Computer Applications Final Study Guide

COMPUTER APPLICATIONS FINAL STUDY GUIDE Time Domain Event Detection Algorithm For Pattern Recognition the Clinical Applications include:      Monitoring – alarm situations in diagnostic monitoring (ex. ICU) in response to change in vital signal(s) o HR, BP, oxygenation, etc. Administration – of treatment by implantable devices o EX : Implantable defibrillators, pacemakers, insulin pumps Detection – of arrhythmias (EX: long term ECG recorder (Holter recorder)), heart rate variability analysis Cardiac gating – for cardiac imaging o EX: determining heart boundaries Processing – of large amounts of data o EX: imaging long term monitoring *Challenge: designing an algorithm that avoid missing true “events” (false negatives) while not falsely calling something on an event that is not (false positives) Amplitude Threshold Algorithms for Event Detection:     Simplest method “rate estimate”  ex. heart rate Algorithm: o Set amplitude threshold level o Once crossed the threshold an event is detected & event registered o Blanking period  Once the threshold crossing is detected, “blank” detection for a period of time (skip over a time interval) before searching for the next event (threshold crossing)  Amplitudes in the blanking period will be ignored if they are above the threshold Challenges: o How do you set the amplitude threshold?  % of max, % of mean, else? o How to set the width of the blanking period?  Ex. QRS in an ECG signal o o o o o  Width of QRS (<0.12 sec)?  Max HR? Range(45-400bpm)? How to handle changing amplitude of a signal, especially if precise timing of the onset of an event is important  Imprecision in detecting the onset of an event (jitter) using threshold crossing and blanking period algorithm  Sensitive to noise, baseline wander, change in amplitude, motion artifact. Change in signal polarity, morphology (waveform) Biphasic signals (pressure wave) Baseline wander (due to motion, respiration) Noise (white noise) Properties of the first and second derivatives:    Derivative: o High pass digital filter (1st deriv)  Amplified high frequency (rapid rates of changes) portions of the original signal (QRS in ECG)  Attenuates low frequency portions of the signal (P and T waves in ECG) o Small errors (small amplitude, high frequency noise) get magnified when taking the derivative First Derivative: o First derivative (slope) of a signal has zero values at max and min time points o High pass filter Second Derivative: o Smooth the data first (low pass digital filter, moving average, and/or coherent averaging) to reduce the effects of taking derivative of data with high frequency noise  o Tells you if it is diastolic or systolic pressure  Systolic – concave down  Diastolic – concave up Note: o Small errors (EX: small amplitude and high frequency noise) get magnified when taking the derivative of a signal o Smith data first to reduce effects of taking derivative of data with high frequency noise  Low pass digital filter, moving average, and/or coherent averaging Correlation Methods: Notes from tutor:       Autocorrelation: o Detection of repeating events in 1 channel or across much channels  ECG Correlation Coefficient: o Plot 2 signals against each other and look at relationship between x and y o Positive and negative correlation Coherent Averaging: o Put ECG on top of each other and match them up see if they are similar Cross Correlation: o Finding similarities (correlation) of a set of signals as a function of time displacement  Shift one signal through the other and find correlation at each shift value o If the signals match each other and are the same then it is essentially the auto correlation Correlation between two signals (x[n] and y[n]) implies that they change with respect to each other in a structured way Types of correlations: o Correlation coefficient o Cross correlation function o Autocorrelation function Correlation Coefficient   Measure of the strength and direction of correlation between two signals Answers if two signals correlate to each other      -1 <= rxy <=1 o rxy= is proportion of the variance of y[n] that is predictable from the variance x[n] o determines the degree of match (the strength and direction of the correlation) between the shapes of the two signals  NOT THE MAGNITUDE Random noise tends to be self-canceling Correlation coefficient is of use to us when the relationship between y[n] and x[n] is linear or approximately linear o Linear regression o If you cross correlate with someone that is not linear you are misinterpreting the data and underestimating Correlation function: o Plots x and y if there was a time delay o Detects common rhythms even when there is a time delay Correlation coefficient in a template matching algorithm o Estimate the CC between measures signal and some reference waveform (template) o If CC exceeds some threshold, detect the event o Blanking period o