retinal surface. It is further shaped by:

the optics of the eye;
monocular vision;
binocular vision;
eye movements;
head and body orientation; and
perceptual interpretation.

Retinal Image Space and perceived Visual Space are therefore closely related but not identical.

Visual Image Space

Visual Image Space is the image or perspective space formed and experienced through the human visual system.

It belongs particularly to Visual Perspective Type 2.

Visual Space concerns the experienced spatial organisation of such properties as:

direction;
position;
distance;
depth;
apparent size;
apparent shape; and
orientation.

It should nevertheless be distinguished from the retinal image itself, the momentary visual field and the physical Object Space towards which vision is directed.

The physical environment, retinal image and perceived visual organisation therefore constitute related but analytically distinct spatial stages.

Photographic and Instrument Image Space

A camera forms a photographic or instrument-generated Image Space from the Object Space towards which it is directed.

A three-dimensional physical scene may therefore become a two-dimensional photographic image.

The camera projection can be geometrically valid for the selected camera model while nevertheless preserving only part of the original spatial information.

The resulting image may also contain optical characteristics associated with:

lens geometry;
field of view;
magnification;
focus;
resolution;
aberration;
distortion; and
sensor geometry.

Photographic Image Space is therefore the result of a particular imaging process rather than a direct duplication of physical space.

Graphical Image Space

Graphical Image Space is produced when spatial objects or scenes are constructed or represented through drawing, painting, diagramming, drafting or related graphical methods.

A graphical perspective system establishes rules for locating image points, lines, planes and forms on a picture surface.

In Linear Perspective, for example, an assumed three-dimensional Object Space can be transformed into a two-dimensional graphical Image Space containing projected lines, vanishing points, horizon relationships and apparent spatial recession.

The graphical image may correspond closely to an existing physical scene, represent an imagined scene, or be constructed independently according to geometrical rules.

Primary and Secondary Geometry

The distinction between Primary Geometry and Secondary Geometry helps clarify the relationship between Object Space and Image Space.

Primary Geometry concerns the three-dimensional arrangement of object and projection relationships in Object or Target Space.

Secondary Geometry concerns the relationships between the points, lines and shapes of the resulting projection in Image Space.

In simplified form:

Primary Geometry → Object Space
Secondary Geometry → Image Space

A graphical perspective drawing can therefore be constructed using rules of Secondary Geometry even when the complete Primary Geometry of the original three-dimensional scene is not explicitly constructed.

Mathematical Image Space

Image Space may also be mathematical.

A mathematical perspective process can represent spatial information through coordinates, transformations, projection matrices and geometrical relationships.

The resulting Image Space may be:

two-dimensional;
three-dimensional;
Cartesian;
spherical;
Euclidean;
non-Euclidean; or
another mathematically defined coordinate space.

Mathematical Image Space provides a means of describing and calculating spatial transformations independently of whether the result is ultimately displayed as a conventional visible picture.

Digital and New Media Image Space

New Media Perspective extends Image Space into computational environments.

A digital Image Space may contain:

bitmap images;
vector graphics;
computer-generated imagery;
three-dimensional models;
point clouds;
virtual environments;
augmented-reality imagery; and
interactive spatial representations.

Unlike a fixed two-dimensional picture, a digital Image Space can potentially be navigated, transformed, scaled, rotated, linked to other views or explored from changing viewpoints.

Three-dimensional computer modelling, for example, develops a coordinate-based representation of spatial forms within a 3-D image or model space.

Image-Space position

Position in Image Space is defined according to the coordinate or reference system of the image.

In a conventional two-dimensional image, position may be specified using:

X and Y coordinates;
distance from the image borders;
pixel coordinates;
angular position;
image-centred coordinates; or
another local reference system.

These Image-Space positions should not automatically be confused with corresponding physical positions in Object Space.

For measurement or mapping purposes, local Image-Space coordinates may need to be transformed back into Object-Space coordinates or related to a known physical scale.

Image Scale

An image normally represents Object Space at some particular projection or image scale.

A large physical object may occupy only a small region of Image Space, while a small nearby object may occupy a comparatively large image area.

The relationship may be expressed as:

Object-Space dimension → projection scale → Image-Space dimension

Maps, technical drawings and other measured representations may provide an explicit scale allowing Image-Space measurements to be related to corresponding Object-Space dimensions.

In central perspective, however, local projected scale may also vary with distance and position within the scene.

