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Wavelets (Chapter 7)
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What are Wavelets? Wavelets are functions that “wave” above and below the x-axis, have (1) varying frequency, (2) limited duration, and (3) an average value of zero. This is in contrast to sinusoids, used by FT, which have infinite energy. Sinusoid Wavelet
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What are Wavelets? (cont’d)
Like sines and cosines in FT, wavelets are used as basis functions ψk(t) in representing other functions f(t): Span of ψk(t): vector space S containing all functions f(t) that can be represented by ψk(t).
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What are Wavelets? (cont’d)
There are many different wavelets: Haar Morlet Daubechies
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What are Wavelets? (cont’d)
(dyadic/octave grid)
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What are Wavelets? (cont’d)
j scale/frequency localization space localization
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Continuous Wavelet Transform (CWT)
Scale parameter (measure of frequency) Translation parameter, measure of time Normalization constant Forward CWT: Continuous wavelet transform of the signal f(t) Mother wavelet (window) Scale = 1/j = 1/Frequency
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CWT: Main Steps Take a wavelet and compare it to a section at the start of the original signal. Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity.
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CWT: Main Steps (cont’d)
3. Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.
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CWT: Main Steps (cont’d)
4. Scale the wavelet and repeat steps 1 through Repeat steps 1 through 4 for all scales.
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Coefficients of CTW Transform
Wavelet analysis produces a time-scale view of the input signal or image. space
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Continuous Wavelet Transform (cont’d)
Inverse CWT: double integral!
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FT vs WT weighted by F(u) weighted by C(τ,s)
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Properties of Wavelets
Simultaneous localization in time and scale - The location of the wavelet allows to explicitly represent the location of events in time. - The shape of the wavelet allows to represent different detail or resolution. space
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Properties of Wavelets (cont’d)
Sparsity: for functions typically found in practice, many of the coefficients in a wavelet representation are either zero or very small. Linear-time complexity: many wavelet transformations can be accomplished in O(N) time.
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Properties of Wavelets (cont’d)
Adaptability: wavelets can be adapted to represent a wide variety of functions (e.g., functions with discontinuities, functions defined on bounded domains etc.). Well suited to problems involving images, open or closed curves, and surfaces of just about any variety. Can represent functions with discontinuities or corners more efficiently (i.e., some have sharp corners themselves).
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Discrete Wavelet Transform (DWT)
(forward DWT) (inverse DWT) where
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DFT vs DWT FT expansion: WT expansion one parameter basis or
two parameter basis
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Multiresolution Representation using
fine details narrower, small translations j coarse details
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Multiresolution Representation using
fine details j coarse details
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Multiresolution Representation using
fine details wider, large translations j coarse details
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Multiresolution Representation using
high resolution (more details) j … low resolution (less details)
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Approximation Pyramid (revisited)
high resolution j = J low resolution j=0 high resolution j = J low resolution j=0 scale=1/j
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Image Pyramids
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Prediction Residual Pyramid (revisited)
Prediction residual pyramid can be represented more efficiently. In the absence of quantization errors, the approximation pyramid can be reconstructed from the prediction residual pyramid.
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Subband coding In subband coding, an image is decomposed into a set of bandlimited components, called subbands. Since the bandwidth of the resulting subbands is smaller than that of the original image, the subbands can be downsampled without loss of information.
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Perfect Reconstruction Filter
Z transform: Goal: find H0, H1, G0 and G1 so that
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Perfect Reconstruction Filter: Conditions
If Then
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Perfect Reconstruction Filter Families
QMF: quadrature mirror filters CQF: conjugate mirror filters
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2-D
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Efficient Representation Using “Details”
details D3 details D2 details D1 L0
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Efficient Representation Using Details (cont’d)
representation: L0 D1 D2 D3 in general: L0 D1 D2 D3…DJ (analysis) A wavelet representation of a function consists of a coarse overall approximation detail coefficients that influence the function at various scales.
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Reconstruction (synthesis)
H3=L2+D3 H2=L1+D2 details D3 details D2 H1=L0+D1 details D1 L0
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Example - Haar Wavelets
Suppose we are given a 1D "image" with a resolution of 4 pixels: [ ] The Haar wavelet transform is the following: L0 D1 D2 D3
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Example - Haar Wavelets (cont’d)
Start by averaging the pixels together (pairwise) to get a new lower resolution image: To recover the original four pixels from the two averaged pixels, store some detail coefficients.
