Projections used in panoramic phtography - part 1

Most programs for the assembly of multi-image panoramic photographs offer a choice which projection is represented by the final result. I thought I would offer a little technical background in this matter.

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In this context, projection is the process by which object points in three-dimensional space are mapped onto locations on a (flat or flattenable) two dimensional surface. A projection is a particular scheme for doing so.

Projection is involved in any photography, but we most often hear the term in connection with panoramic photography. This is likely because the implications of different projections become most significant to us in the case of a large field of view.

The matter of projections was first highly developed in the field of cartography, that matter of making maps of the earth's surface. The issue arises here because the points on the earth surface (even if we ignore local changes in altitude) must be treated as a three dimensional field, while of course maps are expected to be flat and two dimensional.

Many of the projections we use were first developed in this contexts, and often their names reflect that history.

Four projections we often hear of used in this work are:

• Rectilinear
• Cylindrical ("classical")
• Mercator
• Equirectangular

The rectilinear projection is ordinarily the one we aspire to have followed in ordinary photography at "moderate" fields of view. Its two most important properties are:

• If we have an object surface that lies in a plane, perpendicular to the axis of the lens, then the image is just a "scaled" version of the object surface, with all distances between points uniformly proportional to the distances between the corresponding object points. All angles between lines we might construct between object points are preserved in the image. (We say that the image array of points is similar, in the formal geometric sense, to the object array of points.)

• All straight lines in the three-dimensional object space are represented as straight lines in the image.

The use of the rectilinear projection is useful in multi-image panoramic photography in such cases as the photography of a building with a "mostly planar" facade, spanning a substantial field of view from the camera location, when we want the image to look like an architect's "elevation" drawing of the facade.

A rectilinear projection is absolutely limited to a field of view of 180°, and various considerations make it unsuitable for fields of view even approaching that.

We can understand the concept of the rectilinear projection with a pinhole camera metaphor. We have a pinhole in an opaque plate. The film is flat and its plane is parallel to the plane of our "test object surface".

From each object point, we consider that a single ray passes through the pinhole onto the film. The triangles on the object and image sides of the pinhole, formed of the ray of interest, the axis of the camera extending to the object plane or film, and the lines connecting their ends, are similar (in the formal geometric sense, all angles being the same and all side lengths being uniformly proportional) and this leads to the image array being formally similar to the object array.

Now that we have practiced our pinhole metaphor, we will use it to explain the definition of another projection, the cylindrical projection. (I mean here what I call the "classical" cylindrical projection, since other related projections are often categorized as "cylindrical" (including the other two we will visit here shortly).

Again imagine our pinhole. Her, we need to imagine that it has a supernatural property: light will pass through it in the same way whether "head on" of from a substantial angle to the side.

We bend out sheet of film into a partial cylinder, whose axis is vertical and passes through the pinhole. Again, we consider rays from various points in the scene.

Imagine that we have a very narrow "billboard", whose face is perpendicular to the line from the pinhole to its center. We can see that with respect to this narrow object field, the result will be just the same as with the flat film, and thus with respect to that narrow object the behavior of the system is just like with the rectilinear projection.

We find that, if we consider a set of equally spaced points in a vertical line on our billboard, they will be equally spaced on the film. Thus the cylindrical projection preserves the vertical "scale" of object points at any particular distance (where by distance I mean distance measured parallel to the "camera" axis, not radially from the pinhole to the point - this is a critical distinction here as we consider significant vertical fields of view).

We find that, if we had a building facade that was semicircular, with its center at the camera, the cylindrical projection would make the image (as we flattened out the film) similar (in the geometric sense to the actual facade of the building (if we flattened it out).

But if we had a "wide" building, with its facade perpendicular to a line to its center from the camera, its portrayal on the image will not be similar to the facade itself. The reason is that the ends of the building are farther from the camera than the center. As a result, the vertical scale for them in the image is less than the vertical scale for points in the "center" of the facade.

Thus the image of the building facade on the film will "droop" as we go away from the center.

Note that just as the rectilinear projection is not good for large fields of view (horizontal or vertical, in fact), the cylindrical projection is not good when really large vertical fields of view are involved. If we wanted to embrace a vertical half field of view of 90°, we would need an infinitely-high piece of film.

[continued in part 2]

Projections used in panoramic phtography - part 2

[continued]

Now, let's move to another of our candidate projections, the equirectangular projection. Its properties relate directly to the matter of cartography - the mapping of the surface of the earth.

We cannot understand in terms of a physical projection (as with our pinhole metaphor). It is defined mathematically, as a variant of the (classical) cylindrical projection. To understand how it is defined, we have to take a different outlook of the scene.

Here, we consider the points in the scene to lie on the surface of a large hemisphere, with its center at the camera. We will inscribe on the inside of this hemisphere, as we do on an Earth globe, meridians and parallels.

The meridians lie at equal horizontal angles as seen from the camera. They actually converge overhead and at the nadir (the point that lies directly below "overhead"), as at the north and south poles of the Earth.

