Design and Polyphase Implementation of Rotationally Invariant 2D FIR Filter Banks Based on Maximally Flat Prototype

Abstract

This paper presents a design approach for a class of rotationally invariant 2D filters of finite impulse response (FIR) type, which may form circular filter banks with imposed specifications. The design is conducted analytically in the frequency domain and starts from a maximally flat low-pass prototype based on a trapezoidal function with specified width and slope. Its trigonometric approximation is derived using the Fourier series expressed analytically, truncated to a number of terms depending on the imposed accuracy. The chosen trapezoidal function leads to significantly smaller ringing oscillations compared to the approximation of an ideal square characteristic. By shifting the LP prototype to various frequencies, the desired filter bank is generated, where the component filters have a specified bandwidth, steepness, and overlap. The 2D circular filter bank results by applying a specific frequency mapping to the factored frequency response of the prototype filter. Thus, the frequency responses of the 2D filter bank components will also result in factored form, which is an advantage in implementation. The circular filter bank is designed in two versions, a uniform and a non-uniform (dyadic) filter bank. The designed filter banks have accurate shapes and relatively low order for the specified parameters. These filter banks are then used in a sub-band image decomposition application. Finally, an efficient implementation of these filters at the system level is proposed based on polyphase decomposition and the block filtering technique with a high degree of parallelism, resulting in a lower computational complexity.

1. Introduction

Two-dimensional digital filters have long been and continue to be a cornerstone and a fundamental topic of research in the field of digital image processing [1]. They find many applications in image processing due to their unique capabilities to select specific regions in the frequency plane and thus to perform a desired image filtering task. There are many types of 2D filters with various shapes of their frequency response, such as circular, elliptical, straight directional, wedge- or fan-shaped, square- or parallelogram-shaped, etc. These can be designed both in the FIR and IIR version. Two-dimensional FIR filters are easier to design due to their inherent stability, while IIR filters require supplementary conditions for stability to be imposed in the design process. Furthermore, unlike IIR filters, the FIR filters are able to ensure a linear phase frequency response on the entire frequency range in order to prevent phase distortions, which is a major requirement in image processing.

There are mainly two fundamental approaches to the design of 2D filters. One of them relies on global numerical optimization techniques and in principle leads to optimal filters. The other approach is essentially an analytical design technique, which starts from a so-called 1D prototype filter with imposed characteristics (for instance low-pass, maximally flat, with a specified bandwidth, etc.). Then, depending on the desired shape for the 2D filter, a particular 1D to 2D frequency transformation is applied to the given prototype, which directly yields the desired 2D filter. The essential advantage of the analytical design approach is that the 2D frequency response results in a factored closed-form and is usually parametric; therefore, the filter characteristic can be easily adjusted (in bandwidth, steepness, orientation, etc.). The most widely known and commonly used in the design of 2D FIR filters is the McClellan transformation, applied by researchers in papers such as [2,3]. Some design procedures for computationally efficient 2D FIR filters, with low arithmetic complexity, were based on sampling kernel interpolation [4] or tunable Farrow structure [5]. Various advanced mathematical tools have also been applied to 2D FIR filter design, such as positive trigonometric polynomials [6] and matrix-based algorithms for constrained least-squares [7]. A recent paper [8] summarized the transformations for FIR and IIR filter design. A synthesis technique for 2D FIR filters with improved selectivity is given in [9].

A particular important class of 2D filters is circularly symmetric filters, with a circular or rotationally invariant shape of their frequency response. They have been extensively used, in particular as circular filter banks, due to their capability to select circular regions in the frequency plane. Wide-band circular filters have been approached in papers like [10,11]. Other design approaches for 2D circular filters employed the Farrow structure and multiplier block [12], semi-definite programming [13], or harmony search algorithm and multiplier-less structures [14]. Design and applications of adjustable 2D digital filters with elliptical and circular symmetry were covered in [15]. Some analytical design methods for circular filters, in particular circular filter banks of FIR type with Gaussian shape, implemented using the polyphase technique, were also proposed in [16]. An analytic design procedure was elaborated in [17] for zero-phase directional filters, and their main application was demonstrated, namely the detection and extraction of oriented straight lines from images. Some advanced applications of circular Gabor filters, for instance in SAR interferograms, were studied in [18].

