Digital Image Processing Fundamentals

<BODY ><DIV ALIGN=LEFT><BR></DIV> <A NAME="Heading1"></A><H1 ALIGN=RIGHT> 5. Digital <A NAME="Index1"></A>Image Processing Fundamental<A NAME="Index2"></A>s </H1> <DIV ALIGN=LEFT><BR></DIV><DIV ALIGN=LEFT><BR></DIV><DIV ALIGN=LEFT><BR></DIV><DIV ALIGN=RIGHT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>There’s more to it than meets the eye. </I></FONT><BR></DIV><DIV ALIGN=RIGHT><FONT FACE="Times" SIZE=3 COLOR="Black">– 19th century proverb </FONT><BR></DIV><DIV ALIGN=RIGHT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=5 COLOR="Black">D</FONT><FONT FACE="Times" SIZE=3 COLOR="Black">igital image processing is electronic data processing on a 2-D array of numbers. The array is a numeric representation of an </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>image</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. A </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>real</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> image is formed on a sensor when an energy emission strikes the sensor with sufficient intensity to create a sensor output. The energy emission can have numerous possible sources (e.g., acoustic, optic, etc.). When the energy emission is in the form of electromagnetic radiation within the band limits of the human eye, it is called visible light [Banerje<A NAME="Index3"></A>e]. Some objects will reflect only electromagnetic radiation. Others produce their own, using a phenomenon called </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>radiancy</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. Radiancy<A NAME="Index4"></A> occurs in an object that has been heated sufficiently to cause it to glow visibly [Resnic<A NAME="Index5"></A>k]. Visible light images are a special case, yet they appear with great frequency in the image processing literature. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Another source of images includes the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>synthetic</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> images of computer graphics. These images can provide controls on the illumination and material properties that are generally unavailable in the real image domain. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">This chapter reviews some of the basic ideas in digital signal processing. The review includes a summary of some mathematical results that will be of use in Chapter 15. The math review is included here in order to strengthen the discourse on sampling. </FONT><BR></DIV> <A NAME="Heading2"></A><H2 ALIGN=LEFT> 5.1. The Human Visual Syste<A NAME="Index6"></A>m </H2> <DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">A typical human visual system consists of stereo electromagnetic transducers (two eyes) connected to a large number of neurons (the brai<A NAME="Index7"></A>n). The neurons<A NAME="Index8"></A> process the input, using poorly understood emergent properties (the min<A NAME="Index9"></A>d). Our discussion will follow the eye, brain and mind ordering, taking views with a selective focus. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The ability of the human eye to perceive the spectral content of light is called color vision. A typical human eye has a spectral response that varies as a function of age and the individual. Using clinical research, the CIE<A NAME="Index10"></A> (Commission Internationale de L’Eclairag<A NAME="Index11"></A>e) created a statistical profile of human vision called the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>standard observe<A NAME="Index12"></A>r</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. The response curves of the standard observer indicate that humans can see light whose wavelengths have the color names red, green and blue. When discussing wavelengths for visible light, we typically give the measurements in </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>nanometers</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. A nanometer is </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=22 SRC="ch500.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> meters and is abbreviated </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>nm</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. The wavelength for the red, green and blue peaks are about 570-645 nm, 526-535 nm, and 444-445 nm. The visible wavelength range (called the mesopic<A NAME="Index13"></A> range) is 380 to about 700-770 nm [Netraval<A NAME="Index14"></A>i] [Cohe<A NAME="Index15"></A>n]. </FONT><BR></DIV><DIV ALIGN=CENTER><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=148 WIDTH=226 SRC="ch501.gif"></FONT><BR></DIV><DIV ALIGN=CENTER><FONT FACE="Times" SIZE=3 COLOR="Black">Fig. 5-1. Sketch of a Human Ey<A NAME="Index16"></A>e</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Fig. 5-1 shows a sketch of a human eye. When dimensions are given, they refer to the typical adult human eye unless otherwise stated. Light passes through the cornea and is focused on the retina by the lens. Physiological theories use biological components to explain behaviour. The optical elements in the eye (corne<A NAME="Index17"></A>a, lens<A NAME="Index18"></A> and retin<A NAME="Index19"></A>a) form the primary biological component<A NAME="Index20"></A>s of a photo sensor. Muscles are used to alter the thickness of the lens and the diameter of the hole covering the lens, called the iri<A NAME="Index21"></A>s. The iris diameter typically varies from 2 to 8 mm. Light passing through the lens is focused upon the retin<A NAME="Index22"></A>a. The retina contains two types of photo sensor cells: rods<A NAME="Index23"></A> and cone<A NAME="Index24"></A>s. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">There are 75 to 150 million rod cell<A NAME="Index25"></A>s in the retina. The rods contain a blue-green absorbing pigment called rhodopsi<A NAME="Index26"></A>n. Rods are used primarily for night vision (also called the scotopic rang<A NAME="Index27"></A>e) and typically have no role in color vision [Gonzalez and Wood<A NAME="Index28"></A>s].</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Cones are used for daylight vision (called the photopic range). The tristimulus theory of color perception is based upon the existence of three types of cones: red, green and blue. The pigment in the cones is unknown [Hun<A NAME="Index29"></A>t]. We do know that the phenomenon called adaptation<A NAME="Index30"></A> (a process that permits eyes to alter their sensitivity) occurs because of a change in the pigments in the cones [Netraval<A NAME="Index31"></A>i]. The retina cells may also inhibit each another from creating </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><B> </B></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">a high-pass filter for image sharpening. This phenomenon is known as lateral inhibitio<A NAME="Index32"></A>n [Myler<A NAME="Index33"></A>s].