Fullcolor: acquired with a fullcolor sensor
Pseudocolor: assigned colors to a particular monochrome intensity or range of intensities.
$\quad\quad$Some of the gray-scale methods are directly applicable to color images.
Intensity for achromatic light
gray level for black-to-grays-to-white.
Radiance from the source, luminance by the observer, subjective brightness
$\quad\quad$It is important to keep in mind that having three specific primary color wavelengths for the purpose of standardization does not mean that these three fixed RGB components acting alone can generate all spectrum colors.
Primary colors of light: RGB
Primary colors of pigments/colorants: magenta, cyan, yellow
$\quad\quad$CRT: array of triangular dot patterns of electron-sensitive phosphor that produce different colors. In the case of LCD, light filters are used to produce the three primary colors of light.
$\quad\quad$Color characteristics: brightness, hue (an attribute associated with the dominant wavelength in a mixture of light waves, like yellow, red, orange), saturation (the relative purity or the amount of white light mixed with a hue).
Chromaticity: Hue, saturation
Tristimulus
CIE chromaticity diagram
color gamut: a range of colors produced by RGB monitors
$\quad\quad$A color model (color space, color system) is a specification of a coordinate system and a subspace within that system where each color is represented by a single point.
pixel depth: the number of bits used to represent each pixel in RGB space.
216 colors out of the 256 colors are de facto standard for safe colors. The safe-color cube has only valid colors on the surface cube.
$\quad\quad$Most devices that deposit colored pigments on paper such as color printers and copiers, require CMY data input or perform an RGB to CMY conversion. $$ \pmatrix{C\M\ Y}=\pmatrix{1\1\1}-\pmatrix{R\G\B} $$ A fourth color, black, combined with C M Y is added.
Practical for human interpretation.
Hue: the angle by which a point rotates around the black-white axis, from red, through yellow, green, cyan, blue, magenta and back to red. Undefined for a saturation of zero (white, black, and pure grays).
Brightness (Intensity): the axis
Saturation: distance from the axis
$\quad\quad$For human visualization and interpretation. Humans can discern thousands of color shades and intensities, compared to only two dozen or so shades of gray.
Gray scale
E.g.
Sometimes it is of interest to combine several monochrome images into a single color composite.
$\quad\quad$The results of individual color component processing are not always equivalent to direct processing in color vector space.
where the pixel values here are triplets or quartets.
$$
s_i=T_i(r_1,r_2,...,r_n)
$$
where
$\quad\quad$The hues directly opposite one another on the color circle are called complements.
Color complements are useful for enhancing detail that is embedded in dark regions of a color image, particularly when the regions are dominant in size.
The RGB complement transformation here do not have a straightforward HSI space equivalent.
$\quad\quad$Highlighting a specific range of colors in an image.
Approaches:
- display the colors of interest so that they stand out from the background
- use the region defined by the colors as a mask for further processing
$\quad\quad$One of the simplest way to slice a color image is to map the colors outside some range of interest to a nonprominent neutral color.
$$
s_i=\begin{cases}0.5 &\text{if } Dist(r,a)>D_0\
r_i&\text{otherwise}\end{cases}
$$
where
Tints, Shades and Tones
Hue, Tint, Tone and Shade
Color Balance
Most common use in photo enhancement and color reproduction.
CIELAB
Tonal range, key-type, high-key, low-key, middle-key
Tonal transformations: The idea is to adjust the image's brightness and contrast to provide maximum detail over a suitable range of intensities. In the RGB and CMYK spaces this means mapping all three or four color components with the same transformation function; in the HSI color space, only the intensity component is modified.
Color balancing: determined objectively by analyzing with a color spectrometer a known color in an image, accurate visual assessments are possible when white areas where the RGB components are present. Skin tones also are excellent subjects for visual color assessments.
$\quad\quad$The gray-level histogram processing transformations can be applied to color images in an automated way. It is reasonably successful at handling low-, high-, and middle-key images. It is generally unwise to histogram equalize the components of a color image independently, which results in erroneous color. ==The HSI color space is ideally suited to this type of approach, which can be performed by equalizing only the intensity component==.
$\quad\quad$The intensity equalization process do not alter the values of hue and saturation of the image, it does impact the overall color perception.
Smoothing by neighborhood averaging can be carried out on a per-color-plane basis. One can only smooth the intensity component in HSI space.
That is, to compute the Laplacian of each component image separately. Still, one can sharpen only the intensity component in HSI space.
It is natural to think first of the HSI space because color is conveniently represneted in the hue image. Typically, saturation is used as a masking image in order to isolate further regions of interest in the hue image.
