DOCSLIB.ORG

Noise, Image Reconstruction with Noise

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Noise, Image Reconstruction with Noise Noise, Image Reconstruction with Noise EE367/CS448I: Computational Imaging and Display stanford.edu/class/ee367 Lecture 10 Gordon Wetzstein Stanford University What’s a Pixel? photon to electron converter à photoelectric effect! source: Molecular Expressions wikipedia What’s a Pixel? • microlens: focus light on photodiode • color filter: select color channel • quantum efficiency: ~50% • fill factor: fraction of surface area used for light gathering source: Molecular Expressions • photon-to-charge conversion and ADC includes noise! ISO (“film speed”) sensitivity of sensor to light – digital gain bobatkins.com Noise • noise is (usually) bad! • many sources of noise: heat, electronics, amplifier gain, photon to electron conversion, pixel defects, read, … • let’s start with something simple: fixed pattern noise Fixed Pattern Noise • dead or “hot” pixels, dust, … • remove with dark frame calibration: I − I I = captured dark Iwhite − Idark on RAW image! not JPEG (nonlinear) Emil Martinec Noise • other than that, different noise follows different statistical distributions, these two are crucial: • Gaussian • Poisson Gaussian Noise • thermal, read, amplifier • additive, signal-independent, zero-mean + = Gaussian Noise x ∼ N (x,0) • with i.i.d. Gaussian noise: b = x +η η N 0,σ 2 (independent and identically-distributed) ∼ ( ) additive Gaussian noise Gaussian Noise x ∼ N (x,0) • with i.i.d. Gaussian noise: b = x +η η ∼ N 0,σ 2 (independent and identically-distributed) ( ) b ∼ N (x,σ 2 ) 2 (b−x) 1 − 2 p(b | x,σ ) = e 2σ 2πσ 2 Gaussian Noise x ∼ N (x,0) • with i.i.d. Gaussian noise: b = x +η η N 0,σ 2 (independent and identically-distributed) ∼ ( ) • Bayes’ rule: b ∼ N (x,σ 2 ) 2 (b−x) 1 − 2 p(b | x,σ ) = e 2σ 2πσ 2 Gaussian Noise - MAP x ∼ N (x,0) • with i.i.d. Gaussian noise: b = x +η η N 0,σ 2 (independent and identically-distributed) ∼ ( ) • Bayes’ rule: • maximum-a-posteriori estimation: b ∼ N (x,σ 2 ) 2 (b−x) 1 − 2 p(b | x,σ ) = e 2σ 2πσ 2 Gaussian Noise – MAP “Flat” Prior • trivial solution (maximum likelihood, not useful in practice): p(x) = 1 Gaussian Noise – MAP Self Similarity Prior • Gaussian “denoisers” like non-local means and other self- similarity priors actually solve this problem: General Self Similarity Prior • generic proximal operator for function f(x): • proximal operator for some image prior General Self Similarity Prior • can use self-similarity as general image prior (not just for denoising) • proximal operator for some image prior General Self Similarity Prior • can use self-similarity as general image prior (not just for denoising) σ 2 = λ /σ σ = λ /σ (h parameter in most NLM implementations is std. dev.) • proximal operator for some image prior Image Reconstruction with Gaussian Noise x ∼ N (Ax,0) • with i.i.d. Gaussian noise: b = Ax +η η ∼ N 0,σ 2 (independent and identically-distributed) ( ) Image Reconstruction with Gaussian Noise x ∼ N (Ax,0) • with i.i.d. Gaussian noise: b = Ax +η η ∼ N 0,σ 2 (independent and identically-distributed) ( ) b ∼ N (Ax,σ 2 ) 2 (b−Ax) 1 − 2 p(b | x,σ ) = e 2σ 2πσ 2 Image Reconstruction with Gaussian Noise x ∼ N (Ax,0) • with i.i.d. Gaussian noise: b = Ax +η η N 0,σ 2 (independent and identically-distributed) ∼ ( ) • Bayes’ rule: • maximum-a-posteriori estimation: b ∼ N (Ax,σ 2 ) 2 (b−Ax) 1 − 2 p b | x,σ = e 2σ ( ) 2 2πσ • regularized least squares (use ADMM) Scientific Sensors • e.g., Andor iXon Ultra 897: cooled to -100° C • scientific CMOS & CCD • reduce pretty much all noise, except for photon noise What is Photon (or Shot) Noise? • noise caused by the fluctuation when the incident light is converted to charge (detection) • also observed in the light emitted by a source, i.e. due to particle nature of light (emission) • same for re-emission à cascading Poisson processes are also described by a Poisson process [Teich and Saleh 1998] Photon Noise • noise is signal dependent! • for N measured photo-electrons • standard deviation is σ = N , variance is σ 2 = N • mean is N N ke− N • Poisson distribution f (k; N) = k! Photon Noise - SNR signal-to-noise ratio N SNR = N SNR in dB N photons: σ = N 2N photons: σ = 2 N 1,000 10,000 N nonlinear! Photon Noise - SNR signal-to-noise ratio N SNR = N N photons: σ = N 2N photons: σ = 2 N nonlinear! wikipedia Maximum Likelihood Solution for Poisson Noise • image formation: • probability of measurement i: • joint probability of all M measurements (use notation trick ): Maximum Likelihood Solution for Poisson Noise • log-likelihood function: • gradient: Maximum Likelihood Solution for Poisson Noise • Richardson-Lucy algorithm: iterative approach to ML estimation for Poisson noise • simple idea: 1. at solution, gradient will be zero 2. when converged, further iterations will not change, i.e. Richardson-Lucy Algorithm • equate gradient to zero = 0 • rearrange so that 1 is on one side of equation Richardson-Lucy Algorithm • equate gradient to zero = 0 • rearrange so that 1 is on one side of equation • set equal to Richardson-Lucy Algorithm • for any multiplicative update rule scheme: • start with positive initial guess (e.g. random values) • apply iteration scheme • future updates are guaranteed to remain positive • always get smaller residual • RL multiplicative update rules: Richardson-Lucy Algorithm - Deconvolution Blurry & Noisy Measurement RL Deconvolution Richardson-Lucy Algorithm • what went wrong? • Poisson deconvolution is a tough problem, without priors it’s pretty much hopeless • let’s try to incorporate the one prior we have learned: total variation Richardson-Lucy Algorithm + TV • log-likelihood function: • gradient: Richardson-Lucy Algorithm + TV • gradient of anisotropic TV term • this is “dirty”: possible division by 0! Richardson-Lucy Algorithm + TV • follow the same logic as RL, RL+TV multiplicative update rules: • 2 major problems & solution “hacks”: 1. still possible division by 0 when gradient is zero à set fraction to 0 if gradient is 0 2. multiplicative update may become negative! à only work with (very) small values for lambda (A Dirty but Easy Approach to) Richardson-Lucy with a TV Prior RL RL+TV, lambda=0.005 RL+TV, lambda=0.025 Log Residual Mean Squared Error Measurements Signal-to-Noise Ratio (SNR) mean pixel value µ signal SNR = = standard deviation of pixel value σ noise PQ t = e 2 PQet + Dt + Nr P = incident photon flux (photons/pixel/sec) Qe = quantum efficiency t = eposure time (sec) D = dark current (electroncs/pixel/sec), including hot pixels Nr = read noise (rms electrons/pixel), including fixed pattern noise Signal-to-Noise Ratio (SNR) mean pixel value µ signal SNR = = standard deviation of pixel value σ noise PQ t = e signal-dependent 2 PQet + Dt + Nr signal-independent P = incident photon flux (photons/pixel/sec) Qe = quantum efficiency t = eposure time (sec) D = dark current (electroncs/pixel/sec), including hot pixels Nr = read noise (rms electrons/pixel), including fixed pattern noise Next: Compressive Imaging • single pixel camera • compressive hyperspectral imaging • compressive light field imaging et al., 2013 Marwah Wakin et al. 2006 References and Further Reading • https://en.wikipedia.org/wiki/Shot_noise • L. Lucy, 1974 “An iterative technique for the rectification of observed distributions”. Astron. J. 79, 745–754. • W. Richardson, 1972 “Bayesian-based iterative method of image restoration” J. Opt. Soc. Am. 62, 1, 55–59. • N. Dey, L. Blanc-Feraud, C. Zimmer, Z. Kam, J. Olivo-Marin, J. Zerubia, 2004 “A deconvolution method for confocal microscopy with total variation regularization”, In IEEE Symposium on Biomedical Imaging: Nano to Macro, 1223–1226 • M. Teich, E. Saleh, 1998, “Cascaded stochastic processes in optics”, Traitement du Signal 15(6) • please also read the lecture notes, especially for the “clean” ADMM derivation for solving the maximum likelihood estimation of Poisson reconstruction with TV prior!.