Reference (template) signals Types:  Simulated waveforms that look like desired signal  Average (coherent avg) waveform estimated from measured data o Threshold for correlation coefficient  Training data  Use collection of actual measured data  Calculate correlation coeff  Pick correlation coeff threshold that separates event from eventless o Is there enough data o Is the data represented  Bootstrapping  Simulate a large number of random eventless signals and correlate with the reference signal (event)  Pick threshold from distribution of correlation eff (the 99% point at the tail of the distribution)  To create simulated eventless signal o Scramble event signals many times over o Scramble measured noise (eventless) signals many times over o Generate many time series from a known distribution using random number generators Autocorrelation and cross correlation  Auto correlation and cross correlation o Acquired over multiple channels o Detected of repetitive events in one channel or across multiple channels o Ex. Detection of EEG rhythms  Cross correlation o Takes into account the possibility that the two signals may have correlated components, but that these components in one signal may be delayed in time with respect to those components in the other signal  Use two signals that you think are related (at diff time)  Put on top of each other  Coherent averaging  Dominant o Determined by the successive values of the correlation coefficient taken at different time lags o Most common use is to determine whether x[n] and y[n] contain periodic components of the same freq. o Displays at the period (lag) of any periodic patter present in both x[n] and y[n] Cross correlation  make it the same  autocorrelation (check yourself) Cross correlation finds the error between the two signals  Two approaches for estimating o Direct Approach    Autocorrelation function o When the two signals are the same, the cross correlation function becomes the autocorrelation function of that signal o Autocorrelation is a cross correlation of itself o Is it the measure of rime related properties in the signal (time series) that are separated by fixed time delays o Most common use is to find periodicities in a signal:  Evaluates how well a signal correlated with itself various time lags  Attempts to locate the dominant periodic element (rhythm) within a signal Signal Averaging (Coherent Averaging)     Most biomed signals are a combination of constant signal component s[n] and an uncorrelated random (noise) signal component r[n] o Y[n]=s[n]+r[n] Filters (low pass) work well when the frequency contents (bandwidth) of the constant signal and noise don’t overlap o If there is overlap (broadband noise) between constant signal and random noise, removing noise with filtering will also attenuate (distort) the constant signal the solution is signal averaging – digital technique for separating a repetitive signal from random noise without introducing signal distortion o take first signal then put on top of next signal to see if its repetitive o uses one signal o the original signal has NOISE o average the two o at the end you want to see the dominant rhythm (dominant rhythm = where the two signals are the same) How to determine the coherent average for a signal   o Align multiple measured events with respect to some time marker o Add up corresponding data points across multiple events and divide each some by the total number of events o Result: template of desired event Assumptions o Noise must be random, stationary, uncorrelated with the signal o Temporal position of each signal waveform must be accurately known o Signal must be repetitive, with No change in signal morphology o If the noise of the signal is random, it can be mathematically shown that averaging “m” repetitions of a signal  Maintains the amplitude of the constant signal component, s[n]  Reduces the variances of the noise component by factor of m  Improves the signal to noise ratio (SNR) by a factor of √ m  SN Rm =√ m∗SNR Challenges o Time alignment of multiple events to create template  Possible choices  External time signal  Some feature of the event (fudicial point)  cross correlation function o lag time with the highest correlation between two measured events Frequency Domain Analysis    characterize signals in the frequency domain o express signal energy as a function of frequency instead of time o provides information about signal periodicity and morphology Generally o Rapidly changing signals (steep slopes, sharp inflection, etc.) have relatively high frequency content o Slowly changing signals have relatively low frequency content Why is it important to determine the frequency content (bandwidth) of a biomedical signal? o Provides quantitative information about signal morphology and periodicity o Analog to digital (A/D) conversion: sampling frequency and the Nyquist Criterion    Reconstruction of an analog signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth  Fs>2fmax o Design of digital filters to separate signal from noise (increase SNR) o Design of instrumentation for faithful detection of a physiologic signal o Discriminatory features (automated detection algorithm) o Frequency domain signal processing algorithms o Identification of internal properties and response of linear, time invariant systems Types of Fourier Analysis o Fourier Series o Discrete Fourier Series o Fourier Transform o Discrete Fourier Transform (DFT)  Fast Fourier Transform (FFT) computationally efficient algorithms for computing DFT Which type Fourier Analysis to use o Deterministic signals  Fourier Series  Continuous & periodic  Answers the question : given a continuous periodic signal, how do we find its trigonomic series presentation  Discrete Fourier Series (DFS)  Discrete & periodic  Fourier Transform (transform = nonperiodic)  Continuous & nonperiodic o Can be any point in time o Goes from neg infinity to infinity  Discrete Fourier Transform / Fast Fourier Transform  Discrete & finite (non-periodic) o Random signals  DFT or FFT o Biomedical (deterministic & random) signals  Usually finite duration, digital signals  Use Discrete Fourier Transform or FFT Fourier Series    Asks the question  given a continuous periodic signal, how do we find it trigometric series representation The coefficents an and bn can be used to plot the frequency spectrum o Frequency content, bandwidth, spectral content, or periodogram For plotting frequency spectrum (spectral representation) it is convenient to represent x(t) by introducing a phase angle in the argument of the cosine term o Amplitude and phase o frequency harmonic Discrete Fourier Series    Used to calculate the Fourier series coefficient of a discrete and periodic signal, x[n] Cannot detect frequencies above fs/2 Amplitude spectrum is obtained by plotting the o Amplitude of X[K]=|X[K]| vs. K*fundamental frequency K is the discrete Resolution of the resulting spectrum, X[K]      Spatial resolution: frequency between two spectral lines o Fs/N 1 1 = o T total N∗∆ t Depends only on Ttotal = total duration of signal being analyzed o does not depend on sampling frequency (fs) so long as Nyquist criterion is satisfied As we increase the period, Ttotal, we improve the spatial resolution Zero padding – increases length of a signal by appending zero values to the signal o Increases the spectral resolution, not by adding no information, but by interpolating existing information Windowing – applied to short time signals prior to DFT to reduce “spectral leakage” due to edge discontinuities by gradually attenuating the signal edges Fast Fourier Transform (FFT)    Computationally efficient algorithms for computing DFT The principle – the decomposition of the transform of a sequence of N samples into a number of transforms of shorter sequences leads to great reduction in computation efforts Determines the frequency content of an array data[N] of length N and sampling frequency, fs A/D Conversion    Types of signals o Continuous (analog)  Most analog signals are analog  continuous in both dependent and independent variable (time and amplitude)  Defined for every instant of time o Discrete signals  Data is only defined at discrete intervals of time (discrete in time, continuous in amplitude)  Finite resolution in time o Digital signals  Both time and amplitude are discrete  Finite resolution in time and amplitude ‘ Ultimate goal: convert the analog signal to a digital signal with a minimum loss of information Two important aspects of data acquisition and A/D conversion o Sampling rate (Nyquist criterion) o Type of A/D converter (# bits, polarity, full scale) A/D Conversion steps: 1. Amplification a. The signal from the electrode/sensor is usually quite small in amplitude b. Signal amplification is needed to bring the amplitude of the signal into the range of the A/D converter i. Maximize A/D conversion resolution 2. Analog low pass filter (anti-aliasing filter) a. Aliasing – information that messes with the overall signal b. Highest frequency in the analog signal may be unknown c. Filtering of the analog signal using an analog low pass filter is needed to band limit the analog signal d. Band limiting a signal is needed to decide on the proper sampling frequency (fs) based on the Nyquist criterion to avoid aliasing i. Fs> 2Fmax e. Upper limit on sampling frequency is set by the time needed for the A/D conversion process f. Tutor Notes i. Filtering of the analog signal using an analog low pass filter is needed to bandlimit the analog signal 1. Bandlimiting is needed to decide on the proper sampling frequency based on Nyquist criterion 2. Fs>2fmax 3. Sample and hold circuit (continuous to discrete converter) a. A/D conversion process is not instantaneous b. Sample and hold circuit i. Sample the filtered analog signal (xa(t)) as nearly instantaneous as possible and ii. Holds the sampled value (xs(t)) constant during the subsequent A/D conversion process c. Goes from analog to discrete i. Discrete- finite in time, infinite in frequency 4. A/D converter (ADC) a. Changes the analog voltage values stored by the sample-and-hold circuit to a digital representation b. Quantization i. Assigning the sampled signal values to discrete amplitude levels (quantization level) ii. Number of quantization  2N iii. Unipolar operators: 1. 