Singular spectrum analysis-based image sub-band decomposition filter banks

This paper presents a novel type of a filter bank for image decomposition that incorporates “singular spectrum analysis” (SSA). SSA is based on “singular value decomposition” (SVD).The presented filter banks provide good directional selectivity. Sub-band images highlight exposed directional features such as 0∘,90∘,+ 45∘, and − 45∘ of the input image. The proposed method can be seen as a tool to separate an image into layers or to present an image as a sum of its high-frequency directional components and a low-frequency approximation.


Introduction
Filter banks play an important role in signal processing and are used in many areas. In general, the main purpose of filter banks is to divide an input signal into sub-bands containing distinct frequency domains. They are known for their applications in the area of image processing, such as image compression and denoising, and many more. Different filter bank designs can be implemented depending on the application. This process is famous under different names: sub-band decomposition, sub-band transform, sub-band filtering, and multiscale transform.
In general, a filter bank consists of an analysis and a synthesis stage. During the analysis stage, the input signal is divided into sub-bands depending on the requirements of the application. The respective synthesis stage is the inverse transform to reconstruct the input signal from the sub-bands. Depending on the use case, there is a processing stage between the analysis and the synthesis stage. The processing stage is tailored to one of many possible applications, such as extraction of edges, removal of details, image compression, and compressed sensing.
A comprehensive book [1] of 2018 surveys widely used types of multiscale (sub-band) transform methods according to the historical development process: fourier analysis and fourier transform, wavelets together with different types of wavelet transform (discrete, continuous, undecimated, biorthogonal, complex, dual-tree complex, and quaternion), the reverse transform is implemented as a trivial addition of the created sub-band images.
The property of good directional selectivity is useful for image processing applications such as edge and object detection, face and gesture recognition, and feature extraction in machine learning.
To demonstrate the mode of operation of filter banks, the well-known synthetic "Octagon" image shown in Fig. 1a is used as the input. It was chosen for all experiments. It is a black-and-white PNG image with 600 × 600 pixels with a shape providing horizontal, vertical, and diagonal directional features.
Additionally, the natural "Cameraman" image, a black and white PNG image with 256 × 256 pixels shown on Fig. 1b, is used to demonstrate the performance of the developed algorithms. All experiments were conducted with the MATLAB toolkit.
The remainder of this paper is structured as follows: Section 2 observes the image sub-band decomposition methods with directional selectivity properties, regarding the directions 0 • , 90 • , + 45 • , and − 45 • . Section 3.1 provides the algorithm of SSA. Section 3.2 reviews how SSA is applied to vectors (one-dimensional data) and to two-dimensional (2020) 2020: 29 Page 3 of 18 Fig. 1 The input test images image data. Four schemes of how to apply one-dimensional Basic SSA to two-dimensional image data are introduced in Section 3.3. In Section 3.4, a novel scheme of creating "SSA-iSBD 2-channel filter banks" is presented. The outputs of the respective filter banks are visualized and discussed in Section 4, where "SSA-iSBD multi (4 and 6)-channel filter banks" are introduced, offering good directional selectivity. The paper concludes with Section 5.

