ELEC Dr Reji Mathew Electrical Engineering UNSW

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1 ELEC 4622 Dr Reji Mathew Electrical Engineering UNSW

2 Dynamic Range and Weber s Law HVS is capable of operating over an enormous dynamic range, However, sensitivity is far from uniform over this range Example: for a luminance dynamic range of 10 6 Will require approximately 20 bits to capture this rage 20 bits to capture with an increment step size of 1 Makes digitising images a difficult task How can we get away with 8 or 10 bits per pixel?

3 Dynamic Range and Weber s Law HVS is capable of operating over an enormous dynamic range, However, sensitivity is far from uniform over this range Example: for a luminance dynamic range of 10 6 HVS operates in a non-linear mode So that the number of actually discernible intensity levels is vastly smaller than 10 6

4 Dynamic Range and Weber s Law HVS responds approximately logarithmically to luminous intensity Weber s law: Smallest detectable intensity smallest change in luminous intensity δy, which can be detected against a background of intensity y, is proportional to y. δy y is a constant. This makes sense in terms of the logarithmic model above. Differentiating y above with respect to y

5 Weber s law: Dynamic Range and Weber s Law can be understood in terms of uniform sensitivity to changes in y We can represent image intensities in the log domain Then uniform changes in the digital representation correspond to approximately uniform changes in perceived intensity That is quantization (e.g. into 8 bits per pixel) takes into account perception of the HVS In practice, this is rarely done Image intensities: typically represented in gamma corrected domain Gamma correction: possesses similar properties as the log representation

6 Dynamic Range and Weber s Law Quantize or discretise after applying the Gamma correction Quantization error Larger error but remember Webber s law

7 Dynamic Range and Weber s Law Typical gamma values for RGB representations γ = 2.2 to γ = 2.8 For a different representation Lab or CIE Lab γ = 3.0 this tracks Weber s law reasonably closely over the most useful range of intensities

8 Standard Colour Image Representations XYZ representation representation of the effect of colour stimuli on the HVS linear representation, so that a large number of bits are usually required to represent the three colour coordinates common to use floating point representations Other standard colour spaces can be derived from XYZ Can easily convert between any of these representations

9 Standard Colour Image Representations Linear srgb Linear so NO Gamma correction s standard whitepoint and primary chromaticities are chromaticity vectors: points on the normalised x y plot

10 Standard Colour Image Representations Gamma Corrected RGB derived from the relevant linear RGB colour space by applying a gamma correction function Gamma function is identical for each colour channel standard srgb colour space uses γ = 2.4 and β = Can you specify the steps in converting from XYZ to srgb? Remember convert to linear RGB (Matrix multiplication) followed by Gamma correction

11 Standard Colour Image Representations YIQ, YUV and YCbCr opponent colour spaces developed initially for television broadcast linear mappings of gamma corrected RGB coordinates YCbCr popular colour space for digital image processing Intensity Colour channel difference signals

12 CIE Lab Standard Colour Image Representations developed in order to reflect psychometric observations concerning the sensitivity of the HVS to changes in colour. opponent colour space linear mapping of gamma corrected primaries, where the primaries are whitepoint normalized XYZ values γ = 3 and β = 0.16

13 CIE Lab Standard Colour Image Representations developed in order to reflect psychometric observations concerning the sensitivity of the HVS to changes in colour. opponent colour space linear mapping of gamma corrected primaries, where the primaries are whitepoint normalized XYZ values White point normalisation γ = 3 and β = 0.16

14 Standard Colour Image Representations X 0, Y 0, Z 0 are the XYZ tri-stimulus values corresponding to the whitepoint of the illuminant to which the assumed viewer is adapted X 0, Y 0, Z 0 are given by Surface reflectance s λ = 1 CIE Lab (or just Lab) is actually a family of colour spaces Due to L a λ Adaptation illuminant

15 Standard Colour Image Representations CIE Lab colour space can be used to express the perceptual significance of colour differences [L 1, a 1, b 1 ] t and [L 2, a 2, b 2 ] t be two slightly different colours Perceptual significance of colour difference is CIE δe CIE Lab colour space: Perceptual difference between similar colours - approximated by the Euclidean distance between their Lab coordinates δe = 1 corresponds to a just noticeable difference (JND) between the colours.