</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The current model for the retinal cell<A NAME="Index34"></A>s shows a cone cell density that ranges from 900 </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=51 SRC="ch502.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> to 160,000 </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=51 SRC="ch502.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> [Gibson]. There are 6 to 7 million cone cell<A NAME="Index35"></A>s, with the density increasing near the fove<A NAME="Index36"></A>a. Further biological examinatio<A NAME="Index37"></A>n indicates that the cells are imposed upon a noisy hexagonal array<A NAME="Index38"></A> [Wehmeie<A NAME="Index39"></A>r]. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Lest one be tempted to count the number of cells in the eye and draw a direct comparison to modern camera equipment, keep in mind that even the fixated eye is constantly moving. One study showed that the eyes perform over 3 fixations per second during a search of a complex scene [William<A NAME="Index40"></A>s]. Further more, there is nearly a 180-degree field of view (given two eyes). Finally, the eye-brain interface enables an integration between the sensors’ polar coordinate scans, focu<A NAME="Index41"></A>s, iris<A NAME="Index42"></A> adjustments and the interpretation engine. These interactions are not typical of most artificial image processing systems [Gonzalez<A NAME="Index43"></A> and Woods]. Only recently have modern camcorders taken on the role of integrating the focus and exposure adjustment with the sensor. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The optic nerve has approximately 250,000 neurons connecting to the brain. The brain has two components associated with low-level vision<A NAME="Index44"></A> operations: the lateral geniculate nucleus and the visual corte<A NAME="Index45"></A>x. The cells are modeled using a circuit that has an inhibit input, capacitive-type electrical storage and voltage leaks, all driving a comparitor with a variable voltage output. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The capacitive storage elements are held accountable for the critical fusion frequency response of the eye. The critical fusion frequency is the rate of display whereby individual updates appear as if they are continuous. This frequency ranges from 10-70 Hz depending on the color [Teeva<A NAME="Index46"></A>n] [Netraval<A NAME="Index47"></A>i]. At 70 Hz, the 250,000-element optic nerve should carry 17.5 million neural impulses per second. Given the signal-to-noise ratio of a human auditory response system (80 dB), we can estimate that there are 12.8 bits per nerve leading to the brain [Shamm<A NAME="Index48"></A>a]. This gives a bit rate of about 224 Mbps. The data has been pre-processed by the eye before it reaches the optic nerve. This preprocessing includes lateral inhibition between the retinal neurons. Also, we have assumed that there is additive white Gaussian noise on the channel, but this assumption may be justified. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Physiological study has shown that the response of the cones is given by a Gaussian sensitivity for the cone center and surrounding fields. The overall sensitivity is found by subtracting the surrounding response from the center response. This gives rise to a difference of Gaussian expression which is discussed in Chap. 10. Further, the exponential response curve of the eye is the primary reason why exponential histogram equalization was used in Chap. 4. </FONT><BR></DIV> <A NAME="Heading3"></A><H2 ALIGN=LEFT> 5.2. Overview of Image Processing </H2> <DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">An image processing system consists of a source of image data, a processing element and a destination for the processed results. The source of image data may be a camera, a scanner, a mathematical equation, statistical data, the Web, a SONAR system, etc. In short, anything able to generate or acquire data that has a two-dimensional structure is considered to be a valid source of image data. Furthermore, the data may change as a function of time. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The processing element is a computer. The computer may be implemented in a number of different ways. For example, the brain may be said to be a kind of biological computer that is able to perform image processing (and do so quite well!). The brain consumes about two teaspoons of sugar and 20 watts of power per hour. An optical element can be used to perform computation and does so at the speed of light (and with very little power). This is a fascinating topic of current research [Fietelso<A NAME="Index49"></A>n]. In fact, the injection of optical computing elements can directly produce information about the range of objects in a scene [DeWitt and Lyo<A NAME="Index50"></A>n].</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Such computing elements are beyond the scope of this book. The only type of computer that we will discuss in this book is the digital computer. However, it is interesting to combine hybrid optical and digital computing. Such an area of endeavor lies in the field of </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>photonic<A NAME="Index51"></A>s</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">.