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Chapter 2:
Fourier Integral Representation of an Optical Image

This chapter describes optical transfer functions. The concepts of linearity and shift invariance were introduced in Chapter 1. This chapter continues that discussion by applying those concepts to optical imaging components and systems.

Images are two-dimensional and are accurately described by twodimensional Fourier integrals. Common practice, however, is to analyze or measure horizontal and vertical frequency response and then use the results to characterize imager performance. This chapter describes the errors that result from assuming that an imager is characterized by its horizontal and vertical frequency response.

Throughout this tutorial, the mathematics is at the introductory calculus level. However, the descriptive arguments require some familiarity with the concepts of Fourier analysis and complex functions.

2.1 Linear Shift-Invariant Optical Systems

In Fig. 2.1, a simple optical system is imaging a clock onto a screen. For simplicity, unity magnification is assumed. If each point in the scene is blurred by the same amount, then the system is shift invariant. If the image intensity profile equals the sum of the individual blurs from each point in the scene, then the system is linear.

The optical blur is called the point spread function (psf). The psf is illustrated in the lower left corner of the image. Each point source in the scene becomes a psf in the image. The psf is also called the impulse response of the system. Each point in the scene is blurred by the optics and projected onto the screen. This process is repeated for each of the infinite number of points in the scene. The image is the sum of all of the individual psf's.

Two considerations are important here. First, the process of the lens imaging the scene is linear and, therefore, superposition holds.

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