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## Über dieses Buch

th Although photography has its beginning in the 17 century, it was only in the 1920’s that photography emerged as a science. And as with other s- ences, mathematics began to play an increasing role in the development of photography. The mathematical models and problems encountered in p- tography span a very broad spectrum, from the molecular level such as the interaction between photons and silver halide grains in image formation, to chemical processing in ?lm development and issues in manufacturing and quality control. In this book we present mathematical models that arise in today’s p- tographic science. The book contains seventeen chapters, each dealing with oneareaofphotographicscience.Eachchapter,exceptthetwointroductory chapters, begins with general background information at a level understa- able by graduate and undergraduate students. It then proceeds to develop a mathematical model, using mathematical tools such as Ordinary Di?erential Equations, Partial Di?erential Equations, and Stochastic Processes. Next, some mathematical results are mentioned, often providing a partial solution to problemsraisedby the model.Finally,mostchaptersinclude problems.By the nature of the subject, there is quite a bit ofdisparity in the mathematical level of the various chapters.

## Inhaltsverzeichnis

### Introduction

Abstract
Although photography has its beginning in the 17th century, it was only in the 1920’s that photography emerged as a science. And as with other sciences, mathematics began to play an increasing role in the development of photography. The mathematical models and problems encountered in photography span a very broad spectrum, from the molecular level such as the interaction between photons and silver halide grains in image formation, to chemical processing in film development and issues in manufacturing and quality control.
Avner Friedman, David S. Ross

### 1. History of Photography

Abstract
This is a book of mathematical models related to photographic science and engineering. More specifically, it is book of mathematical models of several of the chemical aspects of traditional photography. We say traditional photography to contrast our subject matter from digital photography, in which images are captured by arrays of CCDs, charge coupled devices — computer chips. We note that we are concerned with the chemical aspects of traditional photography, because we will not cover the other important aspect of traditional photography, namely, optics.
Avner Friedman, David S. Ross

### 2. An Overview

Abstract
Gelatin is a bland jellylike protein derived by boiling the skin, bones and other tissues of animals — primarily cows, in the case of the gelatin used in photographic film. It is used as a thickener and a stabilizer in food and pharmaceutical products, as well as in photographic film.
Avner Friedman, David S. Ross

### 3. Crystal Growth — Ostwald Ripening

Abstract
The silver halide crystals of photographic emulsions are precipitated from solutions of halides and silver salts. For example, if we mix solutions of silver nitrate (AgNO3) and potassium bromide (KBr), the following reaction takes place:
AgNO3 (solution)+KBr (solution) → AgBr (solid)+KNO3 (solution).
Avner Friedman, David S. Ross

### 4. Crystal Growth-Sidearm Precipitation

Abstract
In Chap. 3 we described Ostwald ripening, an important process of the crystallization. In this section we describe a model of another process of crystal growth.
Avner Friedman, David S. Ross

### 5. Gelatin Swelling

Abstract
In Chap. 2 we described emulsion layers, which contain silver halide grains and oil droplets in gelatin. The silver halide grains capture light, i.e., they record the arrival of photons that enter the shutter when a photograph is taken. The oil droplets contain dye couplers which, during the development of the film, produce colored dye. Emulsion layers also contain other ingredients. Whereas the silver halide grains and oil droplets occupy a substantial volume of the emulsion, the other materials occupy a negligible volume.
Avner Friedman, David S. Ross

### 6. Gelation

Abstract
As we described earlier, photographic film is made up of a polymeric base onto which emulsion and gelatin layers are coated; the emulsions contain silver halide grains, oil droplets (containing coupler) and other chemicals. In this chapter we consider the process of making the solid gelatin from solution of aqueous gelatin; this process is called gelation. It proceeds by molecular cross-linking in the initial gelatin solution.
Avner Friedman, David S. Ross

### 7. Polymeric Base

Abstract
The base of photographic film is a transparent polymeric material upon which several thin layers of photographic emulsion are coated. When a flat polymeric film is bent to some fixed curvature, held at this state for some time, and then released, its curvature is observed to drop instantaneously to some non-zero value and then gradually decrease with time. This phenomenon is generally referred to as bending recovery. Bending of polymer films and the associated recovery phenomena are critical in vaious packaging, forming and finishing operations. One of the considerations in choosing material for film base is to achieve desired bending recovery.
Avner Friedman, David S. Ross