</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The output of the processing may be a display, created for the human visual system. Output can also be to any </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>stream</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. In Java, a stream is defined as an uninterpreted sequence of bytes. Thus, the output may not be image data at all. For example, the output can be a histogram, a global average, etc. As the output of the program renders a higher level of interpretation, we cross the fuzzy line from image processing into the field of </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>vision</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. As an example, consider that image processing is used to edge detect the image of coins on a table. Computer vision is used to tell how much money is there. Thus, computer vision will often make use of image processing as a sub-task. </FONT><BR></DIV> <A NAME="Heading4"></A><H3 ALIGN=LEFT> 5.2.1. Digitizing a Signal<A NAME="Index52"></A> </H3> <DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Digitizing is a process that acquires quantized samples of continuous signals. The signals represent an encoding of some data. For example, a microphone is a pressure transducer that produces an electrical signal. The electrical signal represents acoustic pressure waves (sound). </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The term </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>analog</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> refers to a signal that has a continuously varying pattern of intensity. The term </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>digital</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> means that the data takes on discrete values. Let </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>s</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">(</FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>t</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">) be a continuous signal. Then, by definition of continuous, </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=30 WIDTH=113 SRC="ch503.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.1)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">We use the symbol </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>R</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> to denote the set of real numbers. Thus </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=134 SRC="ch504.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black">, which says that </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>R</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is the set of all </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> such that </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is a real number. We read (5.1) saying, in the limit, as </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>t</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> approaches </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>a,</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> such that </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>a</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is a member of the set of real numbers, </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=52 SRC="ch505.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black">. The expression </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=16 WIDTH=49 SRC="ch506.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> is read as “the set of all </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">’s such that </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>P(x)</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is true” [ </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">Moore<A NAME="Index53"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">64].</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">This is an </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>iff</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> (i.e., if and only if) condition. Thus, the converse must also be true. That is, </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=20 SRC="ch507.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">is not continuous iff there exists a value, </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>a</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> such that: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=30 WIDTH=70 SRC="ch508.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.2)</FONT><BR></DIV><DIV ALIGN=LEFT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black">is true. </FONT></SUB><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">For example, if </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=20 SRC="ch509.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> has multiple values at </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>a</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">, then the limit does not exist at </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>a</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">.</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The analog-to-digital conversion consists of a sampler and a quantizer. The quantization is typically performed by dividing the signal into several uniform steps. This has the effect of introducing </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>quantization<A NAME="Index54"></A> noise </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. Quantization noise is given, in dB, using </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=12 WIDTH=75 SRC="ch510.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.3)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">where </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>SNR </I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is the signal-to-noise ratio and </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>b</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is the number of bits. To prove (5.3), we follow [ </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">Moor<A NAME="Index55"></A>e</FONT><FONT FACE="New York" SIZE=3 COLOR="Black">] and assume that the input signal ranges from -1 to 1 volts. That is, </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=156 SRC="ch511.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3a)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">Note that the number of quantization intervals is </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=13 WIDTH=13 SRC="ch512.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black">. The least significant bit has a quantization size of </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=18 WIDTH=44 SRC="ch513.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> . Following [ </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">Mitr<A NAME="Index56"></A>a</FONT><FONT FACE="New York" SIZE=3 COLOR="Black">], we obtain the bound on the size of the error with: </FONT><BR></DIV><DIV ALIGN=LEFT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> <IMG HEIGHT=16 WIDTH=66 SRC="ch514.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3b)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">The variance of a random variable, </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>X</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">, is found by </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=17 WIDTH=107 SRC="ch515.