Segmentation is one area in which better results generally are obtained by using RGB color vectors. Given a set of sample color points representative of the colors of interest, we obtain an estimate of the "average" color points that we wish to segment. By introducing a distance measure
Processing three individual planes to form a composite gradient image can yield erroneous results. It may yields accpetable results for decting edges, but accuracy is an issue.
Let
$\quad\quad$It is possible for color channers to be affected differently by noise. Filtering of full-color images can be carried out on a per-image basis or directly in color vector space. Filters like the class of order statistics filters can not be used this way because of the essence of vectors of each pixel in a color image.
$\quad\quad$Wavelet transforms are based on small waves, called wavelets, of varying frequency and limited duration, which provide the equivalent of a musical score for an image, not only what frequencies but also when.
$\quad\quad$If both small and large objects or low- and high-contrast objects are present simultaneously, it can be advantageous to study them at several resolutions.
$\quad\quad$A powerful yet conceptually simple structure for representing images at more than one resolutution is the image pyramid. An image pyramid is a collection of decresing resolution images arranged in the shape of a pyramid.
Approximations form an approximation pyramid and predicition residuals form a prediction residual pyramid, which are computed iteratively.
Approximation filter candidates: neighborhood averaging (mean pyramids), lowpass Gaussian filtering (Gaussian pyramids), no filtering (subsampling pyramids).
Interpolation: nearest-neighborhood, bilinear, bicubic.
$\quad\quad$In the absence of quantization error, the resulting prediction residual pyramid can be used to generate the complementray approximation including the original image without error. Prediction residual images can be highly compressed by assigning fewer bits to the more probable values.
$\quad\quad$In subband coding, an image is decomposed into a set of bandlimited components called subbands. The decomposition is performed so that the subbands can be reasembled to reconstruct the original image without error.
A digital filter:
$$
\hat{f}(n)=\sum\limits^{\infin}_{k=-\infin}h(k)f(n-k)
$$
where
Given a filter
-
$h_2(n)=-h_1(n)$ : sign reversal -
$h_3(n)=h_1(-n)$ : order and time reversal -
$h_4(n)=h_1(K-1-n)$ : order reversal -
$h_5(n)=(-1)^nh_1(n)$ and$h_6(n)=(-1)^nh_1(K-1-n)$ : modulation
The analysis filter bank breaks input sequence into two half-length sequences, the subbands. Synthesis bank filters combine them and produce the approximation, identical to the original input. The system is siad to employ perfect reconstruction fitlers.
$\quad\quad$The synthesis filters are modulated versions of the analysis filter. For perfect reconstruction, there are the following conditions:
Cross-modulation:
$$
g_0(n)=(-1)^nh_1(n)\
g_1(n)=(-1)^{n+1}h_0(n)
$$
or
$$
g_0(n)=(-1)^{n+1}h_1(n)\
g_1(n)=(-1)^{n}h_0(n)
$$
Biorthogonality condition:
$$
\langle h_i(2n-k),g_j(k)\rangle=\delta(i-j)\delta(n),\quad i,j={0,1}
$$
Some may satisfy orthonormality:
$$
\langle g_i(n),g_j(n+2m)\rangle=\delta(i-j)\delta(m),\quad i,j={0,1}
$$
Moreover, they satisfy:
$$
g_1(n)=(-1)^ng_0(K_{even}-1-n)\
h_i(n)=g_i(K_{even}-1-n),\quad i={0,1}
$$
where
$\quad\quad$An orthonormal filter bank can be developped around the impulse response of a single filter called the prototype. For biorthogonal filter banks, two prototypes are required. 1-D orthonormal and biorthogonal filters can be used as 2-D separablefilters for the processing of images, resulting in four subbands: approxiamtion, vertical detail, horizontal detail and diagonal detail.
where
Haar basis functions:
Define the integer
$$
\mathrm{H}_2=\dfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\1&-1\end{bmatrix}
$$
$\quad\quad$Our principal interest in the Haar transform is that the rows of
where
The coefficients
Biorthogonal system in Wikipedia
$$
\langle\phi_j(x),\tilde{\phi}k(x)\rangle=\delta{jk}=\begin{cases}0&j\neq k\1&j=k\end{cases}
$$
If the expansion set is not a basis, then they and their duals are overcomplete or redundant, forming a frame in which
$$
A|f(x)|^2\leq\sum\limits_k|\langle\phi_k(x),f(x)\rangle|^2\leq B|f(x)|^2
$$
for some
If
MRA (MultiResolution Analysis):
- The scaling function is orthogonal to its integer translates (Orthogonality)
- The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales.
- The only function that is common to all
$V_j$ is$f(x)=0$ - Any function can be represented with arbitrarily precision
Thus
$$
\begin{align}\phi_{j,k}&=\sum\limits_k \alpha_n\phi_{j+1,n}(x)
\&=\sum\limits_n h_\phi(n)2^{(j+1)/2}\phi(2^{j+1)}x-n)
\end{align}
$$
Set
Eq.(28) is called the refinement equation, MRA equation, dilation equation
It states that the expansion functions of any subspace can be built from double-resolution copies of themselves, i.e., from expansion functions of the next higher resolution space.