16 Colour and Illuminant Conversion Remember: starting from a monitor signal [m 1, m 2, m 3 ] vector to [XYZ] Provided the monitor transfer matrix is non-singular Constraint m p 0 restricts the set of tri-stimulus responses which can be feasibly induced. Convex combinations of the three chromaticity vectors Solid triangle having these three chromaticities as its vertices

17 Colour and Illuminant Conversion But what is the correct [XYZ] vector to represent to the viewer HVS perceives colour in a manner which is sensitive to the ambient viewing conditions Consider [1] Capturing a surface under original illumination and [2] Displaying this to a viewer adapted to another illumination We want to stimulate the viewer s cones in such a way as to produce sensation of the original scene surfaces, viewed under the illuminant conditions to which the viewer is adapted.

18 Colour and Illuminant Conversion Illumination L o original s(λ) ρ 0 (λ), ρ 1 (λ), and ρ 2 (λ), Illumination L a? adapted s(λ) t x (λ), t y (λ) and t z (λ)

19 Colour and Illuminant Conversion Illumination L o original s(λ) Illumination L a ρ 0 (λ), ρ 1 (λ), and ρ 2 (λ), adapted s(λ) t x (λ), t y (λ) and t z (λ)

20 Colour and Illuminant Conversion To perform colour conversion can we find a matrix A, Highly unlikely that such a colour conversion matrix exists

21 Colour and Illuminant Conversion We could instead find a conversion matrix that minimises the squared error for a collection of surfaces. Or build a statistical model of typical surface reflectance Example: minimize the expected squared error Where the expectation is taken over the statistical distribution of typical surface reflectance functions s(λ)

22 Colour and Illuminant Conversion Problem with the squared error metric Does not account for the fact that errors in certain colours are more important to the human viewer than errors in other colours. That is not all error are the same some are less noticeable to the HVS Of particular importance is the scene whitepoint White surfaces in the original scene should be reproduced as white surfaces under the illuminant to which the viewer is adapted

23 Colour and Illuminant Conversion Of particular importance is the scene whitepoint White surfaces in the original scene should be reproduced as white surfaces under the illuminant to which the viewer is adapted White surface: reflects light uniformly over all wavelengths, i.e. s(λ) = 1, λ viewer s adapted whitepoint original scene whitepoint Whitepoint constraint

24 Colour and Illuminant Conversion How to incorporate the whitepoint constraint? may be imposed on the squared error minimization problem Alternatively, when P = 3, it is common to simplify the colour conversion matrix A to be the diagonal matrix which equalizes the scene and adapted whitepoints set of 3 multiplication factors, one for each colour channel referred to as a white balancing operation white balancing is not generally sufficient by itself

25 Colour and Illuminant Conversion Often in addressing colour conversion problems neither the original scene illuminant L o (λ), nor the adapted illuminant L a (λ), are exactly known. Therefore need to make some assumptions Assume outdoor images are captured under a D50 illuminant, adapted illuminant is a daylight colour known as D65 numeric quantity in this labeling is the colour temperature, D Kelvin D Kelvin

26 Colour and Illuminant Conversion /white-balance-neutral-not-alwaysnatural

27 Summary The course notes are your primary source of reference So refer to the notes to understand the concepts and ideas presented in this course What follows is a list of important points and questions that can help you in preparing for the final exam.

28 Fourier Transform (FT) Summary Discrete Time Fourier Transform (DTFT) Interpolation of a bandlimited continuous signal, f(t), from its impulse samples, x[n], based on the DTFT and FT relationships Sampled sequence x[n], captures all the information in the continuous signal f(t), provided f(ω) = 0, ω π This is the Nyquist s sampling theorem: illustrated by the cycle diagram

29 Summary Properties of the FT Conjugate Symmetry - if input signal is real valued Spatial shifts causing a phase shift in the frequency domain Parseval s relationships Rotation of an image causes rotation of the FT same direction and angle

30 Summary Finite Impulse Response (FIR) Filters Convolution applying for images (2D signals) Application by moving windows Separability Handling of boundaries Filter Design should know the three methods Windowing Rectangular window, Hanning Window Frequency Sampling minimize a weighted squared error between desired and actual frequency responses Transformation Method McClellan Transformation Mapping kernel map 2-D design into a 1-D design problem Block diagram of the basic filter stages

31 Summary Filters you should know Ideal/Rectangular Filter rectangular in the frequency domain What is the impulse response (i.e. response in the image domain)? Gaussian Filter Gaussian in the frequency domain Impulse response? Moving Average filter rectangular in the image domain What is it s frequency response? Implementation- computational complexity is independent of the filter s region of support. How can this be realised?