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> where </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=29 SRC="ch516.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> is a probability distribution function. For the signal whose average is zero, the variance of (5.3b) is </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=33 WIDTH=127 SRC="ch517.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3c).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">The signal-to-noise ratio for the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">quantization<A NAME="Index57"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">power is </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=19 WIDTH=206 SRC="ch518.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3d)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">Hence the range on the upper bound for the signal-to-quantization noise power is </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=12 WIDTH=75 SRC="ch510.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">Q.E.D.</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">In the above proof we assumed that uniform steps were used over a signal whose average value is zero. In fact, a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">digitizer<A NAME="Index58"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">does not have to requantize an image so that steps are uniform. An in-depth examination of the effects of non-linear quantization on SNR is given in [Gersho]. Following Gersho, we generalize the result of (5.3), defining the SNR as </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=34 WIDTH=152 SRC="ch519.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3e)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">where </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=12 WIDTH=112 SRC="ch520.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">and </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=37 SRC="ch521.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> is the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>mean-square distortio<A NAME="Index59"></A>n</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I> </I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> defined by the inner product between the square of the quantization error for value </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> and the probability of value </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">. The inner product between </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>e</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> and </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>p</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> is given by </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=111 SRC="ch522.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3f).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">where</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=18 WIDTH=89 SRC="ch523.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black"> (5.3g).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">The inner product is an important tool in transform theory. We will expand our discussion of the inner product when we touch upon the topic of sampling. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">We define </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>Q(x)</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> as the quantized value for </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>x</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">. Maximizing </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">SNR<A NAME="Index60"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">requires that we select the quantizer to minimize (5.3f), given </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>a priori </I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> knowledge of the PDF (if the PDF is available). Recall that for an image, we compute the PMF (using the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Histogram<A NAME="Index61"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">class) as well as the CMF. As we shall see later, (5.3f) is minimized for </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>k</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">-level thresholding (an intensity reduction to </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>k</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> colors) when the regions of the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">CMF<A NAME="Index62"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">are divided into </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><I>k</I></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> sections. The color is then remapped into the center of each of the CMF regions. Hence (5.3f) provides a mathematical basis for reducing the number of colors in an image provided that the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">PDF<A NAME="Index63"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">is of zero mean (i.e, no DC offset) and has even symmetry about zero. That is </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=67 SRC="ch524.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black">. Also, we assume that the quantizer has odd symmetry about zero, i.e., </FONT><SUB><FONT FACE="New York" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=76 SRC="ch525.gif"></FONT></SUB><FONT FACE="New York" SIZE=3 COLOR="Black">. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">A simple zero-memory 4-point </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">quantizer<A NAME="Index64"></A> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black">inputs 4 decision levels and outputs 4 corresponding values for input values that range within the 4 decision levels. When the decision levels are placed into an array of </FONT><FONT FACE="Times" SIZE=3 COLOR="Black">double precisio<A NAME="Index65"></A>n</FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> numbers, in Java (for the 256 gray-scale values) we write: </FONT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>public</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>void</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> thresh<A NAME="Index66"></A>4(</FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>double</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> d[]) { </FONT></TT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>short</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> lut[] = </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>new</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> short[256]; </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>if</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> (d[4] ==0) </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>for</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> ( </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>int</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> i=0; i < lut.length; i++) { </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>if</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> (i < d[0]) lut[i] = 0; </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>else</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>if</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> (i < d[1]) </FONT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black">lu<A NAME="Index67"></A>t</FONT></TT><FONT FACE="New York" SIZE=3 COLOR="Black">[i] = ( </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>short</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">)d[0];</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>else</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>if</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> (i < d[2]) lut[i] = ( </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>short</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">)d[1];</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>else</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>if</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> (i < d[3]) lut[i] = ( </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>short</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black">)d[2];</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><FONT FACE="New York" SIZE=3 COLOR="Black"><B>else</B></FONT><FONT FACE="New York" SIZE=3 COLOR="Black"> lut[i] = 255; </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> System.out.println(lut[i]); </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> } </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black"> </FONT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black">applyLu<A NAME="Index68"></A>t</FONT></TT><FONT FACE="New York" SIZE=3 COLOR="Black">(lut);</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="New York" SIZE=3 COLOR="Black">}</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">We shall revisit quantization in Section 5.2.2. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Using the Java AWT’s Image class, we have seen that 32 bits are used, per pixel (re<A NAME="Index69"></A>d, gree<A NAME="Index70"></A>n, blue<A NAME="Index71"></A> and alph<A NAME="Index72"></A>a). There are only 24 bits used per colo<A NAME="Index73"></A>r, however. Section 5.2.2 shows how this relates to the software of this book. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Recall also that the digitization process led to sampling an analog signal. Sampling a signal alters the harmonic content (also known as the spectr<A NAME="Index74"></A>a) of the signal. Sampling<A NAME="Index75"></A> a continuous signal may be performed with a pre-filter and a switch. Fig. 5-2 shows a continuous function, </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=25 SRC="ch526.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">, being sampled at a frequency of </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=13 SRC="ch527.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">. </FONT><BR></DIV><DIV ALIGN=CENTER><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=107 WIDTH=296 SRC="ch528.gif"></FONT></SUB><BR></DIV><DIV ALIGN=CENTER><FONT FACE="Times" SIZE=3 COLOR="Black">Fig. 5-2. Sampling System </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The switch in Fig. 5-2 is like a binary amplifier that is being turned on and off every </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=26 SRC="ch529.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> seconds. It multiplies </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=25 SRC="ch526.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> by an amplification factor of zero or one. Mathematically, sampling<A NAME="Index76"></A> is expressed as a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>pulse trai<A NAME="Index77"></A>n, </I></FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=25 SRC="ch530.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">, multiplied by the input signal </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=25 SRC="ch526.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">, i.e., sampling is </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=25 SRC="ch526.gif"><IMG HEIGHT=14 WIDTH=25 SRC="ch530.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">..</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">To discuss the pulse train mathematically, we must introduce the notation for an impulse. The </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>unit impulse<A NAME="Index78"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">, or </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Dirac delt<A NAME="Index79"></A>a,</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> is a generalized function that is defined by </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=116 SRC="ch531.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.4) </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">where </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=9 WIDTH=9 SRC="ch532.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> is arbitrarily small. The Dirac delta has unit area about a small neighborhood located at </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=12 WIDTH=28 SRC="ch533.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">. Multiply the Dirac delta by a function and it will </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>sift</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> out the values where the Dirac delta is equal to zero: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=179 SRC="ch534.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.5)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">This is called the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>sifting<A NAME="Index80"></A> property </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> of the Dirac delt<A NAME="Index81"></A>a. In fact, the Dirac<A NAME="Index82"></A> delta is equal to zero whenever its argument is non-zero. To make the Dirac activate, given a non-zero argument, we bias the argument with an offset, </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=16 WIDTH=59 SRC="ch535.