foueier optics课件c2 analysis of two-dimensional signals and system.ppt

CHAPTER 2 Analysis of Two-Dimensional Signals and Systems,Some Frequently Used Functions Convolution and correlation Fourier Transform Some Useful Fourier Transform Pairs,Frequently Used Functions -Rectangle function,Sinc function,Signum function,sgn(x),x,1,-1,Triangle function,x,tri(x/a),a,-a,1,Circle function,r0,x,y,z,1,Dirac delta function,Another definition of delta function,Limit forms of delta function,Comb function,Convolution,The convolution of f1(x,y) and f2(x,y) is defined as,Correlation,The correlation of f1 and f2 is defined as f1(x,y)f2(x,y)=f1f2 (x,y),Autocorrelation,The autocorrelation of X(x,y) is defined as,Fourier Series Representation of Periodic Signals,Let x(t) be a periodic signal with period T, i.e.,Example: the rectangular pulse train,The Fourier Series,Then, x(t) can be expressed as,where is the fundamental frequency,cn is the spectrum of x(t) c0 is called the constant or dc component of x(t),Dirichlet Conditions,x(t) has only a finite number of maxima and minima over any period x(t) has only a finite number of discontinuities over any period,A periodic signal x(t) has a Fourier series if it satisfies the following conditions: x(t) is absolutely integrable over any period, namely,Spectra of aperiodic function,Periodic signals can be represented with the Fourier series Aperiodic signals can be analyzed in terms of frequency components also Periodic signals discrete spectra Aperiodic signals continuous spectra generally,One-dimensional Fourier transform,Let period T, periodic function becomes aperiodic function,One-dimensional Fourier transform,Fourier analysis in two dimensions,Fourier analysis is a mathematical tool of great utility in the analysis of both linear and nonlinear phenomena It is widely used in the study of electrical networks and communication systems In optics, it appears in form of two dimensions,Definition of Fourier Transform,Fourier transform of function g(x,y) is defined by,Inverse Fourier transform of G(fx,fy) is defined by,Transform pairs for some functions,Function,Function,Transform,Transform,Existence Conditions of Fourier Transform,1. Original function g(x,y) must be absolutely integrable over the infinite (x,y) plane 2. g(x,y) must have only a finite number of discontinuities and a finite number of maxima and minima in any finite rectangle 3. g(x,y) must have no infinite discontinuities,Comparison between existence conditions,Absolutely integrable, finite number of maxima and minima, finite number of discontinuities, no infinite discontinuities Fourier series needs above conditions over one period Fourier transform needs above conditions over infinite (x,y) plane Conclusion: The existence conditions of Fourier transform are stricter than Fourier series,Examples of Unsatisfying Existence Conditions,It is often convenient to represent true physical waveforms by idealized mathematical functions For example,fails to satisfy existence condition 3,both fail to satisfy existence condition 1,Generalized Fourier transform,It is often possible to find a meaningful transform of functions that do not strictly satisfy the existence conditions, provided those functions can be defined as the limit of a sequence of functions that are transformable Generalized transforms can be manipulated in the same manner as conventional transforms, and the distinction between the two cases can generally be ignored,GFT example_1: Delta Function,Delta function is described by limit of a sequence,The sequence can be Fourier transformed,Delta function is Fourier transformed,GFT example_2: constant,Constant is described by limit of rect function,Fourier transform of rect function is,Fourier transform of constant is,The Fourier Transform as a Decomposition,Fourier transform decomposes a function g(x, y) into a linear combination of elementary functions of the form expj2(fxx+ fyy) For any particular frequency pair ( fx, fy) the elementary function has a phase =2(fxx+ fyy)= 2n,This elementary function may be regarded as being “directed” in the (x,y) plane at an angle,Coordinate components of spatial period,Spatial period L,Spatial frequency: fx,fy,Fourier Transform Theorems,The basic definition of the Fourier transform leads to a rich mathematical structure associated with the transform operation FT properties will find wide use in later material These properties are presented as mathematical theorems,Linearity theorem,The transform of a weighted sum of two (or more) functions is simply the identically weighted sum of their individual transforms,Similarity theorem,A “stretch” of the coordinates in the space domain (x, y) results in a “contraction” of the coordinates in the frequency domain( fx,fy), plus a change in the overall amplitude of the spectrum,Proof of similarity theorem,Shift theorem,Translation in the space domain introduces a linear phase shift in the frequency domain,Proof of shift theorem,Convolution theorem,The convolution of two functions in the space domain is entirely equivalent to the more simple operation of multiplying their individual transforms and inverse transforming,Proof of convolution theorem,Conjugation theorem,Correlation theorem,F(fg)(x,y)=F(fx,fy)G*(fx,fy),Autocorrelation theorem,This theorem may be regarded as a special case of the convolution theorem in which we convolve g(x, y) with g*(-x, -y).,Theorem of direct and inverse transform,The successive direct transformation and