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

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