What is a base tracking algorithm?
Basis pursuit algorithm is an algorithm for solving an equation constraint problem that minimizes unknown parameter L1 norm.
Base tracking is a means that is usually used in signal processing. Solving the L0 norm in the optimization problem is the basic idea of the L1 norm is the basic idea of the basics.
For example, I originally an optimization problem:
Min || x || _0 (the minimum of the L0 Norm) Subject to y = ax.
This || x || _0 is how many non-zero elements in X; then we ask Min || X || _0, just want to know what to decide the Up to 0 elements.
However, the L0 norm has non-convexity, not very good to solve, then we turn to solve the Optimization of the L1 norm.
So, the base tracking algorithm is turning to solve
Min || x || _1 (the minimum of the L1 norm) Subject to || Y-AX || _2 = 0 (2 norm)
This || x || _1 is the absolute value of X; then we ask Min || X || _1, what is the solution X whose absolute value is minimized.
More popular, such as I ask a linear equation group
AX = B
X is the unknown amount of our request. This A matrix is not a partial array. It is an unexpected matrix, then this linear equation group has several forms. So which group we want to do?
If in general, a group of least two multipliers can be obtained directly, that is, x = (a'a) ^ (- 1) a'b. But we now use the basis tracking, just want to get a set of 0 elements with the most solutions.
A new base tracking solution algorithm is proposed. Depending on the signal characteristics, the dictionary is adaptively selected; through the approximation of the L1 norm, the constrained extreme value problem is converted to an unconstrained problem, and a new iterative algorithm is quickly solved; several typical signal experiments are verified This method has a good denoising effect. Key words: base tracking dictionary denoising basis tracking method is a new method for signal sparse representation. It seeks the most sparse representation of the signal from a complete (over-complete) function (basis) collection, i.e., using as little base as possible to accurately represent the inherent intrinsic characteristics of the signal. The base tracking method uses the norm of the system as the measurement of the signal sparseness, and the signal sparse indicates that the signal is scattered by minimizing the L1 norm is defined as a class of constrained extreme value problems, which in turn is to be solved by linear planning issues. At present, the base tracking method has a good application in the field of one-dimensional signal processing.
The Squiration Strong University Statistics Working Group represented by David L.donoho has taken a lot of good application results in one-dimensional signal denoising and super resolution. Although a new linear planning algorithm is used - an internal point algorithm, a base tracking method is to minimize a global target function because of all dictionary vectors, its computation amount is still large. It is because of the difficulty of solving large-scale linear planning issues, the current base tracking method is limited to one-dimensional signal denoising and super resolution. This paper proposes a new idea to solve the above constraints. First, the dictionary is adaptively selected according to signal characteristics; through the approximate representation of the L1 norm, the constrained extreme value problem is converted to an unconstrained problem, and a rapid solving of a iterative algorithm; finally passed several types of typical signal denoising experiments To verify the application effect of this method. The experimental results show that the improved basis tracking method can be quickly stabilized. At the same time, there is a good denoising effect.
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1 Dictionary
For the observed discrete signals S∈H, H is Hilbert space, the dictionary φ = {γ φ = = φγ, gamma ∈ γ} in a given H, where the γ is an indicator set, and φγ is the base function, also known as atom. The base tracking method defines the signal sparse representation to the following extreme problem, namely
Among them, αγ (γ∈γ) is a representation coefficient. If the vector in the dictionary represents a matrix φ, the coefficient represents a column vector, then (1) can be represented as min || α || 1 Subject to S = φα (2) in the case of noiseless observation Consider the following model: Y = S + σZ where S is the real signal, y is the observation signal, Z is the standard Gaussia white noise, σ is the noise root mode. Base tracking method Denoising is summed up to solve the following optimization problems:
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The above optimization problem is committed to minimizing signal reconstruction errors, and the representation of the signal is most sparse. The regularization parameter λ controls the balance between allowed errors and sparseness. According to (3), the core problem of the curing method to noise involves three aspects of the selection of atoms, the structure of the dictionary, solving algorithm design. Among them, the structure of the dictionary is an important part of the base tracking method. In order to exactly explicitly indicate signals, the dictionary and signal applications are adaptive, or bytes are obtained from signals. Typically, the dictionary used by the base tracking method is complete, complete, owed, and the like. You can design the dictionary based on the prior information of the signal and actual needs. The typical dictionary is complete or complete. For simple signal denoising, only a complete dictionary is generally required. For composite signals to noise, it is often necessary to construct a complete dictionary. For complete dictionaries, there is a sparse representation of signals, as noise is always standing. The dictionaries used in this article are: (1) Heaviside Dictionary
The atoms in this dictionary are not orthogonal, but the one-dimensional discrete signal S = (S1S2 ... SN) of any length is N, which is said:
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The Heaviside dictionary has a top triangle, a simple structure, and is good at capturing mutation characteristics in a fragmentary constant signal. (2) Time - Scale Dictionary (Wavelet Dictionary) Take the HAAR Wavelet Dictionary: Your Wavelet: ψ = L [0, 1], Mother Cell Cylinder: ψ = 1 [1/2, 1], 1-1 [ 0, 1/2] HAAR wavelet dictionary contains a translation and telescopic transform of a small-wavelength, and a translational transformation of a small-wavelene group. Set ψ = (A, B, V), where α∈ (0, ∞) is a scale variable, b∈ [0, n] characterization position, v ∈ {0, 1} characterizes the gender. HAAR wavelet dictionary is:
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Contains n atoms to form a set of orthogonal groups. Of course, there are other types of wavelet dictionaries, although some small-wavelens do not have a clear wavelet function expression similar to HAAR wavelet, but their dictionaries have similar discrete structures similar to the HAAR wavelet dictionary. Compare usually mainly Daubechies, Coiflet, Symmlet, etc. The wavelet dictionary is applied to a slide smooth signal. (3) Heaviside Dictionary + Wavelet Dictionary For more complex composite signals, the SMS is unable to get the sparse representation of the signal, and several dictionary can be synthesized to get a complete dictionary. For example, Heaviside Dictionary + Wavelet Dictionary. Technology area
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