Load more

Recommended publications
  • Blind Denoising Autoencoder
    1 Blind Denoising Autoencoder Angshul Majumdar In recent times a handful of papers have been published on Abstract—The term ‘blind denoising’ refers to the fact that the autoencoders used for signal denoising [10-13]. These basis used for denoising is learnt from the noisy sample itself techniques learn the autoencoder model on a training set and during denoising. Dictionary learning and transform learning uses the trained model to denoise new test samples. Unlike the based formulations for blind denoising are well known. But there has been no autoencoder based solution for the said blind dictionary learning and transform learning based approaches, denoising approach. So far autoencoder based denoising the autoencoder based methods were not blind; they were not formulations have learnt the model on a separate training data learning the model from the signal at hand. and have used the learnt model to denoise test samples. Such a The main disadvantage of such an approach is that, one methodology fails when the test image (to denoise) is not of the never knows how good the learned autoencoder will same kind as the models learnt with. This will be first work, generalize on unseen data. In the aforesaid studies [10-13] where we learn the autoencoder from the noisy sample while denoising. Experimental results show that our proposed method training was performed on standard dataset of natural images performs better than dictionary learning (K-SVD), transform (image-net / CIFAR) and used to denoise natural images. Can learning, sparse stacked denoising autoencoder and the gold the learnt model be used to recover images from other standard BM3D algorithm.
    [Show full text]
  • A Comparison of Delayed Self-Heterodyne Interference Measurement of Laser Linewidth Using Mach-Zehnder and Michelson Interferometers
    Sensors 2011, 11, 9233-9241; doi:10.3390/s111009233 OPEN ACCESS sensors ISSN 1424-8220 www.mdpi.com/journal/sensors Article A Comparison of Delayed Self-Heterodyne Interference Measurement of Laser Linewidth Using Mach-Zehnder and Michelson Interferometers Albert Canagasabey 1,2,*, Andrew Michie 1,2, John Canning 1, John Holdsworth 3, Simon Fleming 2, Hsiao-Chuan Wang 1,2 and Mattias L. Åslund 1 1 Interdisciplinary Photonics Laboratories (iPL), School of Chemistry, University of Sydney, 2006, NSW, Australia; E-Mails: [email protected] (A.M.); [email protected] (J.C.); [email protected] (H.-C.W.); [email protected] (M.L.Å.) 2 Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, 2006, NSW, Australia; E-Mail: [email protected] (S.F.) 3 SMAPS, University of Newcastle, Callaghan, NSW 2308, Australia; E-Mail: [email protected] (J.H.) * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +61-2-9351-1984. Received: 17 August 2011; in revised form: 13 September 2011 / Accepted: 23 September 2011 / Published: 27 September 2011 Abstract: Linewidth measurements of a distributed feedback (DFB) fibre laser are made using delayed self heterodyne interferometry (DHSI) with both Mach-Zehnder and Michelson interferometer configurations. Voigt fitting is used to extract and compare the Lorentzian and Gaussian linewidths and associated sources of noise. The respective measurements are wL (MZI) = (1.6 ± 0.2) kHz and wL (MI) = (1.4 ± 0.1) kHz. The Michelson with Faraday rotator mirrors gives a slightly narrower linewidth with significantly reduced error.
