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Introduction to image processing and linear filtering
F. Tupin Athens Week
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Introduction to image processing
What is an image ? Spatial resolution and Fourier transform Quantification and histogram Linear filtering Athens Week
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Introduction to image processing
What is an image ? Spatial resolution and Fourier transform Quantification and histogram Linear filtering Athens Week
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What is an image ? Message support
Signal (continuous 2D signal) of a physical measure Passive imagery (Colour intensity) Active imagery (X-ray transmission, electromagnetic radiation, …) Other dimensions: Volumetric measures (3D medical images) 2D ½ (stereo images + intensity) Video (2D+t) 3D+t (images sequences) Athens Week
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What is a numerical image ?
A digital image (2D) is a matrix defined by its resolution (amount of pixels) its depth (amount of potential values for each pixel) its palette (colour look up table (CLUT)) Athens Week
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Digital Image Athens Week
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Different kind of images
Application Size Channels Video Visiophony TV HD TV 256 x 256 720 x 625 1920 x 1150 1 3 Bio-medical Tomography (IRM) Radiography 512x512 256x256x256 1024 x 1024 1-3 Remote Sensing 1990- 2000 x 3000 6000 x 6000 15000 x 15000 3-7 3-20 3-256 Robot vision Quality control Automatic driving 512 x 512 2-3 Defense Surveillance, tracking Athens Week
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From the continuous world to the numerical one
Image digitization Sampling (number of pixels) Quantification (number of bits per pixel) Athens Week
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From the continuous world to the numerical one
Image digitization Sampling (number of pixels) Quantification (number of bits per pixel) Athens Week
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Introduction to image processing
What is an image ? Spatial resolution and Fourier transform Quantification and histogram Linear filtering Athens Week
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Resolution : Vision (physical Image) vs. Digital Image
Human vision is able to look (in stereo) at from very small objects to huge ones The digital image has a fixed resolution determined by the amount of pixels of the image. Sampling Preserving the image frequencies (Shannon theory, 2D Fourier transform) Preserving the image content: depends of each application (psycho-visual tests, size of the smallest element to detect, …) Athens Week
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Resolution of an image Nl : amounts of lines per picture height H
Maximum frequency : Nl/2 cycles per picture height If D is given (in TV, D=6H), there is a direct relationship between cycles per degree and cycles per picture height H Viewer D Athens Week
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Sampling and information intuitive approach with example
Aliasing 1D : false frequency Athens Week
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Sampling and information intuitive approach with example
Aliasing 2D : false frequency and false direction of periodical structures ! Athens Week
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1-D Fourier transform Sampling at Fe spectrum periodization at Fe
Avoid aliasing: low pass filtering in [-Fe/2;Fe/2] All information in [-Fe/2;Fe/2]: normalized frequency Athens Week
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1D discrete Fourier transform
Signal sampling Fourier periodization Frequency sampling signal periodization Numerical signals = implicit periodization in both temporal and frequency domains ! Athens Week
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2D Fourier transform Fourier transform = signal projection on 2D complex sinusoid Frequency domain = new visualization of image information Fourier transform = complex signal (phase + modulus) Athens Week
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2D Discrete Fourier Transform
2D DFT Number of frequencies = number of pixels f(i,j) is the image in the spatial domain. Exponential term is the basis function corresponding to each point, F(k,l), in the Fourier space. Athens Week
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2D DFT - Properties F(0,0) corresponds to the average brightness
F(N-1, N-1) represents the highest frequency Inverse Fourier transform: Athens Week
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2D DFT computation å å Separation of DFT Formula:
Expressing two-dimensional Fourier transform in terms of series of two N one-dimensional transforms decreases the number of required computations. å - = 1 2 ) , ( N i ki e j f k P p å - = 2 1 ) , ( N lj i j e k P l F p Athens Week
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FFT Fast Fourier Transform(FFT) FFT is complex
N2 complexity is reduced to Nlog2N. Limited to image size N = 2n. FFT is complex Real / imaginary or magnitude /phase Magnitude linked to geometric information Phase linked to spatial position To re-transform we need to use both magnitude and phase. If real image, symmetry vs 0 Athens Week
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2-D discrete Fourier transform Simple examples
2 pixel wide vertical stripes Fourier transform Athens Week
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DFT : simple examples Vertical lines : No variation along y axis
Variation along x axis linked to the spatial frequency of the lines Fourier peak P(k,l) interpretation: |P(k,l)| related to signal frequency (O,P(K,l)) perpendicular to the pattern direction Athens Week
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DFT simple examples Athens Week
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Logarithmic operator applied
DFT Real examples Fourier transform Original image Logarithmic operator applied Athens Week
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DFT Real examples Athens Week
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DFT Real examples Athens Week
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Re-transform using only magnitude
DFT Phase information Phase image Re-transform using only magnitude The value of each point determines the phase of the corresponding frequency. The phase information is crucial to reconstruct the correct image in the spatial domain. Athens Week
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DFT properties Athens Week
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Back to the sampling problem…
Must be achieved at a frequency superior to 2 times the highest frequency in the signal (Nyquist frequency) Or the original image has to be filtered at a frequency below the half of the sampling frequency Athens Week
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Original and spectrum Athens Week
