Wavelet-Based Multiresolution Analysis
Wavelet analysis is a branch of mathematics that combines functional analysis, spline analysis, harmonic analysis, Fourier analysis, and numerical analysis. It has provided significant breakthroughs in nonlinear fields, such as signal processing, image processing, speech analysis, pattern recognition, and quantum physics (Sun & Cui, 2020). In 1988, Stéphane Georges Mallat and Yves Meyer introduced the concept of multiresolution analysis for the construction of orthogonal wavelet bases, clearly illustrating the wavelet’s multiresolution characteristics from a spatial perspective (Yan et al., 2020). Mallat and Myer combined all previous methods for constructing orthogonal wavelet bases to propose a method for constructing orthogonal wavelets and a fast orthogonal wavelet transform method, which is known as the Mallat fast wavelet algorithm. This algorithm holds a seminal position in wavelet analysis that is similar to that of the fast Fourier transform algorithm in classical Fourier analysis (Sun et al., 2020).
Denoising plays a crucial role in the field of image processing and computer vision. First, denoising helps to enhance image quality and visual perception, making images clearer and easier to understand. Second, it contributes to improving the accuracy and robustness of subsequent image processing tasks, such as object detection and image segmentation.
However, existing denoising techniques still have some shortcomings. Some methods may remove noise at the expense of losing image details, resulting in blurred or distorted images. Other methods may not perform well on certain types of noise, such as texture or speckle. Additionally, some methods may require a significant amount of computational resources or complex parameter tuning, making them less efficient or easy to implement. Therefore, discussing the existing denoising techniques and their shortcomings is crucial for identifying the development direction of new methods and improving existing ones. By analyzing the pros and cons of existing techniques, we can place the contribution of new methods within the broader research framework, providing guidance and insights for further advancements in the field of image processing (Yan, 2023).
The structure diagram of a three-layer multiresolution tree is shown in Figure 1. The purpose of this tree is to provide guidance for constructing orthogonal wavelet bases that closely approximate the L2(R) space in terms of frequency, while filters with different bandwidths are equivalent to orthogonal wavelet bases with varying frequency resolutions. The figure shows that the multiresolution analysis does not consider the high-frequency part, but only deeply decomposes the low-frequency part. The relationship is H = C3 + B3 + B2 + B1.
Figure 1. Structure Diagram of a Multiresolution Tree
Defining Multiresolution
Multiresolution analysis in space L2(w) refers to a sequence of spaces: Gk}k∈ x in L2(w)(L2(W) is a square-integrable real space, as well as a signal space with finite energy, that satisfies the following conditions: approximability, scalability, monotonicity, translation invariance, and existence of Riesz bases.
Approximability is calculated using the formula shown in equation (1):
∩k∈xGk=0,close∪−∞∞Gk=L2W(1)Scalability is calculated using the formula shown in equation (2):
f(t)∈Gk⇔f(2t)∈Gk+1(2)This condition reflects the consistency of scale changes, nearly orthogonal wavelet transformations, and spatial variations.
Monotonicity is calculated using the formulas shown in equations (3) and (4):Gk⊂Gk+1(3) for any:
k∈X(4)Translation invariance is calculated using the formula shown in equation (5).
For any l ∈ X, there are:
δk2−k/2t∈Gk⇒δk2−k/2t−1∈Gk(5)Existence of Riesz bases is calculated using the formulas shown in equations (6) and (7):
Exist δ(t)∈ Go(6)Lead to δ2−k/2t−11∈X(7)
Equations (6) and (7) yield the Riesz basis that constitutes Gk.
Suppose that the low-frequency and high-frequency parts Bk and CK in the analysis tree decomposition are represented by Gk and Pk, respectively. Subsequently, Pk is the orthogonal complement of Gk in Gk+1, as shown in equation (8):
Gk⊕Pk=Gk+1k∈X(8)Therefore, the formula in equation (9) can now be used:
Gk⊕Pk⊕Pk+1⊕…⊕Pk+m=Gk+m(9)From equation (9), it can be deduced that the subspace Go of the multiresolution analysis can be approximated by a finite subspace, as shown in equation (10):
G0=G1⊕P1=G2⊕P2⊕P1=GN⊕PN⊕PN−1⊕…⊕P2⊕P1(10)Thus, space sequence{Pk│k ∈ X}has the nature shown in equations (11)–(13):
f(t)∈Pk⇒ft−2kn∈Pk...k,n∈X(11)f(t)∈Pk⇔f2t∈Pk+1...k∈X(12)Hp,f⇒0,whenk⇒∞(13)Similar to Gk, a function for the set ∮(t)∈ P0 is sought by randomly selecting from f ∈ L2(W). For each k ∈ X, a space Pk is formed with a standard orthogonal basis function set {∮k, n│n∈X}, as calculated in the formula shown in equation (14):
∮k,n(t)=2−k/2∮2−kt−n(14)Now, let fk ∈ Gk represent the approximation of f ∈ L2 (W) at a resolution 2-k obtained using Gk. It also represents the low-frequency component of the function f and corresponds to a “coarse image.”
When Ck∈Pk is used, the error of the approximation also represents the high-frequency component of the function f, which corresponds to the image “details.”
Therefore, equation (10) can be interpreted as shown in equation (15):
f0=f1+fC=f2+C2+C1=⋯=fN+CN+CN−1+CN−2+⋯+C2+C1(15)Because f = f0, equation (15) can be simplified as shown in equation (16):
f=fN+∑i=1NCi(16)The above equations illustrate the concept of the Mallat pyramid algorithm—namely, the concept of multiresolution analysis (Hu, et al., 1999, p. 19). Therefore, any function f ∈ L2 (W) can be completely reconstructed using the low-frequency component; i.e., the “coarse image” part of f at resolution 2-k and the high-frequency component; i.e., the “detail” part of f at resolution 2-k(1≤k≤N).