Two-dimensional filter banks of various shapes have been widely applied in major applications, such as texture segmentation and classification, or various tasks of feature extraction. Two-dimensional filter banks separate the image frequency spectrum into sub-bands. They are currently used in essential applications such as sub-band coding and also the compression of images or video sequences. As shown in the review in [19], the currently used 2D filter banks are mainly directional, such as square-shaped (diamond), parallelogram, wedge/fan filters, etc. Multi-dimensional stable perfect reconstruction filter banks were presented in [20]. In their well-known paper [21], Bamberger and Smith proposed the first directional uniform filter bank with wedge-shaped channels capable of directional decomposition and reconstruction of images. Directional filter banks (DFBs) have been studied by many researchers, for instance, DFBs with an arbitrary number of sub-bands [22], non-uniform DFBs with arbitrary frequency partitioning [23], or multiresolution DFBs [24].

Directional filter banks have many other useful applications, such as fingerprint quality enhancement [25], where the fingerprint image is decomposed in sub-bands, and fingerprint image quality is evaluated based on similarity between different sub-bands. Regarding various techniques dedicated to diminishing computational complexity and increasing the processing speed, the fast block implementation with systolic arrays for 2D digital FIR filters was elaborated in early papers, like [26]. A high-performance 2D parallel block-filtering system envisaged for real-time applications was proposed in [27].

In this paper, an efficient analytic design procedure is developed for a class of 2D filter banks, namely 2D FIR maximally flat circular filter banks (CFBs).

To the extent of the authors’ knowledge, a systematic design of circular filter banks, of FIR or IIR type, uniform or non-uniform, in particular using analytical design techniques, has not been approached by other researchers in the field. This is the main motivation behind our approach, and also the possibility of using such a filter bank in image analysis, by decomposing its spectrum in concentric band regions of the frequency plane, corresponding to sub-band images. As will be shown, the original image can be reconstructed from these sub-band images, the result being visually correct and also confirmed by calculating some specific metrics. Also, a novel system-level implementation structure of the designed filters is proposed based on the polyphase technique, which has a high degree of parallelism and low arithmetic complexity.

Two versions of CFBs will be designed, specifically a uniform CFB and also a non-uniform (dyadic) CFB, each with a given number of component filters. As a prototype for the desired 2D filter, we chose a 1D maximally flat low-pass filter and a specified bandwidth; its frequency response is accurately approximated by a trigonometric polynomial, using a Fourier series, with imposed precision. The corresponding band-pass filters of the FB prototype result by a simple shifting of the LP frequency response to give the peak frequencies.

Once we have the convenient prototype filter bank, the next step is to apply the specific 1D to 2D frequency transformation derived from the well-known McClellan transform [2,3], which yields directly the desired circular filter components of the CFB. Similarly, the non-uniform (dyadic) CFB will be designed. For both filter bank versions, the derived filters’ characteristics will have a precise circular shape, with some distortions towards frequency plane margins, due to the low-order McClellan approximation.

In order to demonstrate the designed FB capabilities, as an image filtering example, both circular filter banks were tested on a grayscale image, which in each case is decomposed into a set of sub-band images. If the FB output images are summed again, they reconstruct almost perfectly the original input test image. This simulation suggests a possible use of the designed CFBs in a sub-band coding scheme.

The current paper is related to some extent to [16], both approaches being based on low-pass zero-phase prototype filters. However, this paper differs from [16] in some essential respects. The chosen trapezoidal prototype with specified slope of the transition region can be easily expressed analytically as a Fourier series, whose coefficients contain explicitly the imposed specifications (p, s) which describe the bandwidth and steepness. This is a major advantage which simplifies the design. When the prototype specifications are changed, the approximation up to a desired order results directly, which renders this design method more flexible. This approach combines the maximally flat property of an ideal filter with the adjustable slope, which significantly reduces ringing, as the provided examples show. Of course, other maximally flat functions could be used as well, such as a smooth hyperbolic tangent (tanh), but the drawback is that its Fourier coefficients cannot be calculated analytically. For instance, in papers [15,17], the approximation is obtained using Chebyshev series and Chebyshev–Pade method, which can only be calculated numerically for each set of specifications.