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Example - Haar Wavelets (cont’d)
Repeating this process on the averages gives the full decomposition:
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Example - Haar Wavelets (cont’d)
The Harr decomposition of the original four-pixel image is: We can reconstruct the original image to a resolution by adding or subtracting the detail coefficients from the lower-resolution versions. 1 -1 2
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Example - Haar Wavelets (cont’d)
Note small magnitude detail coefficients! Dj Dj-1 How to compute Di ? D1 L0
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Multiresolution Conditions
If a set of functions can be represented by a weighted sum of ψ(2jt - k), then a larger set, including the original, can be represented by a weighted sum of ψ(2j+1t - k): high resolution j scale/frequency localization low resolution time localization
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Multiresolution Conditions (cont’d)
If a set of functions can be represented by a weighted sum of ψ(2jt - k), then a larger set, including the original, can be represented by a weighted sum of ψ(2j+1t - k): Vj: span of ψ(2jt - k): Vj+1: span of ψ(2j+1t - k):
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Nested Spaces Vj Basis functions: Vj : space spanned by ψ(2jt - k)
f(t) ϵ Vj ψ(2t - k) V1 … Vj ψ(2jt - k) Multiresolution conditions nested spanned spaces: i.e., if f(t) ϵ V j then f(t) ϵ V j+1
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How to compute Di ? f(t) ϵ Vj IDEA: define a set of basis
Functions that span the differences between Vj
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Orthogonal Complement Wj
Let Wj be the orthogonal complement of Vj in Vj+1 - i.e., all functions in Vj that are orthogonal to Wj Vj+1 = Vj + Wj
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How to compute Di ? (cont’d)
If f(t) ϵ Vj+1, then f(t) can be represented using basis functions φ(t) fromVj+1: Vj+1 Alternatively, f(t) can be represented using two basis functions, φ(t) from Vj and ψ(t) from Wj: Vj+1 = Vj + Wj
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How to compute Di ? (cont’d)
Think of Wj as a means to represent the parts of a function in Vj+1 that cannot be represented in Vj differences between Vj and Vj+1 Vj, Wj
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How to compute Di ? (cont’d)
using recursion on Vj: Vj+1 = Vj + Wj Vj+1 = Vj-1+Wj-1+Wj = …= V0 + W0 + W1 + W2 + … + Wj if f(t) ϵ Vj+1 , then: V W0, W1, W2, … basis functions basis functions
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Wavelet expansion (Section 7.2)
Efficient wavelet decompositions involves a pair of waveforms (mother wavelets): The two shapes are translated and scaled to produce wavelets (wavelet basis) at different locations and on different scales. encode low resolution info encode details or high resolution info φ(t) ψ(t) φ(t-k) ψ(2jt-k)
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Wavelet expansion (cont’d)
f(t) is written as a linear combination of φ(t-k) and ψ(2jt-k) : scaling function wavelet function Note: in Fourier analysis, there are only two possible values of k ( i.e., 0 and π/2); the values j correspond to different scales (i.e., frequencies).
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1D Haar Wavelets φ(t) ψ(t) Haar scaling and wavelet functions:
computes average computes details
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1D Haar Wavelets (cont’d)
Think of a one-pixel image as a function that is constant over [0,1) We will denote by V0 the space of all such functions. Example:
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1D Haar Wavelets Think of a two-pixel image as a function having two constant pieces over the intervals [0, 1/2) and [1/2,1) We will denote by V1 the space of all such functions. Note that Examples: ½ + =
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1D Haar Wavelets (cont’d)
V j represents all the 2j-pixel images Functions having constant pieces over 2j equal-sized intervals on [0,1). Note that width: 1/2j Examples: ϵ Vj ϵ Vj
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1D Haar Wavelets (cont’d)
V0, V1, ..., V j are nested i.e., VJ … V2 V1 fine details coarse details
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1D Haar Wavelets (cont’d)
Mother scaling function: Let’s define a basis for V j : 1 note alternative notation:
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1D Haar Wavelets (cont’d)
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1D Haar Wavelets (cont’d)
Suppose Wj is the orthogonal complement of Vj in Vj+1 i.e., all functions in Vj which are orthogonal to Wj
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1D Haar Wavelets (cont’d)
Mother wavelet function: Note that φ(x) . ψ(x) = 0 (i.e., orthogonal) 1 -1 / 1 1 = 0 . -1 /
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1D Haar Wavelets (cont’d)
Mother wavelet function: Let’s define a basis ψ ji for Wj : 1 -1 note alternative notation:
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1D Haar Wavelets (cont’d)
basis for V 1 : Note that inner product is zero! basis W 1 : j=1
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1D Haar Wavelets (cont’d)
Basis functions ψ ji of W j Basis functions φ ji of V j form a basis in V j+1
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1D Haar Wavelets (cont’d)
V3 = V2 + W2
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1D Haar Wavelets (cont’d)
V2 = V1 + W1
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1D Haar Wavelets (cont’d)
V1 = V0 + W0
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1D Haar Wavelets (cont’d)
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Example - Haar basis (revisited)
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Decomposition of f(x) f(x)= φ0,2(x) φ1,2(x) φ2,2(x) φ3,2(x) V2
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Decomposition of f(x) (cont’d)
V1and W1 V2=V1+W1
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Example - Haar basis (revisited)
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Decomposition of f(x) (cont’d)
V0 ,W0 and W1 V2=V1+W1=V0+W0+W1
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Example - Haar basis (revisited)
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Example
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Example (cont’d)
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Filter banks (analysis)
The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm (filter bank). Example: ψ(2t - k) a1k (j=1) ψ(t - k) a0k (j=0)
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Filter banks (analysis) (cont’d)
The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm (filter bank). Subband encoding! h0(-n) is a lowpass filter and h1(-n) is a highpass filter
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Example - Haar basis (revisited)
[ ] low-pass, down-sampling high-pass, down-sampling (9+7)/2 (3+5)/ (9-7)/2 (3-5)/2 V1 basis functions
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Filter banks (analysis) (cont’d)
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Example - Haar basis (revisited)
[ ] low-pass, down-sampling high-pass, down-sampling V1 basis functions (8+4)/ (8-4)/2
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Convention for illustrating 1D Haar wavelet decomposition
x x x x x x … x x average detail … re-arrange: V1 basis functions re-arrange:
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Convention for illustrating 1D Haar wavelet decomposition (cont’d)
average x x x x x x … x x detail …
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Orthogonality and normalization
The Haar basis forms an orthogonal basis It can become orthonormal through the following normalization: since
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Examples of lowpass/highpass analysis filters
Haar h1 h0 Daubechies h1
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Filter banks (synthesis)
The higher resolution coefficients can be calculated from the lower resolution coefficients using a similar structure.