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Sub-sampled image and spectrum
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Zoom of the sub-sampled image
Image comparison Original image Zoom of the sub-sampled image Athens Week
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Zoom of the sub-sampled image spectrum
Spectrum comparison Original spectrum Zoom of the sub-sampled image spectrum Athens Week
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Explanation Original image spectrum is defined in [-0.5,0.5]
The 2-D pattern spectrum [-0.5,0.5] x [-0.5,0.5] is periodically repeated in (0,1), (O,2),…(p,q),… When doing the sub-sampling: The 2-D pattern spectrum [-0.5,0.5] x [-0.5,0.5] is periodically repeated in (0,0.5), (O,1),…(p/2,q/2),… The different patterns overlap in the Fourier domain The new spectrum corresponds to the [0.25,0.25] x [-0.25,0.25] area Athens Week
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Initial spectrum and pattern of the sub-sampled spectrum
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(0,0) (1,0) (-1,0) Periodization on (p,q) Athens Week
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Periodization on (p/2,q/2)
(0,0) (1,0) (-1,0) Periodization on (p/2,q/2) Athens Week
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(0,0) (1,0) (-1,0) Selection of [-0.25,0.25]x [-0.25,0.25] Athens Week
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Zoom of the sub-sampled image spectrum
Spectrum comparison Original spectrum Zoom of the sub-sampled image spectrum Athens Week
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Sub-sampling To avoid aliasing when sub-sampling
Low-pass filtering (suppression of all the frequencies higher than Fe’/2) Blurring effect but no artefacts are introduced Athens Week
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Introduction to image processing
What is an image ? Spatial resolution and Fourier transform Quantification and histogram Linear filtering Athens Week
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Depth of an image Binary representation: b bits, 2b levels white 11 1
10 01 00 black Athens Week
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The depth : Vision vs. Digital Image
Vision sensitivity limited to 6 to 8 bits per colour component i.e. max 24 bits Vision works in the (350nm) to red (700nm) range Digital images Much larger depth (in medical imaging and in remote sensing) Different wavelengths than vision: NMR imaging (0,001nm) to microwaves images (100000nm) Image visualization: 8 bits = 256 grey-levels (0 for black and 255 for white) Athens Week
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Visible domain rayons cosmiques gammas X ultra violet visible infra
rouge micro ondes TV radio 50 Hz 10-5 nm 10-3 nm 10 nm 500 nm 1500 nm 5 m 1000 m VIOLET BLUE GREEN YELLOW RED 380 nm 500 nm 555 nm 600 nm 720 nm Puech William Université Montpellier II - Nîmes Athens Week
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Image Histogram Histogram: grey level distribution Used to
F(x) = number of pixels having x as grey-level If normalized = probability of the grey-level Used to Contrast enhancement Classification Athens Week
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Histogram: example Histogram modes correspond to region of interests (clouds, parts of the boats, etc.) Athens Week
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Histogram …. be careful ! Athens Week
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Histogram …. be careful ! Athens Week
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Histogram Stretch Linear Stretch Histogram Equalised Stretch
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Colour spaces The RGB space (3 types of phosphors for colour excitation on a screen) B white black G R Athens Week
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Colours (cont.) Printing industry : CYM(K) Cyan Yellow Magenta (can be translated into RGB) Hue Saturation Intensity (HSI): artists, vision I green H white S red blue black Athens Week
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Colours(cont.) YUV : luminance, chrominances in TV
black and white TV backward compatibility Decorrelation of the components Y is containing most of the information Y=0.299R+0.587G+0.114B U=-0.147R-0.289G+0.437B V=0.615R-0.515G-0.100B Athens Week
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Video formats CCIR 601: 576 lines of 720 pixels, 25 images per second, 2 interlaced fields per frame, format 4:2:2 (YUYVYUYV) = 165 Mbit/s HDTV: 16/9 format, 2*576 lignes, 720*2(*4/3) pixels (non interlaced and even higher for digital cinema) TV 16/9 CIF: CCIR 601 /2 /2 /2 QCIF: CIF /2 /2 Athens Week
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Colour Composites A colour composite is a colour image produced through optical combination of multiband images by projection through filters. True Colour Composite: A colour imaging process whereby the colour of the image is the same as the colour of the object imaged. False Colour Composite: A colour imaging process which produces an image of a colour that does not correspond to the true colour of the scene (as seen by our eyes). Athens Week
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TM 7,4,1 TM 5,4,3 TM 5,7,2 TM 4,3,2 Athens Week
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Introduction to image processing
What is an image ? Spatial resolution and Fourier transform Quantification and histogram Linear filtering Athens Week
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Linear filtering Property : F(I+aJ)=F(I)+aF(J)
Image domain: convolution y=x*h Fourier domain : multiplication Y=XH Athens Week
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Convolution: involves moving a 'window' of a few pixels in dimension (e.g. 3x3, 5x5, etc.) over each pixel in the image, applying a mathematical calculation using the pixel values under that window, and replacing the central pixel with the new value. Athens Week
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Linear filtering Athens Week
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Linear Filtering Spatial domain (h convolution kernel)
Frequency domain Athens Week
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Linear Filtering Low pass filters:
noise suppression (= high frequencies) Athens Week
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Linear filtering High or band pass filters:
Selection of frequencies of interest High frequencies (edges) Specific frequencies (texture analysis) Athens Week
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High pass filter = edge detector
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Stripes of the zebra create high energy waves generally along the u-axis; grass pattern is fairly random causing scattered low frequency energy y x v u Athens Week
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Non-linear filtering Simple non-linear filters = median filters: the output of the filter is the median in a given window Min / Max : mathematical morphology Athens Week
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