    [Show full text]
  • 10S Johnson-Nyquist Noise Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 02, 2011)
    Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 02, 2011) Johnson noise Johnson-Nyquist theorem Boltzmann constant Parseval relation Correlation function Spectral density Wiener-Khinchin (or Khintchine) Flicker noise Shot noise Poisson distribution Brownian motion Fluctuation-dissipation theorem Langevin function ___________________________________________________________________________ John Bertrand "Bert" Johnson (October 2, 1887–November 27, 1970) was a Swedish-born American electrical engineer and physicist. He first explained in detail a fundamental source of random interference with information traveling on wires. In 1928, while at Bell Telephone Laboratories he published the journal paper "Thermal Agitation of Electricity in Conductors". In telecommunication or other systems, thermal noise (or Johnson noise) is the noise generated by thermal agitation of electrons in a conductor. Johnson's papers showed a statistical fluctuation of electric charge occur in all electrical conductors, producing random variation of potential between the conductor ends (such as in vacuum tube amplifiers and thermocouples). Thermal noise power, per hertz, is equal throughout the frequency spectrum. Johnson deduced that thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise. http://en.wikipedia.org/wiki/John_B._Johnson 1 ____________________________________________________________________________ Harry Nyquist (February 7, 1889 – April 4, 1976) was an important contributor to information theory. http://en.wikipedia.org/wiki/Harry_Nyquist ___________________________________________________________________________ 10S.1 Histrory In 1926, experimental physicist John Johnson working in the physics division at Bell Labs was researching noise in electronic circuits. He discovered random fluctuations in the voltages across electrical resistors, whose power was proportional to temperature.
    [Show full text]
  • Arxiv:2003.13216V1 [Cs.CV] 30 Mar 2020
    Learning to Learn Single Domain Generalization Fengchun Qiao Long Zhao Xi Peng University of Delaware Rutgers University University of Delaware [email protected] [email protected] [email protected] Abstract : Source domain(s) <latexit sha1_base64="glUSn7xz2m1yKGYjqzzX12DA3tk=">AAAB8nicjVDLSsNAFL3xWeur6tLNYBFclaQKdllw47KifUAaymQ6aYdOJmHmRiihn+HGhSJu/Rp3/o2TtgsVBQ8MHM65l3vmhKkUBl33w1lZXVvf2Cxtlbd3dvf2KweHHZNkmvE2S2SieyE1XArF2yhQ8l6qOY1Dybvh5Krwu/dcG5GoO5ymPIjpSIlIMIpW8vsxxTGjMr+dDSpVr+bOQf4mVViiNai894cJy2KukElqjO+5KQY51SiY5LNyPzM8pWxCR9y3VNGYmyCfR56RU6sMSZRo+xSSufp1I6exMdM4tJNFRPPTK8TfPD/DqBHkQqUZcsUWh6JMEkxI8X8yFJozlFNLKNPCZiVsTDVlaFsq/6+ETr3mndfqNxfVZmNZRwmO4QTOwINLaMI1tKANDBJ4gCd4dtB5dF6c18XoirPcOYJvcN4+AY5ZkWY=</latexit> <latexit sha1_base64="9X8JvFzvWSXuFK0x/Pe60//G3E4=">AAACD3icbVDLSsNAFJ3UV62vqks3g0Wpm5LW4mtVcOOyUvuANpTJZNIOnUzCzI1YQv/Ajb/ixoUibt26829M2iBqPTBwOOfeO/ceOxBcg2l+GpmFxaXllexqbm19Y3Mrv73T0n6oKGtSX/iqYxPNBJesCRwE6wSKEc8WrG2PLhO/fcuU5r68gXHALI8MJHc5JRBL/fxhzyMwpEREjQnuAbuD6AI3ptOx43uEy6I+muT6+YJZMqfA86SckgJKUe/nP3qOT0OPSaCCaN0tmwFYEVHAqWCTXC/ULCB0RAasG1NJPKataHrPBB/EioNdX8VPAp6qPzsi4mk99uy4Mtle//US8T+vG4J7ZkVcBiEwSWcfuaHA4OMkHOxwxSiIcUwIVTzeFdMhUYRCHOEshPMEJ98nz5NWpVQ+LlWvq4VaJY0ji/bQPiqiMjpFNXSF6qiJKLpHj+gZvRgPxpPxarzNSjNG2rOLfsF4/wJA4Zw6</latexit> S S : Target domain(s) <latexit sha1_base64="ssITTP/Vrn2uchq9aDxvcfruPQc=">AAACD3icbVDLSgNBEJz1bXxFPXoZDEq8hI2Kr5PgxWOEJAaSEHonnWTI7Owy0yuGJX/gxV/x4kERr169+TfuJkF8FTQUVd10d3mhkpZc98OZmp6ZnZtfWMwsLa+srmXXN6o2iIzAighUYGoeWFRSY4UkKayFBsH3FF57/YvUv75BY2WgyzQIselDV8uOFECJ1MruNnygngAVl4e8QXhL8Rkvg+ki8Xbgg9R5uzfMtLI5t+COwP+S4oTk2ASlVva90Q5E5KMmocDaetENqRmDISkUDjONyGIIog9drCdUg4+2GY/+GfKdRGnzTmCS0sRH6veJGHxrB76XdKbX299eKv7n1SPqnDRjqcOIUIvxok6kOAU8DYe3pUFBapAQEEYmt3LRAwOCkgjHIZymOPp6+S+p7heKB4XDq8Pc+f4kjgW2xbZZnhXZMTtnl6zEKkywO/bAntizc+88Oi/O67h1ypnMbLIfcN4+ATKRnDE=</latexit>
