影像的定義

1. 影像的定義 • 一般以繪圖、相片或銀幕的顯示，敘述影像的存在。 • 它代表了空間訊息(information)，有實質的意義內涵。 • 世間有更多視而不見的物體、只因為它們不具任何意義。 • 有時也能看到不存在影像。 • 更有主觀性的錯覺。

2. An image is: " A non-uniform distribution of energy or matter“ Types: Aerial Dose Latent Developed • Normally, CW cases • (time-independent solution) is considered

3. 針 孔 映 像

4. 幻 覺 影 像 ?

6. Generalized (cognition) representation of Multi-Dimensional information

7. 座 標 • Conventional system • {x,y} perpendicular to propagation direction • “z” or “s” along propagation axis X Y Z, S

8. 方 式 • Image description • Pixels and Field: • Patterns: ensemble of contiguous non-zero value pixels • Image analysis • Mathematical description • Information content • Independent of image carrier (energy, material) • Limited by pixel size (Shannon)

9. Digital Images • Continuous images A(x,y;s;t) are sampled on a regular grid • Size of grid unit cell defines the “pixel” • Main definitions • Image size: Nx by Ny (pixels) • Image resolution: dX by dY (cm) • Image field: Lx by Ly (cm)

10. 點光源與展體影像 影像的實際定義應該是: 無限組透過光圈的光束聚合 運算？ 描述？

11. Problems with Pinholes • Pinhole size (aperture) must be “very small” to obtain a clear image. • However, as pinhole size is made smaller, less light is received by image plane. • If pinhole is comparable to • wavelength of incoming light, • DIFFRACTION effects blur • the image! • Sharpest image is obtained • when: pinhole diameter • Example: If f’ = 50mm, • = 600nm (red), • d = 0.36mm

12. 全 真 映 像(paraxial) ' f' η z z' η' Inverted Real & Enlarged One-to-One

13. The sinc function This function’s information encoded in the spatial domain, not in the frequency domain.

14. 工 具 The traditional method of describing 3-D imaging properties of a light microscope is by intensity point spread function (PSF) or it’s Fourier transform, the optical transfer function (OTF). However, the more compact way is to use 2-D generalized pupil function. The advantage: the easier way of modification of the observed PSF to introduce known aberrations. The disadvantage: it’s not too easy to determine the complex-valued pupil function from the measured intensity PSF.

15. 困 擾 • Image quality in light microscopy is degraded by aberrations, causing from: • sample’s refractive index ( acting like lens) • the features of the microscope set-up • The result: the image is blurred and not diffraction limited or specifically speaking: • we loose resolution • reduce signal to noise ratio • get distortions in the collected data.

16. Microscope Image Formation and Fourier Optics Image formation in microscope could be described as a linear process in which each point of an object is convolved by the lens PSF to produce a blurred image blurred image original object lens’s PSF

17. In the Fourier plane it turns to be: OTF describes the impulse response of the microscope in the terms of spatial frequency. In most microscopy techniques we measure only light intensities, i.e. : As a result, all information about the light phase is lost!

18. Intensity complex amplitude In Fourier space the intensity OTF is the autocorrelation of the amplitude OTF.

19. The benefits of acquiring phase information for a microscope system are: • quantification of the aberrations/features of an optical system for use in deconvolution of collected data. • using as a means to adjust, correct or compensate for optical problems in the optical system or in the sample using adjustable elements in the optical path. • The pupil function is a powerful way to understand image formation. A phase retrieved pupil function can be used to calculate PSFs that contain key features observed in the measured PSF’s that are not represented in simulated PSFs.

20. 光學系統解像率 Δx = 0.61 λ/N.A.

21. 鑑別率測試靶

22. Johnson 鑑別定義 關 鍵 次 元

23. Johnson 實驗數據 TARGET RESOL. / MIN. DIMEN. IN LINE PAIRS SIDE VIEW DETEC. ORIENT. RECOG. IDENTIF. TRUCK 0.90 1.25 4.5 8.0 M-48 TANK 0.75 1.20 3.5 7.0 STALIN TANK 0.75 1.20 3.3 6.0 CENTURION 0.75 1.20 3.5 6.0 HALF-TRACK 1.00 1.50 4.0 5.0 JEEP 1.20 1.50 4.5 5.5 COMMAND 車 1.20 1.50 4.3 5.5 SOLDIER(站立) .50 1.80 3.8 8.0 105 砲 1.00 1.50 4.8 6.5 AVERAGE 1.0.25 1.4 .35 4.0 .8 6.4  1.5

24. Fourier image Signals are functions of time. There are two ways by which we can represent the signal. Signal Time Domain Representation Frequency Domain Representation Why Use Frequency Representations When We Can Represent Any Signal With Time Functions?

