BIOMEDICAL SIGNAL AND IMAGE. DIGITAL PROCESSING. (Course 055-355-
5501). LECTURE NOTES. Faculty of Engineering, Tel Aviv University ...
Prof. L. Yaroslavsky
BIOMEDICAL SIGNAL AND IMAGE DIGITAL PROCESSING (Course 055-355-5501)
LECTURE NOTES
Faculty of Engineering, Tel Aviv University
MURPHY’S HANDY GUIDE TO MODERN SCIENCE: • • • •
If it’s green or it wriggles, it’s biology If it stinks, it’s chemistry If it doesn’t work, it’s physics To err is human, but to really fool things requires a computer
What is, then, digital signal and image processing?
DIGITAL SIGNAL AND IMAGE PROCESSING: SOLVING a GIGO PROBLEM
DIGITAL Garbage
IN
SIGNAL/IMAGE PROCESSOR
Gold OUT
Main tasks of signal and image processing: • Conversion of signals’ form( signal display, image formation ( reconstruction)) • Correcting sensors; signal calibration, standardization; image restoration • Measuring quantitative data and signal detection and parameter estimation • Interactive signal and image processing (image preparation and enhancement) • Automated signal and image analysis and understanding • Signal and image data coding for storage transmission • Mathematical simulation: design of signal sensors and imaging devices; testing signal and image processing methods; biological growth models, etc.
TYPICAL APPLICATIONS: • Processing ECG, EEG and similar signal • Diagnostic imaging: X-ray, CT, NMR, PET, SPECT, US video • Brain and heart cartography • Digital radiology and mammography • Digital and video microscopy, cell analysis and cytometry • Processing of electron micrographs • Virtual endo- and broncho- scopy • Telemedicine and telesurgery • Augmented reality surgery • Digital image and signal archiving
BIOMEDICAL SIGNAL AND IMAGE PROCESSING Syllabus: Signals, signal representations and mathematical models Digital representation of physiological signals: one-, two- and multi-dimensional discretization and element wise quantization. Fundamentals of signal and image data compression. Signal transformations. Convolution, Fourier and Radon Transforms Discrete representation and efficient computational algorithms of signal transformations. Signal and image interpolation, resampling and geometrical transforms. Signal measurements and parameter estimation. Signal/image reconstruction, restoration, enhancement: linear and nonlinear filters
Prerequisites Elements of linear algebra. Elements of the probability theory and random processes. Signal theory.
References: 1. L. Yaroslavsky. Lecture notes. http://www.eng.tau.ac.il/~yaro 2. L. Yaroslavsky, M. Eden. Fundamentals of Digital Optics. Birkhauser, Boston, 1996 3. L. Yaroslavsky. Digital Picture Processing. An Introduction. Springer Verlag, Heidelberg, New York, 1985 4. R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2-nd Edition, Prentice Hall, 2002 5. R. N. Bracewell, Two-Dimensional Imaging, Prentice-Hall, 1995 6. A. Cohen, Biomedical Signal Processing, v. 1. Time and Frequency Domains Analysis. CRC Press, Boca Raton, Fl.,1986
Biomedical signal and image processing Lecture 1. Introduction. Mathematical models of signals Biomedical information systems and signal processing.
(Course 055-355-5501)
Signals and images in biology and medicine. Sensors and imaging devices. Main tasks of signal and image processing: - conversion of signals’ form( signal display, image formation ( reconstruction)); - correcting sensors; signal standardization; image restoration; - measuring quantitative data and signal detection and parameter estimation; - interactive signal and image processing (image preparation and enhancement); - automated signal and image understanding (“ pattern recognition, image analysis”); - signal and image data coding for storage and long distance transmission. - mathematical simulation: design of signal sensors and imaging devices; testing signal and image processing methods; biological growth models, etc. Digital versus analog signal and image processing: advantages and disadvantages
Mathematical models of signals. Signal space and bases definitions Classification of signals: analogue (continuous), discrete, quantized, digital. Examples. Relationship between continuous, discrete and digital signals in terms of signal space. Signal space. Linear signal space. Linear representation of signals. Bases. Scalar product. a ( x ) = ∑ αrϕ( x , r ); r
αr = ∫ a ( x )φ( x , r )dx X
Orthogonal bases. Kronecker delta and Dirac delta-function as bases orthogonality symbols. r −−s ∫ ϕ( x , r )φ( x , s )dx = δ(r , s ) = 0 X
∑ ϕ(x , r )φ(ξ, r ) = δ( x , ξ) : a ( x ) = ∫ a (ξ)δ ( x , ξ)dξ r
X
Integral transforms: α( f ) = ∫ a ( x )φ( x , f )dx; a ( x ) = ∫ α( f )ϕ( x , f )df ; δ ( x, ξ ) = ∫ ϕ ( x , f )ϕ (ξ , f )df X
F
F
Signal space metrics and processing quality criteria Discrete signals N −−1
L1 :
∑ a (k ) − b (k )
k ==0
1 /2
N −−1 p L p : ∑ a ( k ) − b (k ) k ==0
Continuous signals L1 : ∫ a ( x ) − b ( x ) dx X
1/2
2 L2 : ∫ a ( x ) − b( x ) dx X
1/ P
P L P : ∫ a ( x ) − b ( x ) dx X
Deterministic and statistical treatment of signals Euclidean metric and its statistical justification in signal processing. 2 N −−1 {bk = ak + n k }; P ({nk }) ∝ ∏ exp − (bk − a2k ) = monotonic function of k ==0 2σ n
{
}
(
)
M :max a ( k ) − b (k )
1/ p
N −−1 2 L2 : ∑ a (k ) −− b(k ) k ==0
L1 : sup a ( x ) − b( x ) ; X
2 ∑ (b k − a k ) ⇒ Euclidean metric N-1
k ==0
Local criteria: signal local approximation AVLOSS (k , l ) = AV ∑ LOC (m , n; a ( k , l )) LOSS (a (m , n ), aˆ ( m, n) ) m ,n
Problems: 1. 2. 3. 4. 5. 6.
Give examples of different continuous, discrete, quantized and digital signals and describe their relationships in terms of signal space. What is linear signal space and bases . What are orthogonal basis and delta-functions. Explain and compare deterministic and statistical approaches to signal processing and analysis. Give examples of signal approximation metrics. Justify MSE error metric. Explain the principle of signal local approximation.
Signals as mathematical functions
Function arguments
Function value Scalar (single component) signals
Continuous in values
Vectorial (multicomponent) signals
Continuous in arguments
Continuous signals
Onedimensional (1-D signals)
Multidimensional (2-D, 3-D, 4D, etc. signals)
Quantized in values (quantized) signals
Discrete in arguments (Discrete) signals
Signal digitization Digital signals Signal reconstruction
Classification of signal and mathematical models
Signal space
Equivalency cell
Representative signal
Signal space