Related work
Today, the classical "discrete wavelet transform" (DWT) is still a commonly used scheme for sub-band decomposition of signals (1989, Mallat's dyadic filter tree [2,12]). For image processing, 2D-DWT is an extension of DWT for two-dimensional input signals. Typically used mother wavelets (classes of wavelet bases) are Haar-, Daubechies-, Biorthogonal-, Coiflet-, and Symlet-wavelets [12]. 2D-DWT is suited well for image compression in JPEG2000 [12,13], denoising, and the creation of sparse images for compressed sensing [12,14]. However, 2D-DWT is not good enough for some specific applications such as pattern recognition. The reasons for that are discussed in the following. In Fig. 2, the "Octagon" input image was decomposed using 2D-DWT [15] with "Daubechies D4" as the mother wavelet, showing the first level of 2D-DWT decomposition. Every sub-band image has a size of 300 × 300 pixels as a result of down-sampling in the filter banks (decimation).
The two major drawbacks of 2D-DWT in image processing are: • Shift sensitivity: Shift sensitivity or the lack of shift invariance or shift variance causes unpredictable changes in the output coefficients if the input image is shifted slightly [16,17]. These drawbacks motivated scientists and researches to develop extensions of 2D-DWT as well as other sophisticated wavelet-based filter banks. Many other types of image sub-band decomposition schemes were developed, being beneficial in specific applications each. Only a few of them focus on directional selectivity, such as a clear separation of horizontal and vertical directional features, or even diagonal and antidiagonal directional features. Such algorithms are: • One of DWTs extensions, providing shift invariance, referred to as both "undecimated discrete wavelet transform" (UDWT) and "stationary wavelet transform." Its extension to two dimensions (2D-UDWT) is implemented most efficiently by the "algorithme a'trous" [12]. It was introduced by Holschneider et al., (1989 [18]). Fowler [19] provides an overview of different names for this transform. Without downsampling and aliasing, this filter is shift invariant by the cost of redundancy in the output information. The first level of 2D-UDWT applied to the "Octagon" image is similar to the result of processing the "Octagon" image with 2D-DWT (Fig. 2), with the difference that every sub-band image created with 2D-UDWT has the same size as the decomposed image: 600×600 pixels. 2D-UDWT still suffers from poor directional selectivity, but due to its shift invariance, it was successfully implemented in applications such as feature extraction [20] and image classification [21]. Furthermore, 2D-UDWT offers a substantial performance improvement in denoising [22,23].
• "Dual-tree complex wavelet transform" (DT-CWT) is an extension to DWT capable of approaching both shift invariance and good directional selectivity, proposed by Kingsbury (1998 [24-30]). This complex wavelet-based method offers a combination of properties that was not available in earlier approaches: a principally perfect reconstruction and good directional selectivity together with near shift invariance. According to Kingsbury [29], DT-CWT involves six complex wavelets oriented along the directions ± 15 • , ± 45 • , and ± 75 • . DT-CWT is suited well for numerous applications in image [16] and video [31] processing. A scheme based on DT-CWT in combination with a complex spline, developed by Chaundhury and Unser (2010 [32]), is able to detect the directions 0 • , 90 • , − 45 • , and + 45 • . An "Octagon" image decomposed with DT-CWT, based on a complex spline, was compared to an "Octagon" image decomposed with 2D-DWT [33]. Gabor-like transform is not invertible. An example of Gabor-like transform implementation in the industry is the identification of damaged elements made of polymer and polymeric composites [35]. • "Multiresolution SVD" (MR-SVD), presented by Kakarala and Ogunbona (2001 [6]), is an SVD-based method, demonstrating properties of directionality. Sub-images contain the mix of 0 • , 90 • , − 45 • , and + 45 • directional features. A comparison of the directional selectivity of MR-SVD with that of 2D-DWT, incorporating the "Octagon" image, is presented in [7].

Singular spectrum analysis
"Singular spectrum analysis" (SSA) is a technique initially designed for the analysis of time series. It is a signal processing technique based on "singular value decomposition" (SVD) that decomposes an input signal into several components. These components can be grouped and merged to compose subsequent components. These results can be interpreted as for example a trend, as an oscillation within the input signal, or as noise. SSA can be used for the extraction of trends, for smoothing, for the extraction of periodic components, or for finding structures in time series. It has been applied in physics, signal analysis, and mathematics, but it is also a valuable tool for market research, economics, and meteorology [36].
SSA was initiated by Broomhead and King (1986 [37]), but it was independently introduced by the researchers Danilov and Zhigljavsky in Russia as the "Caterpillar" method [38]. Today, comprehensive descriptions of the theoretical and practical concepts of SSA are available in [11].
In this paper, a preprocessing step is introduced that allows analyzing images with Basic SSA, by transferring the input image into a 1D-array first. This scheme of transforming 2D-arrays into 1D-arrays and vice versa is discussed later.
Both Basic SSA and 2D-SSA rely on the same four processing steps, as described in [39] and [41]. They only differ in step 1 and 4. This difference will also be discussed afterwards.
1. Compute the trajectory matrix X from the input array • The case of an input 1D-array 1D-array is represented as a vector F of length N: Here, the trajectory matrix X 1D is a Hankel matrix constructed from the input 1D-array F. The dimensions of this Hankel matrix are defined only by the window length L that must confirm to 1 ≤ L ≤ N. Let K = (N − L + 1) to get a Hankel matrix of dimensions L × K. The elements of the input 1D-array F are arranged into the Hankel matrix X 1D according to the following scheme [39]: (2020) 2020:29 Page 6 of 18 • The case of an input 2D-array Computing the trajectory matrix regarding a 2D-array, we have a matrix I of size N x × N y : The trajectory matrix is the "Hankel-block-Hankel" (HbH) matrix X 2D constructed from the input 2D-array I. The dimensions of the HbH matrix are defined by two window sizes L x and L y , chosen that The input 2D-array I is arranged into the HbH matrix X 2D of size (L x L y × K x K y ) according to the following rule [41]: Each H j is a Hankel matrix of size L x × K x consisting of elements from the input 2D-array I : 2. Singular value decomposition (SVD) of the trajectory matrix X. Denote λ 1 , . . . , λ L as eigenvalues of XX T taken in the decreasing order (λ 1 ≥ · · · ≥ λ L ≥ 0) and U 1 , . . . , U L denote as corresponding eigenvectors. Set d = rank(X), for simplicity is considered d = L.
where V i is defined as SVD of the trajectory matrix X is represented by λ i -is the i th singular value of X ; U i and V i -are the associated left and right singular vectors of X, for i = 1, . . . , d.
The collection ( is defined as the i th eigentriple of the SVD.

