Digital Holography and Digital Image Processing

The history of science is, to a considerable degree, the history of invention, development and perfecting imaging methods and devices. Modern science began with the invention and application of optical telescope and microscope in the beginning of 17-th century by Galileo Galilei (1564–1642), Anthony Leeuwenhoek (1632–1723) and Robert Hook (1635–1703).

Signals and signal processing methods are treated mathematically through mathematical models. The very basic model is that of a mathematical function. For instance, optical signals are regarded as functions that specify relationship between physical parameters of wave fields such as intensity, and phase and parameters of the physical space such as spatial coordinates and/or of time. Fig. 2.1.1 provides classification of signals as mathematical functions and associated terminology.

As it was discussed in Sect. 2.1, signal digitization can be treated in general terms as determination, for each particular signal, of an index of the equivalency cell to which the signal belongs in the signal space and signal reconstruction can be treated as generating a representative signal of the cell from its index. This is, for instance, what we do when we describe everything in our life with words in speaking or writing. In this case this is our brain that makes the job of subdividing “signal space” into the “equivalency cells” of notions and recognizing which word (cell index) corresponds to what we want to describe. The volume of our vocabulary is about 10 ⁵ ÷ 10 ⁶ words. The variety of signals we have to deal with in signal processing and especially in optics and holography is immeasurably larger. One can see this from a simple example of the number of different images of, for instance, 500 × 500 pixels with 256 gray levels. This number is 256 ^500×500 . No technical device will ever be capable of storing so many images for comparing them with input images to be digitized.

Two principles lie in the base of digital representation of continuous signal transformations:

As it was stated in Sect. 4.2, basic formula of digital filtering a signal defined by its samples { a _k , k = 0,1 N _a − 1} by a filter defined by its discrete PSF { h _n n = 0,1..., N _h − 1} is

Fast Fourier Transforms (FFT) are algorithms for fast computation of the DFT. The principle of the FFT can easily be understood if one compares 1-D and 2-D DFT. Let { a _n ⁽¹⁾ } and { a _k,l ⁽²⁾ } be 1-D and 2-D arrays with the same number N = N ₁ N ₂ of samples: n = 0,1 N ₁ N ₂ − 1; k = 0,1 N ₁ − 1; l = 0,1 N ₂ − 1. Direct computing of the 1-D DFT of array { a _n ⁽¹⁾ } requires N ² = N ₁ ² N ₂ ² operations with complex numbers while computing the 2-D DFT of array { a _k,l ⁽²⁾ } requires only N ₁ ² N ₂ + N ₁ N ₂ ² = N ₁ N ₂ ( N ₁ + N ₂ ) operations as the 2-D DFT is separable into two 1-D DFTs: Therefore one can accelerate computing the DFT by representing it in a separable multi-dimensional form. One can do it if the size of the array is a composite number. Let, as in the above example, N = N ₁ N ₂ . Represent indices k and r of signal and its transform samples as two-dimensional ones: Then the 2-D DFT in Eq. 6.1.2 can be split into two successive 1-D DFTs: This computation scheme is illustrated in flow diagram of Fig. 6-1.

By definition, the distribution function P ( V ) of a random variable v is the probability that the random variable does not exceed the value V. The derivative of P ( V ) with respect to V is called the probability distribution density of the random variable v. Digital signals are characterized by discrete analogs of the distribution function and distribution density respectively — the relative share R ( m ) of the samples that do not exceed the given quantized value q, and the rate h ( q ) of the samples having the value q. The latter characteristic is referred to as signal distribution histogram,the former one as cumulative distribution histogram.

Imaging systems always have certain technical limitations in their design and implementations and generate images that are not as perfect as they would be if there were no implementation limitations. Deviations of real images from perfect ones may be treated as distortions introduced by imaging systems to hypothetical perfect, or “ ideal” signals. Correction of these distortions is the primary goal of image processing.

Image resampling is required in many image processing applications. It is a key issue in signal and image differentiating and integrating, image geometrical transformations and re-scaling, target location and tracking with sub-pixel accuracy, Radon Transform and tomographic reconstruction, 3-D image volume rendering and volumetric imaging.

Measurement of physical parameters of objects is one of the most fundamental tasks in image processing. It is required in many applications. Typical examples are measuring the number of objects, their orientations, dimensions and coordinates. A special case of this problem is also object recognition when it is required to determine object index in the list of possible objects. Usually measurement devices and algorithms are designed for the use for arbitrary images from multitude of images generated by image or hologram sensors. Therefore the most appropriate approach to solving this problem is statistical one.

In this chapter we discuss the problem of locating targets in images that contain, besides the target object, a clutter of non-target objects that obscure the target object. As it follows from the discussion in Sect. 10.7, background non-target objects represent the main obstacle for reliable object localization in this case. Our purpose therefore is to find out how can one design a localization device that minimizes the danger of false identification of the target object with one of the non-target objects.

Since J.W. Tukey introduced median filters in signal processing ([1]), a vast variety of nonlinear filters and families of nonlinear filters for image processing has been suggested. In order to ease navigating in this ocean of filters we will provide in this chapter a classification of the filters described in the literature and describe some most useful filters for image denoising, enhancement and segmentation. The classification is aimed at revealing general common principles in the filter design and their efficient implementation in serial computers and parallel computational networks.