0V  5V iv. Bipolar operators: 1. -5V  5V v. Amplitude resolution 1. Q=full scale range / 2N c. Encoding i. The quantized samples (decimal numbers) into binary format ii. Uses 1 and 0 d. Goes from (analog) discrete to digital Sampling theory, Nyquist criterion, aliasing error     Nyquist criterion o Fs > 2Fmax Sampling points that are further apart than 1/(2Fmax) will confused low and high frequency components in the original signal  Aliasing error o Loss of high frequency components o Corruption of low frequency components o High frequency components appear as low frequency components Fn= Nyquist frequency = cutoff frequency = fs/2 How to avoid aliasing errors in digital data analysis o Digital filters do not work o Remove frequency content of the original analog data that are above the Nyquist frequency prior to the AD conversion  Use analog low pass filter (anti aliasing filter)  Sample signal at a minimum of 4 times the highest frequency (fmax) Linear, time (shift) invariant (LTI) systems:    An LTI system must be linear and time invariant o A set of linear, constant coefficient differential equations (continuous systems) o A set of linear, constant coefficient differential equations (discrete systems) Why are the desirable o Easy to determine the output o Describe the system by the impulse response function o Convolution of the input and the impulse response function Example: o Continuous LTI system  analog low pass filter o Discrete LTI system  digital filter (ex. Hanning Filter)  Characterized by its impulse response function, h[N] o Knowing h[N] we can compute the system output y[n] for any input x[n] o Convolution summation  Time domain  Y[n]= x[n] * h[n] o Convolution operator is commutative  Y[n] = x[n] * h[n] = h[n] * x[n] LTI system impulse response function     Equals to the time domain response of the system to an input that consists of all the frequencies at equal amplitudes Impulse function δ (t) o Discrete time impulse function The Fourier transform of an impulse function is everywhere unity  all frequencies at equal energies and at zero phase Impulse response of an LTI system = time domain signal which appears at the output of the system when an impulse function is applied to its input For a given LTI system to find its output y[n] for an input x[n] :   If we known h[n] impulse response function o Time domain convolution o Frequency domain convolution If you do not know h[n] o Find h[n] by sending an impulse function through the system, then convolution of h[n] and x[n] to get y[n] 1-D Digital Filters:     Important o Pass certain frequencies and reject all other o Smooth data o Eliminate noise, create signal to noise ratio Digital vs Analog o Digital  Consists of a mathematical operation  Can be used to preform operations such as moving average, differentiation, integration, etc o Analog  Uses electronic component such as resistors, capacitors, amplifiers Order of a digital filter o Highest discrete time delay used in the input output equation of the filter Digital filters are divided into two major types: FIR and IIR Finite impulse response (FIR)   Non- recursive & feedforward o Non- recursive  Filter output depends only on current and past input (ex: unfiltered data), no feedback o Finite  Filter impulse response function or weighing function, h[n], has few terms Advantages o Stable (BIBO)  Output is bounded for every input that is bounded o Linear phase (phase distortion minimized)     Linear phase response – all frequency components of the unfiltered signal are delayed by the same amount of time after going through the filter If any of the frequency components is delayed more than the others after going through the filter, distortion occurs Disadvantages o Can be difficult to synthesize  To determine the required coefficients to perform a specific task  High pass or low pass Examples o Smoothing or “moving average” filters  Used to reduce high frequency noise  Hanning filter (3 point moving average) 1  y [n ] =( )¿ ¿ 4 o Notch (band stop or band reject) filters  Rejects a small frequency band but allows all other frequencies to pass  Removal of noise of particular frequency or frequency range from a signa without significant attenuation of higher and/or lower frequencies 1  y [n ] =( )( x [ n ] + x [ n−1 ] +x [ n−2 ]) 3 Recursive or infinite impulse response filters (vox):   Recursive: o filter output depends not only on the input (unfiltered data) but also on