3.4 Evolution of Defocusing
3.4.1 Separation into Three Lenses
Regardless of the aperture arrangement—whether there are apertures off axis or a single one on the axis—in a camera with an infinite depth of field (pinhole optics), the image of a point is the locus of all points on the ray which forms this image. Said differently, all pointsP between point A on the focal plane and the center of an aperture will have the same image at point B on the image plane—thus a single aperture system does not store depth information in an image since it is impossible to know which pointP actually formed the image (see figure3.4-3, top).
Figure 3.4-3: In pinhole optics, multiple points along the same ray (pointsP) will form an image in the same exact place (pointB) as that of a pointAat the “focal” plane (top). If a second aperture is added off-axis, both apertures will image pointAat pointBbut all other pointsP will have different images through the second aperture as they are no longer on the same ray through that aperture. Thus with two apertures, depth information can be recovered independently of blur.
However the if there are multiple apertures in the system the depth information can be recovered because each image (from each aperture) will represent a different ray in space which, if each image comes from the same particle, should intersect at exactly one point.
Looked at it another way, a point cannot move along two different directions at once and thus there will always be depth information as long as there are at least two apertures.
This depth information can be obtained as in the original defocusing concept (three apertures in one lens) or as in the present implementation (one aperture in each of three lenses—see figure3.4-4).
As long as the three lenses do not share an optical axis (which they could only share if they at the exact same location in space) the “defocusing” effect would still be present. This is even true if the axes of the lenses are parallel (which is the case of DDPIV cameras). In this sense the sensitivity of the system can be interpreted as how far particles can be from the optical axis of each lens versus the
distance from the aperture which conceptually is the same measurement as the aperture separation versus distance to the reference plane was for the sensitivity of a single-lens system.
Figure 3.4-4: The top of this figure shows the same point imaged through two lenses that are offset from each other.
The bottom shows the two lenses’ ray traces superimposed on each other, showing how two offset lenses can obtain depth information exactly as a multiple-aperture single-lens setup would.
Separating the system into three lenses allows for much larger separations while conserving image quality and simultaneously separating the apertures each to one sensor8. Limits of sensitivity are now imposed only by the illumination and geometry of the image space of the lenses chosen. In theory then photographic objectives intended for larger-format photography could yield much higher sensitivity than smaller formats, however, in practice it seems the limiting factor is the light fall-off.
Lenses experience light fall-off as a function of radial distance from the axis, and this effect seems to be amplified by the linearity of CCD exposure9. In the end, image quality is no longer a concern in these cases as the images will be too dark to discern well before the edge of the realm of unacceptable aberrations is reached.
The three-lens arrangement was introduced by Pereira at the ONR workshop in San Diego in February 1999 and subsequently at PIV ’99 (seePereira, Gharib, Modarress, and Dabiri [1999b]).
8which also has the added advantage that at the reference plane three distinct points can still be measured.
9Some argue that the microlenses implemented in interline-transfer CCD’s also contribute to the light fall-off.
The paper from the latter was published in 2000 (seePereira, Gharib, Dabiri, and Modarress[2000a]).
A second paper in 2002 analyzed the method in more detail (seePereira and Gharib[2002]).
Chapter 4
Pinhole Optics Approach
4.1 Introduction
The mathematical explanation of defocusing cameras and the basis of the algorithms of the processing software are based on pinhole optics. In pinhole optics, there are no lenses, and apertures represent points through which only single rays of light can pass. All relationships are then based on similar geometry.
The key quantities discussed are exemplified in the 2-aperture depiction in figure4.1-1.
Figure 4.1-1: Pinhole-optics diagram of a two-aperture defocusing arrangement.
reference plane aperture plane image plane
sensor
camera axis sensoraxis
sensor field of view mappable region
characteristic probe volume L
L ll
Z Z
2d 2d a
L is the distance from the aperture plane to the reference plane, l is the distance between the aperture plane and the image plane, 2d is the distance between the two apertures, a is the side length of the characteristic probe volume, andZ is the distance between a particle and the aperture
plane. In this depiction the apertures are centered about the optical axis of the camera so that the distance between an aperture and the center of the aperture plane isd.
The fields of view of the sensors are shown here as shaded in light grey (they of course continue past the reference plane but are not shaded past it); they are constructed by taking lines from the edges of the sensors and passing them through the center of the respective apertures. The mappable region is the region where the fields of view intersect, and in theory any point in this region is measurable by the camera. It is shown here as the darker grey triangle. Thecharacteristic probe volume (sometimes referred to as just probe volume) is the rectangular prism whose short cross section is the largest inscribable square in the cross section of the mappable region. In this two-dimensional case, then, it is just a square.