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Filter banks (synthesis) (cont’d)
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Examples of lowpass/highpass synthesis filters
Haar (same as for analysis): g1 g0 Daubechies g1 +
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2D Haar Wavelet Transform
The 2D Haar wavelet decomposition can be computed using 1D Haar wavelet decompositions (i.e., 2D Haar wavelet basis is separable). Two decompositions Standard decomposition Non-standard decomposition Each decomposition corresponds to a different set of 2D basis functions.
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Standard Haar wavelet decomposition
Steps (1) Compute 1D Haar wavelet decomposition of each row of the original pixel values. (2) Compute 1D Haar wavelet decomposition of each column of the row-transformed pixels.
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Standard Haar wavelet decomposition (cont’d)
average (1) row-wise Haar decomposition: detail re-arrange terms … … x x x … x … … x x x x … … … … … …
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Standard Haar wavelet decomposition (cont’d)
average (1) row-wise Haar decomposition: detail from previous slide: row-transformed result … … … … … … … … …
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Standard Haar wavelet decomposition (cont’d)
average detail (2) column-wise Haar decomposition: row-transformed result column-transformed result … … … … … … … … … … …
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Example … … … row-transformed result … … … re-arrange terms
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Example (cont’d) column-transformed result … … … … …
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2D Haar basis for standard decomposition
To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions. φ0,0(x) ψ0,0(x) ψ0,1(x) ψ1,1(x) Example: V2=V0+W0+W1
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2D Haar basis for standard decomposition
To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions. φ00(x), φ00(x) ψ00(x), φ00(x) ψ01(x), φ00(x)
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2D Haar basis of standard decomposition
V2
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Non-standard Haar wavelet decomposition
Alternates between operations on rows and columns. (1) Perform one level decomposition in each row (i.e., one step of horizontal pairwise averaging and differencing). (2) Perform one level decomposition in each column from step 1 (i.e., one step of vertical pairwise averaging and differencing). (3) Repeat the process on the quadrant containing averages only (i.e., in both directions).
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Non-standard Haar wavelet decomposition (cont’d)
one level, horizontal Haar decomposition: one level, vertical Haar decomposition: … … x x x … x … … x x x x … … … … … … … … Note: averaging/differencing of detail coefficients shown
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Non-standard Haar wavelet decomposition (cont’d)
re-arrange terms … … one level, vertical Haar decomposition on “green” quadrant one level, horizontal Haar decomposition on “green” quadrant … … … … … … … … … …
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Example re-arrange terms … … … … … …
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Example (cont’d) … … … … …
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2D Haar basis for non-standard decomposition
Defined through 2D scaling and wavelet functions:
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2D Haar basis for non-standard decomposition (cont’d)
Three sets of detail coefficients (i.e., subband encoding) LL: average LL Detail coefficients LH: intensity variations along columns (horizontal edges) HL: intensity variations along rows (vertical edges) HH: intensity variations along diagonals
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2D Haar basis for non-standard decomposition (cont’d)
V2
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Forward/Inverse DWT (using textbook’s notation)
i=H,V,D H LH V HL D HH LL
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2D DWT using filter banks (analysis)
LH HH H LH V HL D HH LL HL
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Illustrating 2D wavelet decomposition
LH HH LL LH HH HL LL HL The wavelet transform can be applied again on the lowpass-lowpass version of the image, yielding seven subimages.
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2D IDWT using filter banks (synthesis)
H LH V HL D HH
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Applications Noise filtering Image compression Image fusion
Fingerprint compression Image fusion Recognition G. Bebis, A. Gyaourova, S. Singh, and I. Pavlidis, "Face Recognition by Fusing Thermal Infrared and Visible Imagery", Image and Vision Computing, vol. 24, no. 7, pp , 2006. Image matching and retrieval Charles E. Jacobs Adam Finkelstein David H. Salesin, "Fast Multiresolution Image Querying", SIGRAPH, 1995.
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Image Querying “query by content” or “query by example”
Typically, the K best matches are reported.
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Fast Multiresolution Image Querying
painted low resolution target queries
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