the previously filtered values  feedback Infinite: o filter impulse response function has infinite number of terms o trapezoidal integration h ( x [ n]−x [ n−1 ] ) y [ n ]= y [n −1 ]+  2 Advantages o Sharp roll offs o Computationally simpler than FIR Disadvantages o Can be unstable due to feedback o Phase issues (usually non linear phase) ()   To filter data using a digital filter 1. Given the filter equation of y[n] as a function of x[n], plug in values of x[n] to determine y[n] 2. Given the impulse response function, h[n] a. Convolution in time domain  y[n]=h[n]*x[n] b. Convolution in frequency domain  Y [ ω] =H [ω ] X [ω ] i. Followed by inverse Fourier Transform of Y [ ω] to get y[n] ii. X [ω ]= FT ( x [ n ]) iii. H [ ω ]=FT (h [ n] ) iv. Y [ ω ] =FT ( y [ n] ) 1. Advantage a. No analytical expression required for the filter frequency response function 2. Disadvantage a. Implementation is complex and time consuming Digital Filter Terminology     Cutoff frequency o The highest passband frequency for low pass filters o The lowest passband frequency for high pass filters o Upper and lower cutoff frequencies for a bandpass filter Bandwidth of a filter o Frequency width of the passband of a filter For a low pass filter o bandwidth = cut-off frequency = fc = -3 dB point (or frequency where amplitude = 0.707 of the maximum amplitude) Amplitude spectrum o AKA frequency magnitude spectrum in dB (bode plot) = 20log10|H(f)| in units of decibels (dB) 2-D digital filters, image analysis Kernel:   2D filter impulse response function Typically a square of odd dimensions (3x5 , 5x5, 7x7, etc.) o Prevents image shifting o Prevents distortion  Its center originated (in most cases)  origin is located at the middle of the kernel o Center point: [0,0] Image resolution:   Pixel – smallest unit of a digital image (2D) o Measure of image resolution o Picture element o Smallest pixel is highest resolution Voxel o Volume element o Smallest sample, or data point, on a regularly spaced, 3D grid o Smallest voxel is highest resolution Edge:    Used for feature detection (ex. Tumor, organ) High pass filter Edge detection o Identify the edges (regions of high spatial frequency) of objects within an image o Edge  Points in a digital image in which image brightness changes sharply due to difference in material properties  Region of high spatial frequencies o Use a high pass filter o Second derivative-based filter o Kernel designed to increase the brightness of the center pixel relative to its neighboring pixels Smoothing an image:   Decrease disparity among pixel values by averaging nearby pixels Low pass digital filters Thresholding:  Convert grayscale image into a binary image Sharpening an image   Enhance detailed information within an image High pass digital filters Computational Modeling:     Model – a simplified representation of a system which mimics its relevant features Mathematical model o Translation of out understanding of how a system function into mathematical language  Set of one or more equations, mostly differential equations o Consists of independent (time/space) and dependent (pressure/flow/concentration) variables which describe the physiological system under study Linearity o If linear  integration and laplace o If nonlinear  numerical methods : Eulers & Runge Kutta Coupling/non-coupling Objectives of Mathematical Modeling  Research o Signal processing: mechanistic interpretation of physiological data   o Answer What If questions:  Predict how a physiological system would react to changes in one of the system parameters  Design experiments that would reveal more information about the physiological system being studied o Communicate our understanding of a given system to other scientists Education: o Large scale models are used for teaching and as training tools  Medical schools use mathematical models of various physiological systems to teach students the underlying mechanisms of these systems, and to answer the what if questions Clinical Applications o Determining drug regimen, design of prostheses, artificial organ, control of implacable devices, etc. Compartmental Modeling        Compartment: amount of material that acts kinetically like a distinct, homogenous, well mixed amount of the material o Homogeneous and well mixed  All particles in a given compartment have the same probability of transition since they are considered indistinguishable by the system o No spatial distribution within the compartment (instantaneous mixing) More than one compartment can co-exist within the same physical space Dynamic, lumped, deterministic, continuous models Intuitively clear and easy to use Widely used in many field including o Biomedical engineering o Physiology o Chemical engineering o ETC Resulting model equations are in the form of ordinary differential equations (ODEs) Example o Glucose insulin kinetics and the pathogenesis of diabetes o A simple model of an