The parallels are lines having a constant elevation angle, equally spaced in terms of that. One lies right on the horizontal (it is our "equator"), one perhaps 10° above that, the next 20° above that, and so forth. The last one is at the "north pole" (90°), and in fact has zero length.

With the equirectangular projection, the image of these lines on the film is a uniform rectangular grid, with the meridians appearing as vertical lines, equally spaced, and the parallels as horizontal lines, equally spaced (and with the spacing in the two directions being the same).

You can see why this projection, used for a map, would be (in one way) attractive for navigation, as it provides a uniform plot of latitude vs. longitude.

Note that with this projection, an object field that extended to an elevation angle of 90° does not require an infinitely-high film.

Now what is its attraction in panoramic photography? Does it make anything we really perceive more realistic? Generally not. Its attraction is mainly (as suggested just above) that it allows us to make a panoramic photograph whose vertical field of view extends to an angle of 90°.

Note that with respect to actual scale (magnification), if we consider object distances on a fanciful hemisphere centered on the camera (not a "billboard"), with the equirectangular projection, the vertical scale is uniform over the entire height of the image, but the horizontal scale varies.

This at first may seem counterintuitive - after all, the meridians are represented by parallel (vertical) lines on the image - equally spaced over their entire height. But recall that in terms of actual distance on the surface of our fanciful sphere, the meridians are spaced more closely as we approach the "pole" (90°). Thus the horizontal scale of the image (with respect to distances on the hemisphere) decreases as we move up (or down, of course) from our "equator".

What is the significance of this in photography? Really not much, since we are not ordinarily dealing with objects painted on a hemisphere.

In fact, in general, with an equirectangular projection, buildings and the like look unnatural. Although the vertical scale of the image is uniform with respect to object distances on our fanciful hemisphere, it is not uniform with respect to objects all of whose points are at a consistent horizontal distance from the camera (such as the facade of a building). The result is that for a building of any substantial height, the vertical scale decreases as we move up the building, a "compression of height".

Now lets move to our last projection, the Mercator projection. Again, this can't be explained in terms of any physical projection metaphor (as with our pinhole). It is defined mathematically, as a variant of the (classical) cylindrical projection.

In it, the vertical scale is modified so that is follows the horizontal scale at the same elevation angle, where both scales are defined in terms of distances on our fanciful hemisphere. You can see that this is advantageous in navigation since a certain distanced on the chart, regardless of its direction, or the location, corresponds to the same distance on the surface of the earth.

Now for what it is worth to us (actually very little in panoramic photography), this means that is we had little circular spots of uniform diameter on the surface of our fanciful hemisphere, they would all end up on the image as circles (but the diamater varies with teh elvation angle).

What is its actual attraction in panoramic photography? Well, mostly that it is a middle ground between the cylindrical and equirectangular projections (one whose mathematical description is well known). If we want to work with a large vertical field of view (so that the cylindrical projection is not handsome), but wish to avoid the substantial "height compression" (decline of vertical scale as we move up in elevation) of the equirectangular projection, then this affords an often-handsome compromise.

I hope this has given some insight into the concepts behind these different projections.

Projections used in panoramic phtography - Appendix

Appendix

There has been mention elsewhere in this forum of the implications of uncorrected parallax shift on the projection used in panoramic photography, or of the influence of the ultimate projection involved on the choice of an axis of rotation of the camera (for multi-image panoramic photography) or an axis of rotation of the lens (for stinging-lens panoramic photography.

In fact, when there is a consequential effect of parallax shift, this does not lead to a new projection. It actually leads to no projection at all.

The reason is that the definition of a projection tells us how each point in the three-dimensional scene is mapped to a point (one and only one point, I need to emphasize here) on the image.

When we have uncorrected parallax shift in multi-image panoramic photography, a result is that a single object point may be mapped to two different points on the (assembled) image - one point on one side of a seam and an inconsistent one, with respect to other scene points, on the other.

That does not even meet the definition of a projection, other than perhaps "a defective one".

The considerations that lead to averting the effect of parallax shift in multi-image panoramic photography (typically, providing for camera rotation about an axis passing through the entrance pupil of the lens) are not influenced by the projection into which the "basic" assembled image is ultimately transformed for delivery.

Similarly, in swinging lens panoramic photography, in which parallax shift is not a controlling issue, the considerations that leads to the use of a lens rotation axis through the second nodal point of the lens are not a creature of the projection of the "taken" image (which, if the lens is rectilinear, will be the classical cylindrical projection).

Best regards,

Doug

Doug,

Unless one could do Rubrick's cubes in one's crib as an infant, most folk work with diagrams to figure and realize relationships in 3D space. So even a few diagrams would be helpful in making the thread more accessible.

If that's possibly something you might want to do, that would be appreciated.

Asher

Hi, Asher,

Asher Kelman said:

Doug,

Unless one could do Rubrick's cubes in one's crib as an infant, most folk work with diagrams to figure and realize relationships in 3D space. So even a few diagrams would be helpful in making the thread more accessible.

If that's possibly something you might want to do, that would be appreciated.

Click to expand...

Indeed. They will be in the actual article (it only adds perhaps another 5 hours to the work).

Best regards,

Doug

Great! That will be a great resources for us!

Asher