Regarding the implementation aspect, the solution proposed in this work for the 2D FIR filter components of the resulting CFB is based on a polyphase decomposition of the 2D filtering operation with a large-sized kernel, combined with a block filtering scheme with matrices of smaller size. It differs from the one developed in [16], having a higher degree of parallelism and resulting in a significantly lower arithmetic complexity.

The remainder of this paper is structured as follows. The main section, Section 2, presents in detail the developed analytical design technique, first deriving the uniform and non-uniform prototype filter bank with the imposed specifications (central frequencies and bandwidths) and then applying the frequency transformation and thus deriving the frequency responses of the 2D CFBs. Also, a full design example is included for both circular FB versions. In Section 3, we present an example of image analysis using CFB, which decomposes a given test image into sub-band images, and then the original image is reconstructed from its components. The proposed efficient implementation structure relying on polyphase and block filtering is described in Section 4. Some discussions regarding the advantages of the proposed approach and the arithmetic complexity of implementation are included in Section 5. Finally, in the last section, conclusions are drawn and future work directions are indicated.

2. Analytical Design Technique for 2D Circular FIR Filter Banks

In this section, a novel analytical design procedure is described for 2D FIR circular filters. The design starts from a specified prototype with imposed parameters (bandwidth, steepness, peak frequency), to which a 1D to 2D frequency mapping is applied, yielding the desired 2D filters, in our case a bank of rotationally invariant filters. For simplicity, we will next use the term “circular” throughout the paper. Prior to obtaining through frequency mapping the desired 2D circular filter bank, first we must obtain a 1D prototype filter bank.

2.1. One-Dimensional Maximally Flat Zero-Phase Prototype Filters

Generally, the chosen prototype is a low-pass filter, which may be digital (with transfer function given in complex frequency z), analog (with transfer function given in complex frequency s), or zero-phase (with real transfer function). We will use in this work the latter version, namely a zero-phase prototype, which by frequency mapping will also lead to a zero-phase 2D filter. While in the previous work [16] we used a Gaussian prototype, which has its advantages, in this paper, we prefer a maximally flat prototype, derived through the approximation of a trapezoidal function with a specified width (imposed by filter bandwidth) and a certain slope (imposed by transition region). The ideal, trapezoidal low-pass prototype function $H_{LPI}\left(\omega \right)$ is displayed in Figure 1a; we notice that the transition region is linear between the edge frequencies $p\pi$ and $s\pi$. By laterally shifting this low-pass prototype to the frequencies $\pm {\omega }_0$, the ideal band-pass prototype $H_{BPI}\left(\omega \right)$ displayed in Figure 1b is derived.

2.2. Approximation of the FIR Filter Prototype Using Fourier Series

We consider here a low-pass prototype filter having the shape of a symmetric trapezoidal function centered in zero, as displayed in Figure 1a. Because this is an ideal frequency response of a digital filter, it is defined on the frequency range $[-\pi ,\pi ]$. As marked in Figure 1a, the transition region has a linear slope between the frequencies $p\cdot \pi$ and $s\cdot \pi$, where p and s are subunitary coefficients, with p < s. For the ideal prototype in Figure 1a, $p\cdot \pi$ is the pass-band edge frequency, while $s\cdot \pi$ is the stop-band edge frequency. Considering this as a periodic function with period $2\pi$ and regarding this trapezoidal-shaped LP function as a generating pulse, we obtain by simple analytical calculation the following expression $H_{LP}\left(\omega \right)$, which is the Fourier series expansion of the ideal trapezoidal function $H_{LPI}\left(\omega \right)$ up to a given order N:

$$H_{LPI}\left(\omega \right)\cong H_{LP}\left(\omega \right)=a_0+{\displaystyle \sum _{n=1}^N{a}_n\cdot \cos n\omega }$$

(1)

where the mean value coefficient is given by $a_0$, while the general coefficient $a_n$ of the series has the expression depending on the order N:

$$a_n={\displaystyle \frac{4}{(s-p)\cdot {\pi }^2\cdot n^2}}\cdot \sin \left((s-p)\cdot {\displaystyle \frac{n\pi }{2}}\right)\cdot \sin \left((s+p)\cdot {\displaystyle \frac{n\pi }{2}}\right)$$