epidemic o Hodgkin Huxely model  What is compartmental model or system o Finite number of compartment with specific interconnections among them o Number of compartments depends on both the complexity of the system and available data Numerical Methods for Solving Nonlinear ODE Blood Glucose Insulin Kinetics and Diabetes using Computational Modeling      the energy needed for cell functions is in the form of the chemical adenosine triphosphate (ATP) the energy needed to make the ATP is obtained in part by the oxidation of glucose o one of the main byproducts of carbohydrate digestion Other sources of energy for ATP production include proteins (amino acids) and fats (fatty acids) Organs such as the brain and retina depend almost exclusively on glucose for energy o Other body organs and tissues can use fats (fatty acids) and proteins (amino acids) as alternative energy sources It is important to maintain the blood glucose concentration at a sufficient level to fulfill these needs Why is it important to regular blood glucose concentration?    blood glucose concentration is tightly controlled o ranges between 80 and 90 mg/dl of blood in the fasting person each morning before breakfast blood glucose concentration can increase to 120-140 mg/dl during the first hour or so following a meal, but within couple of hours after a meal, it is restored to its normal level it is important that blood glucose does not exceed certain levels since: o high blood glucose concentration can lead to high osmotic (colloid or oncotic) pressure in the extracellular fluid which can lead to cellular dehydration (osmosis) o high blood glucose concentration can lead to glucose loss through urine  interferes with the ability of the kidneys to reabsorb fluids and electrolytes  potassium, magnesium, bicarbonate, sodium, chloride, calcium  osmotic diuresis  leads to depletion of the body of its fluids and electrolytes (extracellular dehydration) o high blood glucose can lead to both intracellular and extracellular dehydration can lead to circulatory shock o high blood glucose (diabetes) contribution to a number of serious and potentially fatal disease (including heart disease, kidney failure, stroke blindness, etc.) How does the body regulate blood glucose concentration   pancreas: insulin liver: glycogen  insulin: secreted by the Islets of Langerhans in response to increase in glucose plasma concentration o pancreatic cells Functions of insulin: o Makes liver and muscle cells more permeable to glucose o Increase activity of the enzyme glucokinase which increases glucose uptake  This enzyme catalysis the initial phosphorylation of glucose after it diffuses into the liver cells  Once phosphorylated glucose is temporary tapped o Increase activity of the enzyme glycogen synthase, which catalysis the formation of glycogen o Inhibits the enzyme phosphorylase which catalysis the break down of glycogen to glucose o Liver is an important blood glucose buffer system o In fasting hours when the blood glucose concentration falls and insulin secretion falls, the liver releases the glucose back to the blood through a process called glycogenolysis  And opposite (formation of glycogen)  glycogenesis  C- Programming Question Memory Allocation: Static Variables    “Static”  default initial value of zero Initialized once within the declaration when the function is first called o Not each time the function is called Good use of static variables o If you want to retain the value of a variable from one function call to the next o If you only want to initialize a variable at the beginning of the function execution o To initialize an array for the first time Global Variables      Global : does not belong to any particular function in the program o The value of a global variable can be accessed/changed by any function in the program Declaration of a global variable o Not defined within any particular function (declaration made outside any function in the program) Must be declared before used  in the program Default zero initial value Drawbacks o Reduces function generality, readability  Difficult to know what type of parameters a function requires as inputs or returns as outputs  Reduces flexibility in naming variables Pointers  Pointer: variable that contains the address of a variable or object (or file) Check if the file exists Determine the number of points in a file File name  * is used to declare a pointer variable The output of the fopen function is a file pointer or file address that is assigned to a variable of type “pointer to FILE” Read (r), write (w), append (a)     If the mode is in read mode an error occurs if the file does not exist If the type is in write or append , the program creates the file if it does not exist If the mode is in write mode, if the file already exists, its contents are destroyed If it is in append mode, write operations add data to the end of the file if it already exists