Basic Limitation of Projective Imaging

As we have discussed in detail in Sections 7.6.1 and 7.6.2, a projective optical system is a linear shift-invariant system that can be described by a point spread function (PSF) and optical transfer function (OTF).

The 3-DOTF for geometrical optics shows the limitations of a projec- tive imaging system best (see Section 7.6.2):

 

hˆ (q, k  )

2I₀   .

2            1/2

k

Σ

3

 

Π . k₃  Σ .

3 = π |q tan α |

1 − q2 tan2 α

2q tan α

(8.1)

 

The symbols q and k₃ denote the radial and axial components of the wave number vector, respectively. Two severe limitations of 3-Dimaging immediately follow from the shape of the 3-DOTF.

205

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

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206                                                                                                         8 3-D Imaging

 

±

Complete Loss in Wide Wave Number Range. As shown in Fig. 7.13b, the 3-D OTF is rotationally symmetric around the k₃ axis (z direction) and nonzero only inside an angle cone of α around the xy plane. Struc- tures with a wide range of wave numbers especially around the z axis are completely lost. We can “see” only structures in those directions from which the optics collect rays.

 

Loss of Contrast at High Wave Numbers. According to Eq. (8.1), the OTF is inversely proportional to the radial wave number q. Consequently, the contrast of a periodic structure is attenuated in proportion to its wave number. As this property of the OTF is valid for all optical imaging

— including the human visual system — the question arises why can we see fi ne structures at all.

The answer lies in a closer examination of the geometric structure of the objects observed. Most objects in the natural environment are opaque. Thus, we see only the surfaces, i. e., we do not observe real 3-D objects but only 2-D surface structures. If we image a 2-D surface onto a 2-D image plane, the 3-D PSF also reduces to a 2-D function. Mathe- matically, this means a multiplication of the PSF with a δ plane parallel to the observed surface. Consequently, the 2-D PSF is now given by the unsharpness disk corresponding to the distance of the surface from the lens. The restriction to 2-D surfaces thus preserves the intensity of all structures with wavelengths larger than the disk. We can see them with the same contrast.

We arrive at the same conclusion in Fourier space. Multiplication of the 3-D PSF with a δ plane in the x space corresponds to a convolution of the 3-D OTF with a δ line along the optical axis, i. e., an integration in the corresponding direction. If we integrate the 3-D OTF along the k coordinate, we actually get a constant independent of the radial wave number q:

 

2I₀

 

q tan α

∫

1    1 −

. z'

Σ 2

 

1/2

 

dz' = I₀.             (8.2)

π − q tan α |q tan α |            q tan α    

=

To solve the integral, we substitute z''                 z'/(q tan α ) which yields an integral over a unit semicircle.

In conclusion, there is a signifi cant diff erence between surface imag- ing (and thus depth imaging) and volumetric imaging. The OTF for sur- face structures is independent of the wave number. However, for volu- metric structures, we still have the problem of the decrease of the OTF with the radial wave number. When observing such structures by eye or with a camera, we will not be able to observe fi ne details. Projective imag- ing systems are not designed to image true 3-D objects. Consequently, volumetric imaging requires diff erent techniques.

8.1 Basics                                                                                    207

 

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