    [Show full text]
  • All You Need to Know About SINAD Measurements Using the 2023
    applicationapplication notenote All you need to know about SINAD and its measurement using 2023 signal generators The 2023A, 2023B and 2025 can be supplied with an optional SINAD measuring capability. This article explains what SINAD measurements are, when they are used and how the SINAD option on 2023A, 2023B and 2025 performs this important task. www.ifrsys.com SINAD and its measurements using the 2023 What is SINAD? C-MESSAGE filter used in North America SINAD is a parameter which provides a quantitative Psophometric filter specified in ITU-T Recommendation measurement of the quality of an audio signal from a O.41, more commonly known from its original description as a communication device. For the purpose of this article the CCITT filter (also often referred to as a P53 filter) device is a radio receiver. The definition of SINAD is very simple A third type of filter is also sometimes used which is - its the ratio of the total signal power level (wanted Signal + unweighted (i.e. flat) over a broader bandwidth. Noise + Distortion or SND) to unwanted signal power (Noise + The telephony filter responses are tabulated in Figure 2. The Distortion or ND). It follows that the higher the figure the better differences in frequency response result in different SINAD the quality of the audio signal. The ratio is expressed as a values for the same signal. The C-MES signal uses a reference logarithmic value (in dB) from the formulae 10Log (SND/ND). frequency of 1 kHz while the CCITT filter uses a reference of Remember that this a power ratio, not a voltage ratio, so a 800 Hz, which results in the filter having "gain" at 1 kHz.
    [Show full text]
  • Next Topic: NOISE
    ECE145A/ECE218A Performance Limitations of Amplifiers 1. Distortion in Nonlinear Systems The upper limit of useful operation is limited by distortion. All analog systems and components of systems (amplifiers and mixers for example) become nonlinear when driven at large signal levels. The nonlinearity distorts the desired signal. This distortion exhibits itself in several ways: 1. Gain compression or expansion (sometimes called AM – AM distortion) 2. Phase distortion (sometimes called AM – PM distortion) 3. Unwanted frequencies (spurious outputs or spurs) in the output spectrum. For a single input, this appears at harmonic frequencies, creating harmonic distortion or HD. With multiple input signals, in-band distortion is created, called intermodulation distortion or IMD. When these spurs interfere with the desired signal, the S/N ratio or SINAD (Signal to noise plus distortion ratio) is degraded. Gain Compression. The nonlinear transfer characteristic of the component shows up in the grossest sense when the gain is no longer constant with input power. That is, if Pout is no longer linearly related to Pin, then the device is clearly nonlinear and distortion can be expected. Pout Pin P1dB, the input power required to compress the gain by 1 dB, is often used as a simple to measure index of gain compression. An amplifier with 1 dB of gain compression will generate severe distortion. Distortion generation in amplifiers can be understood by modeling the amplifier’s transfer characteristic with a simple power series function: 3 VaVaVout=−13 in in Of course, in a real amplifier, there may be terms of all orders present, but this simple cubic nonlinearity is easy to visualize.