Define a wavelet function
the set
The scaling and wavelet function subspaces are related by
$$
V_{j+1}=V_j\oplus W_j
$$
And
Therefore $$ L^2(\mathbb{R})=V_\infin=V_{j_0}\oplus W_{j_0}\oplus W_{j_1}\oplus\dots $$ or even $$ L^2(\mathbb{R})=\cdots\oplus W_{-2}\oplus W_{-1}\oplus W_0 \oplus W_1 \oplus W_2\oplus \cdots $$ completely in terms of wavelets.
Since
From Eq.(31), we can define the wavelet series expansion:
Given
If the function being expanded is discrete, the resulting coefficients are called the discrete wavelet transform.
If
The complementary inverse DWT:
$$
f(n)=\dfrac{1}{\sqrt{M}}\sum\limits_{k}W_\phi(j_0,k){\phi_{j_0,k}}(n)+\dfrac{1}{\sqrt{M}}\sum_{j=j_0}^{\infin}\sum\limits_{k}W_\psi(j,k){\psi_{j,k}}(n)
$$
Normally,
CWT transforms a continous function into a highly redundant function of two continuous variables - translation and scale.
Forward continuous wavelet transform:
$$
W_\psi(s,\tau)=\sum^{\infin}{-\infin}f(x)\psi{s,\tau}(x)\ dx
$$
where
$$
\psi_{s,\tau}(x)=\dfrac{1}{\sqrt{s}}\psi\bigg(\dfrac{x-\tau}{s}\bigg)
$$
Inverse continuous wavelet transform:
$$
f(x)=\dfrac{1}{C_\psi}\int^{\infin}0\int^{\infin}{-\infin}W_\psi(s,\tau)\dfrac{\psi_{s,\tau}(x)}{s^2}\ d\tau\ ds
$$
where
$$
C_\psi=\int^{\infin}_{-\infin}\dfrac{|\Psi(\mu)|^2}{|\mu|}d\mu<\infin
$$
and
The Mexican hat wavlet: $$ \psi(x)=\Bigg(\dfrac{2}{\sqrt{3}}\pi^{-1/4}\Bigg)(1-x^2)e^{-x^2/2} $$
The transform provides an objective measure of the similiarity between
The FWT is a computationally efficient implementation of the DWT, also called Mallat's herringbone algorithm.
Starting from the refinement equation, scale
The Fourier basis functions guarantee the existence of the FFT, the existence of the FWT depends upon the availability of a scaling function for the wavelets being used.
Uncertainty Principle in information processing
The FWT basis functions provide a compromise between the time-domain representation and the frequency domain representation.
$\quad\quad$In two-dimensional wavelets, a two-dimensional scaling function
The inverse discrete wavelet transform: $$ \begin{align} f(x,y)&=\dfrac{1}{\sqrt{MN}}\sum_m\sum_nW_\phi(j_0,m,n)\phi_{j_0,m,n}(x,y)\&+\dfrac{1}{\sqrt{MN}}\sum_{i=H,V,D}\sum^{\infin}{j=j_0}\sum_m\sum_nW^i\psi(j,m,n)\psi^i_{j,m,n}(x,y) \end{align} $$ Like the 1-D discrete wavlet transform, the 2-D DWT can be implemented using digital filters and downsamplers.
The filter banks above can be used iteratively.
Applicaitons:
- Edge detection: set the approximation to zero and do the inverse DWT to obtain the edge, which method is able to extract edges in a certain direction.
- Noise removal: threshold certain detail coefficients and do the inverse DWT.
Each horizontal strip of constant height tiles contain the basis functions for a single FWT scale. A generalization of the FWT is made to gain greater control of the partitioning of the time-frequency plane.
Wavelet packets: conventional wavelet transform in which the details are filtered iteratively.
A P-scale, one(two)-dimensional wavelet packet tree supports $$ D(P+1)=[D(P)]^{2(4)}+1 $$ A single wavelet packet tree presents numerous decomposition options, whose number is so large that it is impractical to enumerate them individually. Classical entropy- and energy-based cost functions are applicable in many situations and are well suited for use in binary and quaternary tree searching algorithms.
Most transform-based compression schemes work by truncating or thresholding the small coefficients to zero. Define an additive cost function $$ E(f)=\sum_{m,n}|f(m,n)| $$ Steps:
- Compute both the energy of the node and the energy of its four offspring
- Compare the energy of the parent and the sum of the energy of the offspring, prune or preserve the offspring according to the relative result.