Eigentriple grouping
The grouping procedure splits the set of indices {1, . . . , d} into m subsets I 1 , . . . , I m . Let I = {i 1 , . . . , i p }. Then, corresponding to group I, the resultant matrix X I can be written as X I = X i 1 + · · · + X i p . The resultant matrices are calculated for each group I 1 , . . . , I m and the expansion( 8) provides the decomposition X = X I 1 + · · · + X I m (9) The choosing procedure of the sets I 1 , . . . , I m is defined as eigentriple grouping.
• The case of an input 1D-array: The case of an input 2D-array: Reconstruction of the signal, regarding each group.
• The case of an input 1D-array Opposed to the trajectory matrix X 1D , the approximated matrix X 1D I m is not necessarily a Hankel matrix anymore. To "hankelize" it, all elements of each antidiagonal are replaced with their respective averaged value. Then, the reconstructed 1D-arrayF can be assembled by simply arranging elements from this created Hankel matrix. Finally, the original 1D-array F can be reconstructed by: • The case of an input 2D-array The matrices X 2D I m is not necessarily a HbH matrix anymore. To allow reprojection, which is again a rearrangement of elements to create the reconstructed 2D-arrayĨ, X 2D I m must be transferred to a HbH matrix first. This is performed in two steps. First, each sub-matrix of X 2D I m (see Eq. (4)) is "hankelized" individually by averaging the elements of their antidiagonals. At second, all hankelized sub-matrices with the same index are averaged element-wise.Ĩ M is the 2D-array projected from group X 2D I m .

Grouping of eigentriples
In Section 3.1, the step of grouping the eigentriples was not specified yet. The algorithm proposed in this paper separates the set of eigentriples into two groups. Each of both groups is used to reconstruct an output signal, resulting in two reconstructed output signals for each given input signal. The first group of eigentriples contains only the first eigentriple related to the most significant singular value √ λ i of the trajectory matrix X. The reconstructed signal is referred to as main vectorF main for the 1D-array and main componentĨ main for the 2Darray, respectively. The second group of eigentriples contains all eigentriples except the first one. The reconstructed signal is referred to as residual vectorF res (1D-array) and residual compo-nentĨ res (2D-array).
An interesting feature of this grouping scheme is that F =F main +F res and I =Ĩ main + I res . In consequence, this scheme implements a two-channel filter bank with a simple reverse transform. This property can be exploited to lower the computational efforts: if (2020) 2020:29 Page 8 of 18 onlyF main orĨ main is calculated, thenF res orĨ res can be calculated through a substraction: F res = F −F main andĨ res = I −Ĩ main .