    [Show full text]
  • Thermal-Noise.Pdf
    Thermal Noise Introduction One might naively believe that if all sources of electrical power are removed from a circuit that there will be no voltage across any of the components, a resistor for example. On average this is correct but a close look at the rms voltage would reveal that a "noise" voltage is present. This intrinsic noise is due to thermal fluctuations and can be calculated as may be done in your second year thermal physics course! The main goal of this experiment is to measure and characterize this noise: Johnson noise. In order to measure the intrinsic noise of a component one must first reduce the extrinsic sources of noise, i.e. interference. You have probably noticed that if you touch the input lead to an oscilloscope a large signal appears. Try this now and characterize the signal you see. Note that you are acting as an antenna! Make sure you look at both long time scales, say 10 ms, and shorter time scales, say 1 s. What are the likely sources of the signals you see? You may recall seeing this before in the First Year Laboratory. This interference is characterized by two features. First, the noise voltage is characterized by a spectrum, i.e. the noise voltage Vn ( f ) is a function of frequency. 2 Since noise usually has a time average of zero, the power spectrum Vn ( f ) is specified in each frequency interval df . Second, the measuring instrument is also characterized by a spectral response or bandwidth. In our case the bandwidth of the oscilloscope is from fL =0 (when input is DC coupled) to an upper frequency fH usually noted on the scope (beware of bandwidth limiting switches).
    [Show full text]
  • Estimation from Quantized Gaussian Measurements: When and How to Use Dither Joshua Rapp, Robin M
    1 Estimation from Quantized Gaussian Measurements: When and How to Use Dither Joshua Rapp, Robin M. A. Dawson, and Vivek K Goyal Abstract—Subtractive dither is a powerful method for remov- be arbitrarily far from minimizing the mean-squared error ing the signal dependence of quantization noise for coarsely- (MSE). For example, when the population variance vanishes, quantized signals. However, estimation from dithered measure- the sample mean estimator has MSE inversely proportional ments often naively applies the sample mean or midrange, even to the number of samples, whereas the MSE achieved by the when the total noise is not well described with a Gaussian or midrange estimator is inversely proportional to the square of uniform distribution. We show that the generalized Gaussian dis- the number of samples [2]. tribution approximately describes subtractively-dithered, quan- tized samples of a Gaussian signal. Furthermore, a generalized In this paper, we develop estimators for cases where the Gaussian fit leads to simple estimators based on order statistics quantization is neither extremely fine nor extremely coarse. that match the performance of more complicated maximum like- The motivation for this work stemmed from a series of lihood estimators requiring iterative solvers. The order statistics- experiments performed by the authors and colleagues with based estimators outperform both the sample mean and midrange single-photon lidar. In [3], temporally spreading a narrow for nontrivial sums of Gaussian and uniform noise. Additional laser pulse, equivalent to adding non-subtractive Gaussian analysis of the generalized Gaussian approximation yields rules dither, was found to reduce the effects of the detector’s coarse of thumb for determining when and how to apply dither to temporal resolution on ranging accuracy.
    [Show full text]
  • Image Denoising by Autoencoder: Learning Core Representations
    Image Denoising by AutoEncoder: Learning Core Representations Zhenyu Zhao College of Engineering and Computer Science, The Australian National University, Australia, [email protected] Abstract. In this paper, we implement an image denoising method which can be generally used in all kinds of noisy images. We achieve denoising process by adding Gaussian noise to raw images and then feed them into AutoEncoder to learn its core representations(raw images itself or high-level representations).We use pre- trained classifier to test the quality of the representations with the classification accuracy. Our result shows that in task-specific classification neuron networks, the performance of the network with noisy input images is far below the preprocessing images that using denoising AutoEncoder. In the meanwhile, our experiments also show that the preprocessed images can achieve compatible result with the noiseless input images. Keywords: Image Denoising, Image Representations, Neuron Networks, Deep Learning, AutoEncoder. 1 Introduction 1.1 Image Denoising Image is the object that stores and reflects visual perception. Images are also important information carriers today. Acquisition channel and artificial editing are the two main ways that corrupt observed images. The goal of image restoration techniques [1] is to restore the original image from a noisy observation of it. Image denoising is common image restoration problems that are useful by to many industrial and scientific applications. Image denoising prob- lems arise when an image is corrupted by additive white Gaussian noise which is common result of many acquisition channels. The white Gaussian noise can be harmful to many image applications. Hence, it is of great importance to remove Gaussian noise from images.