### 8. Limited Coalescence

Abstract
In aerosol dynamics one models the evolution of the number density n(x, t) of particles of volume x at time t by the equation
$$\begin{gathered} \frac{{\partial n(x,t)}} {{\partial t}} = - n(x,t)\int_0^\infty {\phi (x,\xi )n(\xi ,t)d\xi } \hfill \\ + \tfrac{1} {2}\int_0^x {\phi (x - \xi ,\xi )n(x - \xi ,t)n(\xi ,t)d\xi } \hfill \\ \end{gathered}$$
(1)
where φ(x, ξ) is the collision rate between particles of sizes x and ξ; here the first term on the right-hand side expresses loss of particles of size x due to coalescence with particles of any size ξ, and the second integral expresses the gain of particles of size x through coalescence of particles of sizes ξ and x — ξ with ξ ≤ x — ξ; the factor ½ is introduced when we remove the restriction ξ ≤ x — ξ.
Avner Friedman, David S. Ross

### 9. Measuring Coalescence

Abstract
Recall that an emulsion layer in photographic film contains silver halide grains and oil droplets in suspension. The oil droplets contain couplers, chemicals that form colored dyes during the development of the film.
Avner Friedman, David S. Ross

### 10. Newtonian Coating Flows

Abstract
Figure 10.1 shows the method of coating a moving substrate by allowing a liquid curtain under the influence of gravity to fall on it.
Avner Friedman, David S. Ross

### 11. Coating Configurations

Abstract
In Chapter 10 we considered curtain coating on a moving substrate. There are actually several different technologies for coating. In this chapter we describe coating by extrusion die; this is one of the most commonly used methods for coating photographic emulsion. The substrate, often referred to as a web, is the polymeric film base.
Avner Friedman, David S. Ross

### 12. Curtain Coating

Abstract
In this chapter we consider several issues related to curtain coating: (i) The stabilization of the curtain through the reduction of the surface tension of the liquid/air interface; (ii) The measurement of the dynamic surface tension of that interface; and (iii) The response of the curtain to pressure fluctuations.
Avner Friedman, David S. Ross

### 13. Shear Thinning

Abstract
Colloidal dispersions consist of small particles, with linear dimensions generally around 0.1 μm to 10 μm, dispersed in a liquid such as water or an organic solvent. The particles interact with one another as well as with the fluid. The interactions include hydrodynamic and non-hydrodynamic forces; the particles may also undergo Brownian motion.
Avner Friedman, David S. Ross

### 14. Latent Image Formation

Abstract
Three basic types of silver halide, silver bromide, silver iodide and silver chloride are used in photographic film and paper. In some applications, combinations, for example, silver-bromo-iodide, are used.
Avner Friedman, David S. Ross

### 15. Granularity

Abstract
Granularity is a measure of fluctuations in light transmittance through an aperture as it is scanned over an image. The quality of an image depends upon its granularity; everyone has seen graining photographs — graininess is particularly noticable in enlargements. The smaller the granularity the better the quality of the image. For images captured on photographic film the underlying image structure is that of a random medium whose properties fluctuate from point to point. In order to improve image quality one needs to analyze the transport, scattering, and absorption of light in such media; absorption depends on the first moment and scattering depends on the second moments of the transmitted light.
Avner Friedman, David S. Ross

### 16. A Reaction-Diffusion System

Abstract
In development the film is immersed in an aqueous solution as shown schematically in Fig. 16.1 which, for simplicity, depicts a three-layer film. The emulsion layers contain silver halide grains and oil droplets. The oil droplets contain chemicals — couplers — that form dye and inhibitor in the course of development.
Avner Friedman, David S. Ross

### 17. Parameter Identification

Abstract
The system (16.1) – (16.8) contains a number of unknowns: The development function, λ, and a number of diffusion coefficients and reaction rates. In this chapter we address the question of how to obtain diffusion and reaction rates experimentally.
Avner Friedman, David S. Ross

### Backmatter

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