Perspective Phenomena in Image Space

Many familiar Perspective Phenomena are observable as transformations within Image Space.

These may include:

diminution of apparent size;
foreshortening;
changes in apparent shape;
line convergence;
vanishing points;
vanishing lines;
changes in apparent position;
occlusion;
gradients of colour;
gradients of clarity or acuity; and
changes in light, shade and contrast.

For example, a set of straight parallel lines in Object Space can remain genuinely parallel there while appearing to converge towards a vanishing point in Image Space.

Perspective analysis therefore requires care in distinguishing an Object-Space property from its Image-Space appearance.

The optical image chain

Image Space frequently forms one stage within a longer optical image chain.

A simplified imaging chain may include:

Object → Imager → Sensor → Image → Processor → Display → Analysis

The object is located in Object Space, while the perspective image is formed in Image Space.

A camera, for example, may transform a physical scene into photographic or digital Image Space. That image may subsequently be processed and displayed within another spatial system before finally being viewed by a human observer.

Complex perspective systems may therefore contain several successive Image Spaces.

Intermediate Image Spaces

Between the original Target Space and the final viewed image there may be one or more intermediate Image Spaces.

For example:

Physical Object Space → Camera Image Space → Digital Processing Space → Display Image Space → Visual Image Space

Each stage may transform the spatial information received from the preceding stage.

Image size, colour, geometry, resolution, scale or other characteristics may therefore change as the image passes through the complete perspective chain.

Understanding a finished perspective image may require identifying these intermediate stages rather than treating the final image as though it were produced directly from the original spatial scene.

When Image Space becomes Object Space

The distinction between Object Space and Image Space depends upon the particular stage of a perspective system being analysed.

One system’s Image Space may become another system’s Object or Target Space.

A photograph provides a simple example. The original physical scene is the Object Space of the camera. The photograph is the camera’s Image Space.

If that photograph is subsequently scanned, photographed again, projected or computationally analysed, the first image can become the target of a second perspective process.

The relationship may therefore be:

Object Space A → Image Space A / Object Space B → Image Space B

This is particularly important in Category Chaining, Composite Perspective and modern digital imaging systems.

The correspondence problem

A central problem of perspective is that Image Space does not necessarily contain enough information to reconstruct Object Space uniquely.

A single two-dimensional projection of a three-dimensional scene may be compatible with several different possible Object-Space arrangements.

This is the correspondence or equivalence problem.

Depth, scale, position and actual three-dimensional form may therefore be ambiguous when only the resulting image is available.

Additional information may be required, including:

known object shapes;
metric grids;
multiple viewpoints;
stereoscopic information;
known scales;
contextual information; or
knowledge of the projection method.

The existence of an apparently convincing perspective image does not therefore guarantee a unique reconstruction of the spatial reality that produced it.

Image Space is not Visual Space

Image Space is a broad concept that includes any space in which an image, view, model or representation is formed, located or perceived.

Visual Space is more specific. It concerns spatial organisation as experienced through the human visual system.

A photograph possesses an Image Space before anyone looks at it. When a human observer views that photograph, its visual information participates in the formation of a further Visual Image Space or perceived visual organisation.

Similarly, the optical image formed on the retina is not identical to the final perceived Visual Space.

These distinctions help separate physical, optical, instrument-generated and perceptual stages of the complete perspective process.

Image Space is not necessarily picture space

Image Space should also not be restricted to the traditional idea of a picture containing an illusion of depth.

A perspective image may exist:

on a flat picture plane;
on a curved surface;
on the retina;
within an optical system;
on a photographic sensor;
on a digital display;
within a mathematical coordinate system;
inside a computer model; or
within an interactive three-dimensional environment.

The concept of Image Space therefore provides a common analytical term across visual, optical, graphical, mathematical, instrument and New Media Perspective.

Why Image Space matters

The distinction between Image Space and Object Space is fundamental to perspective because an image is a transformed spatial result, not the object or scene itself.

Without this distinction, it becomes easy to confuse:

object size with image size;
object position with image position;
actual shape with apparent shape;
physical depth with represented depth;
parallel Object-Space lines with converging Image-Space lines;
physical scale with projection scale;
the geometry of a scene with the geometry of its image; and
the original spatial reality with its visual or represented appearance.

Image Space is therefore a foundational concept in Perspective Category Theory because it identifies the optical, visual, graphical, mathematical, instrument-generated or digital space in which a resulting perspective image, view, model or representation is formed, located or perceived.