Processing images with Basic SSA
An input image I, being a 2D-array, can directly be filtered using 2D-SSA. Thus, it can be presented as a sum of a main and a residual component. In this section, four schemes are proposed that allow processing 2D-arrays with Basic SSA. To achieve that, the 2D-array I is transformed to a 1D-array F first to be able to perform Basic SSA. Then, after grouping the eigentriples into two groups as described in Section 3.2, the resulting 1D-arraysF main andF res are rearranged element-wise into the 2D-arraysĨ main andĨ res , respectively. Thus, Basic SSA is used to represent an input image I as a sum of a main (Ĩ main ) and a residual component (Ĩ res ).
As there is no unique scheme defining how to convert a 2D-array into a 1D-array and vice-versa, four possible ways of creating F from I are introduced now. They allow processing an input 2D-array with Basic SSA. They also offer the respective reverse transform to derive the output images from the resulting 1D-arrays: 1. f i are the elements of all columns of I, transposed and arranged one after the other as depicted in Fig. 3a. This modification of SSA is referred to as "Vertical-SSA." 2. f i are the elements of all rows of I, one after the other, as demonstrated in Fig. 3b. This modification of SSA is referred to as "Horizontal-SSA." 3. f i is the ith element of I regarding the path shown on Fig. 3c. This modification of SSA is referred to as "Diagonal-SSA." 4. f i is the ith element of I regarding the path shown on Fig. 3d. This modification of SSA is referred to as "Antidiagonal-SSA."

Presentation of "SSA-iSBD 2-channel filter bank"
As mentioned in Section 3.2, the proposed SSA-based algorithms decompose an image into a main and a residual component, configurable by selecting one of the introduced eigentriple grouping schemes. The decomposition is considered a "SSA-iSBD 2-channel filter bank as depicted in Fig. 4. Here, and in all following diagrams of this type, the main and residual component are reflected by a dashed and a solid line, respectively. Important to mention, the theory of SSA does not propose an optimal value for the window size/length. Instead, the window size/length depends on the application, and all window sizes/lengths used in this paper were chosen empirically to achieve a good visual presentation. To demonstrate, the "Octagon" input image was processed with the "SSA-iSBD 2channel filter bank, " where 2D-SSA is used as "chosen-SSA, " which can be also called "2D-SSA-iSBD 2-channel filter bank, " with the window sizes (L x , L y ) = (10, 10) and (30,30). The resulting main and residual component are shown in Fig. 5.
Note that the smoothness of main components and the thickness of the lines highlighting the edges of the "Octagon" in the residual components relate to the window size. The "2D-SSA-iSBD 2-channel filter bank" operates as a filter bank that separates the input image into a low-frequency and a high-frequency part. The cutoff frequency relates to the chosen window size.
To present alternatives to incorporating 2D-SSA as the "chosen-SSA" building block, four alternatives are presented here to create a "SSA-iSBD 2-channel filter bank. " These are "Vertical-SSA, " "Horizontal-SSA, " "Diagonal-SSA, " and "Antidiagonal-SSA. " The resulting main and residual components of the "Octagon" input image are shown on Fig. 6. The window length was L = 30 in all cases. A comparison of only the residual components highlights the differences of these filter banks with the bank incorporating 2D-SSA, Fig. 5d: • "Vertical-SSA " (Fig. 6b): all vertical directional features are removed, • "Horizontal-SSA " (Fig. 6d): all horizontal directional features are removed, • "Diagonal-SSA " (Fig. 6f): all diagonal directional features are removed, and • "Antidiagonal-SSA " (Fig. 6h): all antidiagonal directional features are removed.
Note that the "SSA-iSBD 2-channel filter bank" provides a "redundant form" of image decomposition: every component (main and residual) contains the same amount of pixels as the input image (N = N x N y ). Thus, N input pixels are transformed to a total of 2N output pixels. Independent from the type of SSA referred to by "chosen-SSA, " the input image can always be reconstructed by just adding the main and the residual component.

Discussion and results
In Section 3.4, "SSA-iSBD 2-channel filter banks" were presented. It was shown that in the case of using "Vertical-SSA, " "Horizontal-SSA, " "Diagonal-SSA, " or "Antidiagonal-SSA, " the residual component lacks one of the directional features. As such a removal is not the goal, this section introduces that a chaining of multiple "SSA-iSBD 2-channel filter banks" perform an extraction of directional features. These filter banks implement a multi-channel decomposition that deconstruct a given input image into a sum of multiple sub-images.