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">. A pulse train is created by adding an infinite number of Dirac delt<A NAME="Index83"></A>as together: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=114 SRC="ch536.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.6)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=159 SRC="ch537.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.7)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">To find the spectra of (5.7) requires that we perform a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Fourier<A NAME="Index84"></A> transform </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. The Fourier transfor<A NAME="Index85"></A>m, just like any transform, performs a correlation between a function and a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>kerne<A NAME="Index86"></A>l</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. The kernel of a transform typically consists of an </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>orthogonal<A NAME="Index87"></A> basis<A NAME="Index88"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> about which the reconstruction of a waveform may occur. Two functions are orthogonal if their inner produc<A NAME="Index89"></A>t </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=14 WIDTH=38 SRC="ch538.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> =0. Recall that the inner product is given by </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=191 SRC="ch539.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.7a)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">From linear algebra, we recall that a collection of </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>linearly<A NAME="Index90"></A> independent<A NAME="Index91"></A> </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">functions forms a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>basis<A NAME="Index92"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> if every value in the set of all possible values may be expressed as a linear combination of the basis set. Functions are linearly independent </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>iff</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> the sum of the functions is non-zero (for non-zero co-efficients). Conversely, functions are linearly dependent </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>iff</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> there exists a combination of non-zero coefficients for which the summation is zero. For example: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=114 SRC="ch540.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.7b)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The ability to sum a series of sine and cosine functions together to create an arbitrary function is known an the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>super position </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> principle and applies only to periodic waveforms. This was discovered in the 1800’s by Jean Baptiste Joseph de Fourie<A NAME="Index93"></A>r [Hallida<A NAME="Index94"></A>y] and is expressed as a summation of sine and cosines, with constants that are called </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Fourier coefficients<A NAME="Index95"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">.</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=146 SRC="ch541.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.7c)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">We note that (5.7c) shows that the periodic signal has discrete spectral components. We find the Fourier coefficients by taking the inner product of the function, </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>f(x)</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> with the basis functions, sine and cosine. That is: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=33 WIDTH=87 SRC="ch542.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.7d)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">For an elementary introduction to linear algebra, see [Anto<A NAME="Index96"></A>n]. For a concise summary see [Stollnit<A NAME="Index97"></A>z]. For an alternative derivation see [Lyon and Ra<A NAME="Index98"></A>o].</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">It is also possible to approximate an </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>aperiodic<A NAME="Index99"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> waveform. This is done with the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Fourier transform </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. The Fourier transform uses sine and cosine as the basis functions to form the inner product, as seen in (5.7a): </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=181 SRC="ch543.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.8).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">By Euler’s identity, </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> <IMG HEIGHT=14 WIDTH=91 SRC="ch544.gif"> (5.9)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">we see that the sine and cosine basis functions are separated by being placed on the real and imaginary axis. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Substituting (5.7) into (5.8) yields </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=232 SRC="ch545.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.10)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">where</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=166 SRC="ch546.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.11)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The term </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=157 SRC="ch547.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.12)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">defines a convolution. We can write (5.10) because multiplication in the time domain is equivalent to convolution in the frequency domain. This is known as the convolution theorem. Taking the Fourier transform of the convolution between two functions in the time domain results in </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=225 SRC="ch548.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.13)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">which is expanded by (5.8) to yield: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=38 WIDTH=163 SRC="ch549.