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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。 An image is: " A non-uniform distribution of energy or matter“ Types: Aerial Dose Latent Developed A A(x, y,s;t) Normally, CW cases (time-independent solution) is considered 針孔映像 幻覺影像? Generalized (cognition) representation of Multi-Dimensional information 座 標 • Conventional system • {x,y} perpendicular to propagation direction • “z” or “s” along Y propagation axis X Z, S 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) 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) 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? 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 d 2 f ' Example: If f’ = 50mm, = 600nm (red), d = 0.36mm 全 真 映 像(paraxial) ' f' η z z' η' Inverted One-to-One Real & Enlarged The sinc function This function’s information encoded in the spatial domain, not in the frequency domain. 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. 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. 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 i ( x, y , z ) o ( x , y , z ) s ( x , y , z ) blurred image original object lens’s PSF In the Fourier plane it turns to be: I (k , k , k ) O (k , k , k ) S (k , k , k ) x y z x y z x y z S (k x , k y , k z ) OTF F PSF 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! Intensity PSFI PSFA complex amplitude 2 F PSFI F PSFA F PSFA In Fourier space the intensity OTF is the autocorrelation of the amplitude OTF. 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. 光學系統解像率 Δx = 0.61 λ/N.A. 鑑 別 率 測 試 靶 Johnson 鑑別定義 關鍵次元 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 Fourier image Signals are functions of time. There are two ways by which we can represent the signal. Time Domain Representation Signal Frequency Domain Representation Why Use Frequency Representations When We Can Represent Any Signal With Time Functions? 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. “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. 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 ” 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 49th Multiple Multiple 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. 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). 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 Bridge vibrations Stress vibrations in buildings and aircraft 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 Formulas to find all the coefficients in a Fourier Series expansion:- 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. i e cos i sin For a signal or function f(t), the Fourier transform is defined as Inverse Fourier transform is defined as Digital implementation M N g (i, j ) f (m, n) PSF (i m, j n) m 1 n 1 M N r (i, j ) f1 (m, n) f 2 (i m, j n) m1 n1 M N F (k , l ) f (m, n)e m 1 n 1 i 2 km / M ln/ N 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 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. Linear system 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 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. Imaging formation Intensity function C= Optical Transfer function OTF = MTF + PTF Neglect in general 系統性能解讀 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 Mathematical representation of an image Functional dependence of f in x (position vector) : f = f (x) General distortion function: h = h (x,ξ) Implied that f at ξis spread out according to the formula h (x,ξ). For linear distortion system, the blurred information: b(x) = f (ξ) h (x,ξ) d (ξ) 2D, all information, over area d, i.e. Fourier Transformation Power spectrum amplitude Phase change Temporal coherence Spatial coherence Infinitive coherence finite coherence Pin-hole Out-of focus imaging Cylindrical Function F-T Blurred spot ( disk ) Amplitude filter shape? Correction and deblurring Spatial filtering Coherent light transparency Shaded area blocked Improve the image TV image Blurred image processed Overdeveloped linear ( optical spectrum) Improve Astigmatism Amplitude Transfer Function Combined = add + shift Phase Transfer Function Spatial masks Low-pass filters eliminate or attenuate high frequency components in the frequency domain (sharp image details), and result in image blurring. High-pass filters attenuate or eliminate lowfrequency components (resulting in sharpening edges and other sharp details). Band-pass filters remove selected frequency regions between low and high frequencies (image restoration). Filters and their effects High-pass Band-pass BLIND DECONVOLUTION Some applications • Image sorting • Remote sensing • Pattern (character) recognition • Target tracking • Biological imaging • Intelligence communication 光纖與光纜 將許多根光纖綑在一起,外圍再包一層 塑膠,便可形成光纜,可傳送更多資訊 光纖通訊所用的零組件 光纖在內視鏡中的應用 • 各種醫療用內視鏡如胃鏡、大腸鏡等,都使用 光纖傳輸訊號 • 內視鏡的基本結構─以胃鏡為例(左圖) ,前端 為一個迷你攝影機,而所拍攝到的胃內部影像 訊號透過光纖傳送到外部螢幕上(右圖) 各種顯示器的應用 軟性電子材料顯示器 數位和傳統相機的差別 • 傳統相機是利用光線讓底片感光,而將影像記錄在底 片上,無法直接連接電腦作處理 • 數位相機是利用電荷耦合元件(Charge Coupled Device , CCD)或是互補式氧化物金屬半導體(Complementary Metal Oxide Semiconductor, CMOS)的影像感應功能, 將光線轉換為數位訊號,這些訊號可儲存於內建的記 憶體晶片上,並且可直接連上電腦作影像處理 追蹤熱源之響尾蛇飛彈 • 響尾蛇飛彈於1953年由美國試射成功,它使用紅 外線感測器追蹤敵機,可鎖定敵機引擎的位置,因 為飛機引擎的溫度最高,會輻射大量紅外線 • 1958年(民國47年)台海「八二三砲戰」期間,我國 空軍F-86「軍刀式」戰鬥機發射AIM-9B型響尾蛇 飛彈,擊落中共空軍 10 架以上米格15戰鬥機 紅外線防盜器 • 人的體溫與周圍環境不同,會發射特定波長範 圍的紅外線,因此可用紅外線感測器來製作防 盜器,只要感測器偵測到有人靠近,即可發出 警訊 • 紅外線防盜器只能偵測是否有人靠近,但無法 分辨是「好人」還是「壞人」,必須要配合其 他影像處理方法 Conclusion Filters -- Linear and nonlinear Source -- coherent and incoherent H. H. Hopkins (1955) B & W: partial coherent !! Mutual intensity included meaning: Precision computation