25. Advantages of Frequency response methods • Gives a different kind of insight into a system. • It focuses on how signals of different frequencies are represented • in a signal. We think in terms of the spectrum of the signal • Most of us would rather do algebra than solve differential equations • Gives more insight into how to process a signal to remove noise • Easier to characterize the frequency content of a noise signal than • it is to give a time description of the noise. Different treatment of different parts of the electromagnetic spectrum means that you can separate out different signals.

26. “So, give it a shot and try learning about frequency response methods. They can save you time and money in the long run” • Objective • Be able to compute the frequency components of the signal. • Be able to predict how the signal will interact with linear systems and circuits using frequency response methods.

27. The Fourier Series Fourier, doing heat transfer work, demonstrated that any periodic signal can be viewed as a linear composition of sine waves “A periodic signal can always be represented as a sum of sinusoids, This representation is now called a Fourier Series”

28. How a signal can be built from a sum of sinusoids? Example:- Here is a single sine signal The expression for this signal is Sig(t) = 1 * sin(2пt/T) + (1/3)sin(6пt/T) + (1/5)sin(10пt/T) 79th Multiple 49th Multiple

29. In fact, the way we are building this signal we are using Fourier's results. We know the formula for the series that converges to a square wave. Here's the formula. For a perfectly accurate representation, let N go to infinity.

31. Calculating The Fourier Series Coefficients At this point there are a few questions that we need to address. What kind of functions can be represented using these types of series? Actually, most periodic signals can be represented with a series composed of sines and cosines. Even discontinuities (like in the square wave function or the saw tooth function in the simulations).

32. practical implications Functions can be composed of sines and cosines at different frequencies, Various linear systems process sinusoidal signals is frequency dependent, The response of a system with a periodic input can be predicted using frequency response methods. Signals can be analyzed using frequency component concepts.  Special computational techniques (FFT) have been developed to calculate frequency components quickly for various signals.  Examples: Sound signals in earthquakes Stress vibrations in buildings and aircraft Bridge vibrations

33. The series for a given function • Periodic signal can be represented as a sum of both sines • and cosines • Also, since sines and cosines have no average term, periodic • signals that have a non-zero average can have a constant • component This series can be used to represent many periodic functions The coefficients, an and bn, are what you need to know to generate the signal

34. Formulas to find all the coefficients in a Fourier Series expansion:-

35. Fourier Transforms • The Fourier transform (FT) is a generalization of the Fourier series. • Instead of sines and cosines, as in a Fourier series, the Fourier transform uses exponentials and complex numbers. • For a signal or function f(t), the Fourier transform is defined as Inverse Fourier transform is defined as

36. Digital implementation

37. 影 像 處 理 • Why should an image be processed prior to analysis? • It suffers from noise • It fails to highlight the particular feature in which we are interested • In image processing, we remove noise & unnecessary features while highlighting the required features • Filtering

38. Optical image Ronchi (1961): Ethereal – physical nature Calculated – mathematical representation (resolution, PSF,.., etc.,), it is noise free! Detected – practice image , source energy & sensitivity included. Resolution is limited by systematic & random errors due to inadequacy of description.

39. Linear system

40. 描述影像的兩條途徑 傅氏轉換 物體 影像頻譜 點展包容積分 乘傳遞函數 影像 調幅頻譜 傅氏返轉換

41. Two points resolution • PSF behavior: 0.8 overlape ; ¼ λRayleigh criteria. A rule conveniently to define resolution. • Depends nothing more than size & shapes of aperture + wavelength of light. The radius of the Airy disk: 1.22λF/# The measurement can never be S/N free.

42. Imaging formation Intensity function

43. C =

44. Optical Transfer function OTF = MTF + PTF Neglect in general

45. 系統性能解讀

46. 映 像 品 質 規 格 • For photographic films, namely modulation transfer function (MTF), ISO speed, granularity, and D-plot, which users can relate to certain image qualities • For digital sensors, signal-to-noise ratio (SNR), dark current, fill factor, full-well capacity, and sensitivity interact with image quality

47. Mathematical representationof an image Functional dependence off in x (position vector) : f = f （x） General distortion function: h = h (x,ξ) Implied that f at ξisspread out according to the formula h (x,ξ). For linear distortion system, the blurred information: b(x) = (ξ) h (x,ξ) d(ξ) 2D, all information, over area d, i.e.

48. Fourier Transformation Power spectrum amplitude Phase change

49. Temporal coherence

50. Spatial coherence Infinitive coherence finite coherence Pin-hole