inverse transformation of a function yields that function again, except at points of discontinuity,Successive transformation theorem,Proof,Separation theorem,If f(x,y) =f1(x)f2(y) and Ff1(x)=F1(u), Ff2(y)=F2(v) then Ff(x,y)= F1(u)F2(v),Rayleighs theorem (Parsevals theorem).,The integral on the left-hand side of this theorem can be interpreted as the energy contained in the waveform g(x, y). This in turn leads us to the idea that the quantity |G(fx,fy)|2 can be interpreted as an energy density in the frequency domain,Proof of Rayleighs theorem_1,Proof of Rayleighs theorem_2,Fourier-Bessel transform_1,Its Fourier transform in a system of rectangular coordinates is,Circularly symmetric function,Fourier-Bessel transform_2,Making a transformation to polar coordinates,Fourier-Bessel transform_3,finally, we get Fourier-Bessel transform,With help of Bessel function identity,The transform becomes,Fourier Transform of Periodic Signals,Since existence conditions of Fourier series are easily satisfied, we firstly expand periodic function into Fourier series and then Fourier transform the series Let x(t) be a periodic signal with period T, it can be represented with its Fourier series,Since,Global property of spatial frequency,Each frequency component of Fourier component of a function extends over the entire (x, y) domain It is not possible to associate a spatial location with a particular spatial frequency,Local frequency,In practice certain portions of an image could contain parallel grid lines at a certain fixed spacing Particular frequency represented by these grid lines are localized to certain spatial regions of the image,Definition of local frequency,Any complex-valued function, which may represent monochromatic optical waves, can be expressed in the form,Local frequency of g is defined as,Example 1 of local frequency,A plane wave,Its local frequency are,Example 2 of local frequency_1,Finite chirp function,Performing the differentiations called for by the definitions of local frequencies,Example 2 of local frequency_2,The one-dimensional spectrum,Windowed Fourier transform,Conventional FT Windowed FT,Independence of space and frequency in FT,When f(x) is transformed to F, the frequency information is maintained but spatial information is buried deeply and can hardly be recognized From F, we know what frequencies appear from the spectrum, but we do not know where they occur in the signal,Dependence of space and frequency in WFT,From Fw, we know not only the spectrum components but also where a component appears in the spatial domain Example of application Analysis of fringe pattern in interferogram Relatives Gabors transform; Wavelet transform,Properties of delta function,Picking property under integration,Similarity,Multiplication,Convolution of delta function,Linear system,Operator Sg1(,)=g2(x,y) S operates on input function g1(,) to produce output function g2(x,y),Decomposition of function,Arbitrary function can be decomposed to certain “elementary“ functions, e.g., (x,y),This means that g1(x,y) is a linear combination of weighted and displaced ,Impulse response function,Impulse response function,Response of the system to the input,The impulse response is also called the point-spread function Output now can be written,The superposition integral means a linear system is completely characterized by its responses to unit impulses,Physical meaning of the superposition integral,For linear imaging system, the effects of imaging elements (lenses, stops, etc.) can be fully described by specifying the images of point sources located throughout the object field.,Linear spatial invariation,If a system is spatial invariant, then for any value of xo and yo we have,Linear spatial invariation,Invariant Linear Systems,The necessary condition of shift invariation for entire input is shift invariant impulse spread function h(x,y;,)= h(x-x0,y-y0;-x0,-y0) Space-invariant linear system has impulse response function,The superposition integral takes on the particularly simple form,Transfer Function,The integral is just convolution of the object function with the impulse response,Fourier transforming the above gives,Transfer function is defined,Comparison between space domain and frequency domain,Space domain,Frequency domain,Discretization,Represent a function g(x, y) by an array of its sampled values taken on a discrete set of points in the (x, y) plane The sampled data can be reconstructed with considerable accuracy by simple interpolation The reconstruction of bandlimited functions can be accomplished exactly,Bandlimited functions,Bandlimited functions mean functions with Fourier transforms that are nonzero over only a finite region of the frequency space,Sampling of function,Samples of the function g(x,y),Fourier transform of the sampled function,Fourier transforming the both sides of the sampled function gives,Fourier transform of comb,Image of Gs(fx,fy),Observation to Gs(fx,fy),The spectrum of gs, can be found simply by erecting the spectrum of g about each point (n/X, m/Y) in the (fx, fy) plane If X and Y are sufficiently small ( i.e. the samples are sufficiently close together ), then 1/X and1/Yof the various spectral islands will be great enough to assure that the adjacent regions do not overlap,Recovering g from gs,X and Y small enough1/X and 1/Y large enough the adjacent regions do not overlap A lowpass filter transmits the term (n = 0, m = 0) of Gs, i.e. G(fx, fy), while perfectly excluding all other terms F-1G