(2)

From this low-pass (LP) prototype, we easily obtain a band-pass (BP) prototype by shifting it laterally around the frequencies $\pm {\omega }_0$:

$$\begin{array}{l}{H}_{BP}\left(\omega \right)=H_{LP}\left(\omega -{\omega }_0\right)+H_{LP}\left(\omega +{\omega }_0\right)\cong \\ \cong (p+s)+{\displaystyle \frac{8}{(s-p)\cdot {\pi }^2}}\cdot {\displaystyle \sum _{n=1}^N\hspace{0.17em}\frac{1}{n^2}\cdot \sin \left((s-p)\cdot \frac{n\pi }{2}\right)\cdot \sin \left((s+p)\cdot \frac{n\pi }{2}\right)}\cdot \cos (n{\omega }_0)\cdot \cos (n\omega )\end{array}$$

(3)

For more clarity, all constants were extracted as a factor in front of the sum. Thus, for the band-pass filter with a specified central frequency ${\omega }_0$, the mean value component will be $b_0=2a_0=p+s$, while the current n-th coefficient will be $b_n=2a_n\cdot \cos (n\cdot {\omega }_0)$.

In order to show the advantage of a trapezoidal filter compared to the ideal square LP filter, we plotted in Figure 2a–d the characteristics of some trapezoidal filters with various values of the parameters p, s, and order N. For a very steep transition, with p = 0.2, s = 0.21, order N = 32 like in (a), the ripple and ringing have a high level; by doubling the order (N = 64) as in (b), the improvement is not significant; however, with p = 0.2, s = 0.3, and N = 32, as in (c), the ripple is much lower; if the order is also increased (N = 64), the ripple is almost zero, so the filter is almost maximally flat, as the ideal trapezoidal function.

2.3. Design of a Uniform FIR Filter Bank Prototype

Next, a uniform filter bank prototype with nine components will be designed, namely one low-pass filter, seven band-pass filters, and one high-pass filter. In this uniform filter bank, the peak frequencies are equally spaced on the frequency axis. In the design of this type of filter bank, once the central frequencies of the filters are defined, the next step is to adopt convenient values for the bandwidth (of width $2p\pi$) and the transition regions (each of width $(s-p)\cdot \pi$). There are mainly two approaches on how to design the filters’ characteristics to cover uniformly the whole frequency range [19]. One would be to allow no overlap of the characteristics between consecutive filters.

However, for the chosen trapezoidal protype, it would be difficult to ensure in this case a maximally flat pass-band and also the convenient slope of the lateral transition regions. With no overlap, the transition regions would be too narrow and steep; therefore, the trapezoidal characteristic would be almost square, which would require keeping more terms of the Fourier series expansion to reduce the ringing level, and therefore the filter will result in being a very high order, which is not practical. The second approach is to allow a certain degree of overlap between the adjacent filters of the bank by imposing that their characteristics intersect each other at a given value of attenuation, which could be, for example, 0.5 (half of peak value). In our case, with an FB with nine components, choosing this value would still lead to rather narrow filters; this is why, here, we will allow for the adjacent frequency responses to intersect at $1/\sqrt{2}\cong 0.7071$, which is the usual bandwidth definition in most filters. Thus, according to this definition, the filter bandwidths will cover tightly the frequency interval $[-\pi ,\pi ]$. In our case, with this requirement, we obtain the coefficient values for the pass-band and stop-band edge frequencies, namely p = 0.05 and s = 0.09. The bandwidth of each band-pass filter, defined as above, will be $B=\pi /8=0.125\hspace{0.17em}\pi$, while the LP and HP filters will be half this bandwidth, namely $\pi /16$. For the BP filters, the maximally flat peak portion will have width $2p\pi =0.1\pi$. The ideal prototype filter bank that resulted in using these parameter values is displayed in Figure 2e. We notice that, indeed, the adjacent filters overlap at the mentioned value.

