Doug Kerr
Well-known member
In the context of interest to us, projection is the process of mapping points in three-dimensional space onto locations on a (flat) two-dimensional surface. A projection is a definable scheme for doing so.
We often hear about projection(s) in connection with panoramic photography, but rarely in connection with "ordinary" photography. But the concept is equally applicable to both. Perhaps that difference in usage comes from:
• As field of view becomes great, the matter of projection becomes a "bigger deal" insofar as the appearance of the image is concerned.
• Modern tools for combining images for multi-image panoramic photography also give us the opportunity to choose the ultimate projection that will be used. In non-panoramic photography, we are not so often already using tools that give us the same opportunity (or we may not know of them).
Here is a lovely site that gives a very nice description of the concepts of projection, with discussions of several important "named" ones, along with both conceptual illustrations and illustrations as applied to actual photography (principally panoramic):
http://www.cambridgeincolour.com/tutorials/image-projections.htm
Much of the basic mathematical work on the matter of projections was developed in the field of cartography (the science of making maps of the Earth's surface), and several bear the names of workers in that field (Mercator, for example). Of course projection is involved there because the Earth's surface is three-dimensional (and not a "flattenable" surface, even if we leave out local variations, such as for hills and valleys) while maps are expected to be flat.
The terminology used to describe map projections is adapted to photography in this way:
• The world seen by the camera is considered to occupy a sphere of infinite radius, centered on the location of the camera.
One way we become concerned with the matter of projection in "ordinary" photography, without usually using the term, is in the matter of "geometric distortion". By this we mean the degree to which a particular lens fails to accurately execute the projection we (without saying so) expect in most ordinary photography (with moderate fields of view).
That projection has several names, but perhaps the most common is "rectilinear". It has this basic property:
•If we consider the points in a scene that lie in a single plane, perpendicular to the axis of the lens, then the image will be just a scaled version of that plane of points (it will be "fully similar", as the geometricians say).
An important corollary is:
•Any straight line in the three-dimensional object space, regardless of orientation, appears as a straight line in the image.
When we get into lenses producing a large field of view (in particular fisheye lenses), a couple of other projections are often assumed (again, without usually using the term). A common one is the one often called equirectangular (sometimes linear scaled or equidistant).
It is defined by this property:
• The distance to any point in the image from the center is proportional to the angle, off the lens axis, of the object point
In photography (especially in panoramic photography), we are often concerned with these two considerations (among others):
• Do straight lines in the actual three-dimensional world show up as straight lines in the image?
• With regard to the fronts of buildings that are perpendicular to a line from the camera location, does the magnification (the ratio of a distance on the image to the distance on the object) remain constant over the height of the building?
The various criteria to which we might aspire in photography (and especially in panoramic photography), including those two, cannot all be satisfied by the use of any single projection. Some projections have in fact been devised to give a compromise among the various criteria in certain situations. The Mercator projection is one example.
It is beyond the scope of this note to discuss which projections fulfill which criteria, and how the compromises play out. For that I leave you to the many works available on panoramic photographic practice. (A good start is given in the article referenced above.)
I will note in closing that a swinging lens panoramic camera, if its lens in ordinary photographic use would produce a rectilinear projection (that is, "has no geometric distortion"), will give the projection called cylindrical.
We often hear about projection(s) in connection with panoramic photography, but rarely in connection with "ordinary" photography. But the concept is equally applicable to both. Perhaps that difference in usage comes from:
• As field of view becomes great, the matter of projection becomes a "bigger deal" insofar as the appearance of the image is concerned.
• Modern tools for combining images for multi-image panoramic photography also give us the opportunity to choose the ultimate projection that will be used. In non-panoramic photography, we are not so often already using tools that give us the same opportunity (or we may not know of them).
Here is a lovely site that gives a very nice description of the concepts of projection, with discussions of several important "named" ones, along with both conceptual illustrations and illustrations as applied to actual photography (principally panoramic):
http://www.cambridgeincolour.com/tutorials/image-projections.htm
Much of the basic mathematical work on the matter of projections was developed in the field of cartography (the science of making maps of the Earth's surface), and several bear the names of workers in that field (Mercator, for example). Of course projection is involved there because the Earth's surface is three-dimensional (and not a "flattenable" surface, even if we leave out local variations, such as for hills and valleys) while maps are expected to be flat.
The terminology used to describe map projections is adapted to photography in this way:
• The world seen by the camera is considered to occupy a sphere of infinite radius, centered on the location of the camera.
One way we become concerned with the matter of projection in "ordinary" photography, without usually using the term, is in the matter of "geometric distortion". By this we mean the degree to which a particular lens fails to accurately execute the projection we (without saying so) expect in most ordinary photography (with moderate fields of view).
That projection has several names, but perhaps the most common is "rectilinear". It has this basic property:
•If we consider the points in a scene that lie in a single plane, perpendicular to the axis of the lens, then the image will be just a scaled version of that plane of points (it will be "fully similar", as the geometricians say).
An important corollary is:
•Any straight line in the three-dimensional object space, regardless of orientation, appears as a straight line in the image.
When we get into lenses producing a large field of view (in particular fisheye lenses), a couple of other projections are often assumed (again, without usually using the term). A common one is the one often called equirectangular (sometimes linear scaled or equidistant).
It is defined by this property:
• The distance to any point in the image from the center is proportional to the angle, off the lens axis, of the object point
In photography (especially in panoramic photography), we are often concerned with these two considerations (among others):
• Do straight lines in the actual three-dimensional world show up as straight lines in the image?
• With regard to the fronts of buildings that are perpendicular to a line from the camera location, does the magnification (the ratio of a distance on the image to the distance on the object) remain constant over the height of the building?
The various criteria to which we might aspire in photography (and especially in panoramic photography), including those two, cannot all be satisfied by the use of any single projection. Some projections have in fact been devised to give a compromise among the various criteria in certain situations. The Mercator projection is one example.
It is beyond the scope of this note to discuss which projections fulfill which criteria, and how the compromises play out. For that I leave you to the many works available on panoramic photographic practice. (A good start is given in the article referenced above.)
I will note in closing that a swinging lens panoramic camera, if its lens in ordinary photographic use would produce a rectilinear projection (that is, "has no geometric distortion"), will give the projection called cylindrical.