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.13a)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Changing the order of integration in (5.13a) yields </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=38 WIDTH=162 SRC="ch550.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.13b)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">with</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=129 SRC="ch551.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.13c)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">and</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=111 SRC="ch552.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.13d)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">we get </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=35 WIDTH=165 SRC="ch553.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.14).</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">This shows that convolution in the time domain is multiplication in the frequency domain. We can also show that convolution in the frequency domain is equal to multiplication in the time domain. See [Carlso<A NAME="Index100"></A>n] for an alternative proof. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">As a result of the convolution theorem, the Fourier transform of an impulse train is also an impulse train, </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=179 SRC="ch554.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.15)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Finally, we see that sampling<A NAME="Index101"></A> a signal at a rate of </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=13 SRC="ch555.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> causes the spectrum to be reproduced at </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=13 SRC="ch555.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> intervals: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=32 WIDTH=148 SRC="ch556.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.16)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">(5.16) demonstrates the reason why a band limiting filter is needed before the switching function of Fig. 5-2. This leads directly to the sampling theorem which states that a band limited signal may be reconstructed without error if the sample rate is twice the bandwidth. Such a sample rate is called the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>Nyquist</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>rate</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> and is given by </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=15 WIDTH=58 SRC="ch557.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black">.</FONT><BR></DIV> <A NAME="Heading5"></A><H3 ALIGN=LEFT> 5.2.2. Image Digitization </H3> <DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Typically, a camera is used to digitize an image. The modern CCD cameras have a photo diode arranged in a rectangular array. Flat-bed scanners use a movable platen and a linear array of photo diode<A NAME="Index102"></A>s to perform the two-dimensional digitization. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Older tube type cameras used a wide variety of materials on a photosensitive<A NAME="Index103"></A> surface. The materials vary in sensitivity and output. See [Galbiat<A NAME="Index104"></A>i] for a more detailed description on tube cameras. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The key point about digitizing an image in two dimensions is that we are able to detect both the power of the incident energy as well as the direction. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The process of digitizing an image is described by the amount of spatial resolution and the signal -to-noise ratio (i.e., number of bits per pixel) that the digitizer has. Often the number of bits per pixel is limited by performing a </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>thresholding</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">. Thresholding (a topic treated more thoroughly in Chap. 10) reduces the number of color values available in an image. This simulates the effect of having fewer bits per pixel available for display. There are several techniques available for thresholding. For the grayscale image, one may use the cumulative mass function for the probability of a gray value to create a new look-up table. Another approach is simply to divide the look-up table into uniform sections. Fig. 5-2 shows the mandrill before and after thresholding operation. The decision about when to increment the color value was made based on the CMF<A NAME="Index105"></A> of the image. The number of bits per pixel (bpp), shown in Fig. 5-2, ranging from left to right, top to bottom, are: 1 bpp, 2 bpp, 3 bpp and 8 bpp. Keep in mind that at a bit rate of 28 kbps (the rate of a modest Internet connection over a phone line) the 8 bpp image (128x128) will take 4 seconds to download. Compare this to the uncompressed 1 bpp image which will take 0.5 seconds to download. Also note that the signal-to-noise ratio for these images ranges from 10 dB to 52 dB. </FONT><BR></DIV><DIV ALIGN=LEFT><BR></DIV><DIV ALIGN=CENTER><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=254 WIDTH=250 SRC="ch558.gif"></FONT><BR></DIV><DIV ALIGN=CENTER><FONT FACE="Times" SIZE=3 COLOR="Black">Fig. 5-3. Quantizing with Fewer Bits Per Pixel </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The code snippet allows the cumulative mass functio<A NAME="Index106"></A>n of the image to bias decisions about when to increment the color value. The input to the code is the number of gray values, </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>k. </I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black">There are several methods to perform the quantization. The one shown in Fig. 5-3 is useful in edge detectio<A NAME="Index107"></A>n (a topic covered in Chap. 10). The </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>kgreyThresh<A NAME="Index108"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> method follows: </FONT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>public</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>void</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> kgreyThres<A NAME="Index109"></A>h(</FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>double</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> k) { </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> Histogram<A NAME="Index110"></A> rh = </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>new</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> Histogram(r,"red"); </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>double</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> cmf[] = rh.getCM<A NAME="Index111"></A>F();</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> TransformTable<A NAME="Index112"></A> tt = </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>new</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> TransformTabl<A NAME="Index113"></A>e(cmf.length);</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>short</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> lut[] = tt.getLu<A NAME="Index114"></A>t();</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>int</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> q=1; </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>short</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> v=0; </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>short</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> dv = ( </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>short</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black">)(255/k);</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>for</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> ( </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>int</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> i=0; i < lut.length; i++) { </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>if</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> (cmf[i] > q/k) { </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> v += dv; </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> q++; </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><I>//(k == q+1)|| </I></FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> </FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"><B>if</B></FONT></TT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> (q==k) v=255; </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> } </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> lut[i]=v; </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> } </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> tt.setLu<A NAME="Index115"></A>t(lut);</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> tt.cli<A NAME="Index116"></A>p();</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> tt.print(); </FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black"> applyLu<A NAME="Index117"></A>t(lut);</FONT></TT><BR></DIV><DIV ALIGN=LEFT><TT><FONT FACE="Courier" SIZE=3 COLOR="Black">}</FONT></TT><BR></DIV><DIV ALIGN=LEFT><BR></DIV> <A NAME="Heading6"></A><H3 ALIGN=LEFT> 5.2.3. Image Displa<A NAME="Index118"></A>y</H3> <DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">One display device that has come into common use is the cathode-ray tube (CR<A NAME="Index119"></A>T). The cathode ray tube displays an image using three additive colors: red, green and blue. These colors are emitted using phosphors that are stimulated with a flow of electrons. Different phosphors have different colors (spectral radiance). </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">There are three kinds of television systems in the world today, NTS<A NAME="Index120"></A>C, PAL<A NAME="Index121"></A> and SECAM. NTSC<A NAME="Index122"></A> which stands for National Television Subcommitte<A NAME="Index123"></A>e, is used in North America and Japan. PAL stands for phase alternating line and is used in parts of Europe, Asia, South Americ<A NAME="Index124"></A>a and Afric<A NAME="Index125"></A>a. SECAM<A NAME="Index126"></A> stands for Sequential Couleur à Mémori<A NAME="Index127"></A>e (sequential chrominance signal and memory) and is used in France, Eastern Europ<A NAME="Index128"></A>e and Russi<A NAME="Index129"></A>a.</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The gamut<A NAME="Index130"></A> of colors and the reference color known as </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>white</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> (called white balance) are different on each of the systems. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Another type of display held in common use is the computer monitor. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Factors that afflict all displays include: ambient light, brightness (black level) and contrast (picture). There are also phosphor chromaticity differences between different CRTs. These alter the color gamut that may be displayed. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Manufacturers’ products are sometimes adopted as a standard for the color gamut to be displayed by all monitors. For example, one U.S. manufacturer, Conrac, had a phosphor that was adopted by SMPTE (Society of Motion Picture and Television Engineer<A NAME="Index131"></A>s) as the basis for the SMPTE<A NAME="Index132"></A> C phosphors. </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">The CRTs have a transfer function like that of (4.14), assuming the value, </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>v</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> ranges from zero to one: </FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black"> </FONT><SUB><FONT FACE="Times" SIZE=3 COLOR="Black"><IMG HEIGHT=16 WIDTH=47 SRC="ch559.gif"></FONT></SUB><FONT FACE="Times" SIZE=3 COLOR="Black"> (5.3)</FONT><BR></DIV><DIV ALIGN=LEFT><FONT FACE="Times" SIZE=3 COLOR="Black">Typically, this is termed the </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>gamma<A NAME="Index133"></A></I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> of a monitor and runs to a value of 2.2 [Blinn]. As Blinn points out, for a gamma of 2, only 194 values appear in a look-up table of 256 values. His suggestion that 16 bits per color might be enough to perform image processing has been taken to heart, and this becomes another compelling reason to use the Java </FONT><FONT FACE="Times" SIZE=3 COLOR="Black"><I>short</I></FONT><FONT FACE="Times" SIZE=3 COLOR="Black"> for storing image values. Thus, the image processing software in this book does all its image processing as if intensity were linearly related to the value of a pixel. With the storage of 48 bits per pixel (for red, green and blue) versus the Java AWT model of 24 bits per red, green and blue value, we have increased our signal-to-noise ratio for our image representation by 48 dB per color. So far, we have not made good use of this extra bandwidth, but it is nice to know that it is there if we need it. </FONT><BR></DIV><DIV ALIGN=LEFT><BR></DIV> </BODY>