    [Show full text]
  • AN10062 Phase Noise Measurement Guide for Oscillators
    Phase Noise Measurement Guide for Oscillators Contents 1 Introduction ............................................................................................................................................. 1 2 What is phase noise ................................................................................................................................. 2 3 Methods of phase noise measurement ................................................................................................... 3 4 Connecting the signal to a phase noise analyzer ..................................................................................... 4 4.1 Signal level and thermal noise ......................................................................................................... 4 4.2 Active amplifiers and probes ........................................................................................................... 4 4.3 Oscillator output signal types .......................................................................................................... 5 4.3.1 Single ended LVCMOS ........................................................................................................... 5 4.3.2 Single ended Clipped Sine ..................................................................................................... 5 4.3.3 Differential outputs ............................................................................................................... 6 5 Setting up a phase noise analyzer ...........................................................................................................
    [Show full text]
  • ECE 417 Lecture 3: 1-D Gaussians Mark Hasegawa-Johnson 9/5/2017 Contents
    ECE 417 Lecture 3: 1-D Gaussians Mark Hasegawa-Johnson 9/5/2017 Contents • Probability and Probability Density • Gaussian pdf • Central Limit Theorem • Brownian Motion • White Noise • Vector with independent Gaussian elements Cumulative Distribution Function (CDF) A “cumulative distribution function” (CDF) specifies the probability that random variable X takes a value less than : Probability Density Function (pdf) A “probability density function” (pdf) is the derivative of the CDF: That means, for example, that the probability of getting an X in any interval is: Example: Uniform pdf The rand() function in most programming languages simulates a number uniformly distributed between 0 and 1, that is, Suppose you generated 100 random numbers using the rand() function. • How many of the numbers would be between 0.5 and 0.6? • How many would you expect to be between 0.5 and 0.6? • How many would you expect to be between 0.95 and 1.05? Gaussian (Normal) pdf Gauss considered this problem: under what circumstances does it make sense to estimate the mean of a distribution, , by taking the average of the experimental values, ? He demonstrated that is the maximum likelihood estimate of if Gaussian pdf Attribution: jhguch, https://commons. wikimedia.org/wik i/File:Boxplot_vs_P DF.svg Unit Normal pdf Suppose that X is normal with mean and standard deviation (variance ): Then is normal with mean 0 and standard deviation 1: Central Limit Theorem The Gaussian pdf is important because of the Central Limit Theorem. Suppose are i.i.d. (independent and identically distributed), each having mean and variance . Then Example: the sum of uniform random variables Suppose that are i.i.d.
    [Show full text]
  • Topic 5: Noise in Images
    NOISE IN IMAGES Session: 2007-2008 -1 Topic 5: Noise in Images 5.1 Introduction One of the most important consideration in digital processing of images is noise, in fact it is usually the factor that determines the success or failure of any of the enhancement or recon- struction scheme, most of which fail in the present of significant noise. In all processing systems we must consider how much of the detected signal can be regarded as true and how much is associated with random background events resulting from either the detection or transmission process. These random events are classified under the general topic of noise. This noise can result from a vast variety of sources, including the discrete nature of radiation, variation in detector sensitivity, photo-graphic grain effects, data transmission errors, properties of imaging systems such as air turbulence or water droplets and image quantsiation errors. In each case the properties of the noise are different, as are the image processing opera- tions that can be applied to reduce their effects. 5.2 Fixed Pattern Noise As image sensor consists of many detectors, the most obvious example being a CCD array which is a two-dimensional array of detectors, one per pixel of the detected image. If indi- vidual detector do not have identical response, then this fixed pattern detector response will be combined with the detected image. If this fixed pattern is purely additive, then the detected image is just, f (i; j) = s(i; j) + b(i; j) where s(i; j) is the true image and b(i; j) the fixed pattern noise.
    [Show full text]