Singular-value decomposition of 3D imaging systems

If a 3D object is imaged by a conventional optical system, a 2D image is created. Parts of the object are in focus, and will be clearly visible in the image. Other parts are out of focus, and will be blurred. If several images are taken, all at different positions as shown in Fig. 1, most of the object will be in focus at some point. Using image restoration techniques to combine the images and remove the blurring, it is possible to recover a 3D image of the object. This technique is used e.g. in fluorescence microscopy [1], as a faster alternative to confocal microscopy.

Fig. 1. A 3D object yields a number of 2D images, at different distances from the imaging system.

In this context, one question is how well the object can be retrieved. How much of the object information, and what parts of it, can be transmitted to the reconstructed image? This question can be answered by performing singular value decomposition [2]. Singular-value decomposition (or SVD) is well-known in the context of matrix multiplications, but can be used for both discrete and continuous cases, analytically or numerically depending on the context. Fourier analysis is one example of SVD, where the object- and image-space singular functions are complex exponentials and the singular values are given by the optical transfer function. In our case, the object is regarded as a continuous function, while the image is continuous in the lateral directions and discrete in the axial direction. Our image-space singular functions will be vectors of continuous functions, while the object-space singular functions are continuous.

Fig. 2. The telecentric system.

For our analysis, the telecentric system in Fig. 2 was used. The telecentric system has the advantage of uniform magnification, i.e., the magnification will not change with the distance to the imaging system. For simplicity, only one transverse dimension was considered. For three image planes, the object-space singular functions are shown in Fig. 3. Figure 4 contains an example object, and its expansion in the singular functions. This expansion is the measurement component of the object [3]. An image of the measurement component will be identical to an image of the entire object, so it can be said that the measurement component represents the object information that will be transmitted to the image.

Fig. 3. The object-space singular functions for the telecentric imaging system with three image planes placed at ζ₁=-10 mm, ζ₂=0 mm, and ζ₃=10 mm. The focal length is f=100 mm and the diameter of the aperture is D=10 mm.

Fig. 4. Test object (left) and its measurement component (right). The system parameters are the same as in Fig. 3.

[1] C. Preza, J. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593-1601 (2004).

[2] H.H. Barrett and K.J. Myers, Foundations of image science (Wiley, Hoboken, USA, 2004).

[3] H.H. Barrett, J.N. Aarsvold, and T.J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Lect. Notes. Comput. Sc. 11, 211-226 (1991).