"SSA-iSBD 4-channel filter bank"
As a first step to realize such filter banks, the proposed SSA types are organized into orthogonal pairs. These pairs correspond to the properties of the residual components they create, whereupon the removed directional features are orthogonal to each other.
The following orthogonal pairs are introduced: • "Vertical-SSA " and "Horizontal-SSA " with the same window length L and • "Diagonal-SSA " and "Antidiagonal-SSA " with the same window length L.
The first SSA type of such an orthogonal pair is referred to as "chosen-SSA" and the remaining is referred to as "complement-SSA. " Both members of the pair are equivalent, so each can be selected to become the "chosen-SSA. " Figure 7 shows the "SSA-iSBD 4-channel filter bank. " It performs a 2-stage decomposition that decomposes an input image into a sum of four sub-images. Each output sub-image is the result of having the input image passing two consecutive "SSA-iSBD 2-channel filter banks" implementing an orthogonal pair.
The first filter bank implements "Vertical-SSA" and "Horizontal-SSA" as its orthogonal pair. This filter bank is referred to as "SSA-iSBD 4-channel filter bank, " type "vertical/horizontal, " with "Vertical-SSA" picked as the "chosen-SSA. " Figure 8 shows the four generated sub-images, whereas a window length of L = 30 was used.
(2020) 2020:29 Page 11 of 18 Fig. 7 The structure of the "SSA-iSBD 4-channel filter bank" • Sub-image 1 is the approximation, containing the low-frequency components ( Fig. 8a), • Sub-image 2 provides all vertical (90 • ) directional features (Fig. 8b), • Sub-image 3 contains all horizontal (0 • ) directional features with orientation (Fig. 8c), and • Sub-image 4 consists of the remains. They contain all diagonal (+ 45 • and − 45 • ) directional features (Fig. 8d). Fig. 8b and sub-image 3 in Fig. 8c contain the extracted 0 • and 90 • directional features, respectively. Noticeably, sub-image 1 in Fig. 8a contains only low-frequency components. This demonstrates that both "Vertical-SSA" (stage 1) and "Horizontal-SSA" (stage 2) extract not only the desired directional features, but they also let the low-frequency components pass. Furthermore, sub-image 4 in Fig. 8d contains all diagonal directional components. This highlights that the directional selectivity of all stages of the filter bank are good: if the low-frequency components, the vertical, and the horizontal directional features are removed, then the "remains" must contain the omitted diagonal directional features.

Sub-image 2 in
The second filter bank incorporates "Diagonal-SSA" and "Antidiagonal-SSA" as its orthogonal pair, with "Diagonal-SSA" as the "chosen-SSA. " The resulting filter bank, referred to as "SSA-iSBD 4-channel filter bank, " type "diagonal/antidiagonal, " extracts + 45 • and − 45 • directional features. The four resulting sub-images created with this filter bank are as shown in Fig. 9.
• Sub-image 1 is the approximation containing the low-frequency components (Fig. 9a), • Sub-image 2 provides all diagonal (+ 45 • ) directional features (Fig. 9b), • Sub-image 3 contains all antidiagonal (− 45 • ) directional features (Fig. 9c), and • Sub-image 4 consists of the remains, in this case vertical (0 • ) and horizontal (90 • ) directional features (Fig. 9d). Noticeably, the sub-images 2 and sub-images 3 created with both exemplary filter banks highlight not only the desired directional components, but they also provide slightly visible traces of the directional components contained in sub-images 4.
Regarding each of the "SSA-iSBD 4-channel filter banks, " N input pixels are transformed to a total of 4N pixels. The sum of all four sub-images is again equal to the input image.

"SSA-iSBD 6-channel filter bank"
The next target is a clear separation 0 • , 90 • , + 45 • , and − 45 • directional features with a single multi-stage filter bank. The "SSA-iSBD 4-channel filter bank, " type "vertical/horizontal, " ispresented in Section 4.1, is already able to separate 0 • and 90 • directional components and to provide a compound of all diagonal directional components (see Fig. 8). However, there are still two issues: the directional features with + 45 • and − 45 • are still not separated from each other, and the respective sub-image 4 does not explicitly contain the diagonal features, but the remains. Thus, if there were subsequent omitted components, they also would have been mixed into the remains. However, with subsequent filtering steps, it is possible to extract and to separate all diagonal directional features into distinct sub-images.
Obviously, other similar structures of "SSA-iSBD 6-channel filter banks" can be created, for example, a filter bank that is based on a "SSA-iSBD 4-channel filter bank, " type "diagonal/antidiagonal. " Such a filter bank will differ in the order of extraction of the directional features.