32 Summary Filter properties how to determine the following: DC gain BIBO gain Zero phase shift in the output?

33 Summary Image Acquisition Model Virtual system PSF how to obtain this? Problems caused by aliasing when aliasing is present, is the end-to-end system is shift invariant? For alias free sampling we must ensure that the system PSF is Nyquist bandlimited Correction for optical blur Why do this correction?

34 Summary Image interpolation and resampling Increasing sampling density what kernel to use? Decreasing sampling density what kernel to use? Should know Implementation of Sinc kernels for increasing rate (upsampling) and decreasing rate (downsampling) Winowed Sincs using Hanning window Implementation of bi-linear interpolation Advantage & disadvantage in comparison to Sinc kernels? Freq respones of bi-linear?

35 Summary Discrete Fourier Transform (DFT) Relationship to the DTFT DFT is a sampled version of the DTFT Can we obtain one from the other? This sampled representation must contain all the information in the spectrally continuous DTFT, provided that the signal is zero outside the region over which the DFT is taken. Filtering with the DFT - Time-domain aliasing When does this occur? How can we avoid this? Filtering with the DFT Under what condition can filtering be performed by multiplication in the DFT domain? (i.e. avoid aliasing in the spatial domain?) What are the steps to employ the Overlap-Add-Save method?

36 Summary Properties of the DFT Can you explain the circular convolution property? Under what conditions is circular convolution and true convolution identical? Fast Fourier Transform (FFT) Observation on which the basic FFT is based on? Given N = 2 r What is the number of real-valued multiplications? What is the number of real-valued additions? How does this compare with DFT?

37 Summary Correlations and matched filtering where y[n] = y[ n] is the mirror image of the pattern, y [n]. Correlation may equivalently be viewed as filtering the image with an FIR filter having region of support R y = [ M, M] 2 How can use the FFT to achieve efficient implementation?

38 Summary What are the problems with Correlation? Normalized Cross Correlation (NCC) Can you write the equation for NCC? How to achieve efficient implementation of NCC?

39 Gaussian Pyramid Summary Laplacian Pyramid Can you draw block diagrams explaining how to realise these pyramid representations? Applications?

40 Summary Bi-level Morphology Erosion Dilation Opening Closing Can you demonstrate the above given a foreground and structuring element? Morphological filters Filtering what? Extension to Grey scale Examples of Grey-scale opening and closing

41 Summary Motion Estimation Block based motion Metrics used for matching (eg MSE, MAD) Can you determine complexity of full search block based matching for motion estimation? Optical Flow (OF) equation What is the OF constraint? What is the OF equation? What are the assumptions underpinning OF estimation? Implementation strategies for OF?

42 Summary Segmentation and Texture Texture features Multi-Dimensional Similarity Measures and Feature Vectors Power Spectrum Density (PSD) How can it be approximated by using the DFT? Colour Can you explain metamers? Can you derive the monitor transfer matrix given the spectral power distributions associated with the three display channels and the tri-stimulus functions tx(λ), ty(λ) and tz(λ)?

43 Summary Colour (continued) What is the purpose of Gamma correction? Is it possible for a monitor with 3 display primaries to cover the whole gamut of the HVS? Explain? HVS perceives colour in a manner which is sensitive to the ambient viewing conditions. If you knew the ambient lighting conditions of a viewer as well as the original lighting conditions under which an image was captured what can you do to reproduce the correct stimulus to the viewer for a given object in the scene?

44 Summary Colour (continued) Standard Colour Image Representations Linear XYZ Linear RGB Gamma corrected RGB YCbCr CIE Lab

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