Fig. 10
The structure of the "SSA-iSBD 6-channel filter bank" implemented as an extension of the "SSA-iSBD 4-channel filter bank," type "vertical/horizontal" features. To compare, a related transform consisting of a combination of DT-CWT and a complex spline [32] is also able to detect the directions 0 • , 90 • , − 45 • , and + 45 • . DT-CWT is famous for its good directional selectivity. Using that transform, a decomposed "Octagon" image also provides clearly separated 0 • and 90 • directional components with slightly visible traces of the − 45 • and + 45 • directional features [33]. Thus, the "SSA-iSBD 6-channel filter banks" presented in this paper shares this property of providing good directional selectivity. As in the previous examples, all window lengths are L = 30. N input pixels are transformed to a total of 6N pixels. For the inverse transform to get a reconstruction, all six sub-images have to be added. Fig. 11 The sub-images 1-6 of the "Octagon" image produced by the "SSA-iSBD 6-channel filter bank" according to the filter bank structure shown in Fig. 10 (2020) 2020: 29 Page 14 of 18

Demonstration of "SSA-iSBD multi-channel filter banks" using a natural image
So far, all provided examples involved the synthetic "Octagon" test image. In this section, the performance of the "SSA-iSBD multi-channel filter bank" is rated using the natural "Cameraman" image (see Fig. 1b). At first, Fig. 12 presents the results of processing the "Cameraman" image with the "SSA-iSBD 4-channel filter bank, " type "vertical/horizontal" (see Fig. 7). Figure 13 presents the results of processing the "Cameraman" image with the "SSA-iSBD 4-channel filter bank, " type "diagonal/antidiagonal. " In the next experiment, the "Cameraman" image was decomposed with the "SSA-iSBD 6-channel filter bank. " Figure 14 presents the calculated sub-images 1-6 according to the filter bank structure shown on Fig. 10 with a window length L = 12.
As mentioned before, the window length L has to be determined empirically depending on the application. Here, the larger window length of L = 12 was selected to demonstrate its influence on the extraction of directional features.
A side-by-side comparison of Figs. 12a-c and 14a-c shows the influence of the chosen window size L on the cutoff frequency. The larger the value of L is, the smaller becomes the part of the spectrum contained in the low-frequency approximation.
All sub-images of Figs. 12, 13, and 14 have been contrast-stretched. The value of each pixel fits into the range from 0 to 255, calculated in the same way as used for the publication of MR-SVD [6].
To increase visibility, the same output sub-images 1-6 of the "SSA-iSBD 6-channel filter bank" are presented again on Fig. 15 in its normalized form. To achieve that, the value of each pixel was rescaled as min i,jĨi,j → 0 and max i,jĨi,j → 1. They are printed using the standard grayscale colormap. Such presentation is commonly used in related work [40,44]. A comparable decomposition of the "Cameraman" image was achieved with the first level of DT-CWT decomposition [44]. In that publication, Fig. 5 contains sub-band images that contain separated directional features for ± 15 • , ± 45 • , and ± 75 • using the same normalization scheme as used here for the images on Fig. 15. The quality of the performance of both filter banks is visually similar: the + 45 • and − 45 • directional features were extracted successfully by both. DT-CWT divides an input image into its low and high frequency components, with a cutoff frequency depending on the chosen mother wavelet (type and order). Regarding the "SSA-iSBD filter banks, " the window length L provides the means for such adjustments.

Conclusions
The "singular spectrum analysis-based image sub-band decomposition filter banks" ("SSA-iSBD filter banks") are introduced. Four types of "SSA-iSBD 2-channel filter banks" based on Basic SSA serve as building blocks for "SSA-iSBD multi-channel filter banks. " Due to the sub-band separation based on SVD, the proposed filter banks are not affected negatively by shift variance. To remind, shift variance is typical for wavelet-based image sub-band decomposition schemes.
Good directional selectivity is provided by the "SSA-iSBD 6-channel filter bank" that represents an input image as a sum of an approximation containing the low-frequency components and, separated from each other, the high-frequency components. These high-frequency components refer to distinct horizontal (0 • ), vertical (90 • ), diagonal (+ 45 • ), and antidiagonal (− 45 • ) directional features, and the remains. This transform is invertible: a simple addition of all components is sufficient to reproduce the input image. The presented novel filter banks introduce the window length as a parameter that has to be determined empirically depending on the application.
This new method of image sub-band decomposition is presented for the attention of the scientific community. Mathematicians may be interested in proving or explaining this heuristically developed method while engineers may incorporate this algorithm to improve their specific features extraction, edge detection, or face and gesture recognition tasks.