Since the filters of the bank are rather selective, their order will be large enough. Imposing the values p = 0.05 and s = 0.09, we truncate the Fourier series to 32 terms, so the prototype order results N = 32. The entire FB with nine components is displayed in Figure 2f. Compared to its ideal version in Figure 2e, the characteristics have a ripple both in the pass band and in the stop-band, which will not visibly affect the performance of the designed filters. If a lower ripple is desired, we must accept a higher order; this tradeoff depends on the application, but usually in image filtering, this ripple level is not significant. Also, the logarithmic characteristics of the band-pass prototype filters BP1, BP3, and BP6 were plotted as examples in Figure 2d–f, respectively, to make the ripple level more visible, especially in the stop-band.

For instance, the low-pass filter (lowest component of the filter bank) has the following frequency response, resulted by truncating the Fourier series given by (1) to 32 terms:

$$\begin{array}{l}{H}_{LP}(\omega )=0.07+0.138783\cdot \cos \omega +0.135173339\cdot \cos \hspace{0.17em}2\omega +0.129294\cdot \cos \hspace{0.17em}3\omega +0.121344\cdot \cos \hspace{0.17em}4\omega \\ +0.111589\cdot \cos \hspace{0.17em}5\omega +0.100353\cdot \cos \hspace{0.17em}6\omega +0.08799831\cdot \cos \hspace{0.17em}7\omega +0.0749176\cdot \cos \hspace{0.17em}8\omega +0.0615129\cdot \cos \hspace{0.17em}9\omega \\ +0.048181\cdot \cos \hspace{0.17em}10\omega +0.035298\cdot \cos \hspace{0.17em}11\omega +0.023204\cdot \cos \hspace{0.17em}12\omega +0.012193\cdot \cos \hspace{0.17em}13\omega +0.00250103\cdot \cos \hspace{0.17em}14\omega \\ -0.005699\cdot \cos \hspace{0.17em}15\omega -0.01230173\cdot \cos \hspace{0.17em}16\omega -0.0172687\cdot \cos \hspace{0.17em}17\omega -0.020627\cdot \cos \hspace{0.17em}18\omega -0.022462\cdot \cos \hspace{0.17em}19\omega \\ -0.0229\cdot \cos \hspace{0.17em}20\omega -0.0221548\cdot \cos \hspace{0.17em}21\omega -0.0204012\cdot \cos \hspace{0.17em}22\omega -0.017879\cdot \cos \hspace{0.17em}23\omega -0.014823\cdot \cos \hspace{0.17em}24\omega \\ -0.011463\cdot \cos \hspace{0.17em}25\omega -0.0080153\cdot \cos \hspace{0.17em}26\omega -0.0046709\cdot \cos \hspace{0.17em}27\omega -0.001591\cdot \cos \hspace{0.17em}28\omega +0.0010982\cdot \cos \hspace{0.17em}29\omega \\ +0.0033086\cdot \cos \hspace{0.17em}30\omega +0.00499\cdot \cos \hspace{0.17em}31\omega +0.006129\cdot \cos \hspace{0.17em}32\omega \end{array}$$

(4)

which is further factored, yielding the following expression:

$$\begin{array}{r}{H}_{LP}(\omega )=13161308.8\cdot \left(\cos \omega +0.797444\right)\left(\cos \omega +0.752867\right)\left(\cos \omega +0.6794747\right)\left(\cos \omega +0.606838\right)\\ \cdot \left(\cos \omega +0.52695\right)\left(\cos \omega +0.442418\right)\left(\cos \omega +0.35371\right)\left(\cos \omega +0.261689\right)\left(\cos \omega +0.167216\right)\\ \cdot \left(\cos \omega +0.071169\right)\left(\cos \omega -0.02555\right)\left(\cos \omega -0.122039\right)\left(\cos \omega -0.217394\right)\left(\cos \omega -0.310724\right)\\ \cdot \left(\cos \omega -0.4011555\right)\left(\cos \omega -0.487831\right)\left(\cos \omega -0.5699864\right)\left(\cos \omega -0.646489\right)\left(\cos \omega -0.7186612\right)\\ \cdot \left(\cos \omega -0.777079\right)\left(\cos \omega -1.016423\right)\left(\cos \omega -1.03401\right)\left({(\cos \omega )}^2+2.009927\cdot \cos \omega +1.010208\right)\\ \cdot \left({(\cos \omega )}^2+1.913374\cdot \cos \omega +0.916469\right)\left({(\cos \omega )}^2+1.755162\cdot \cos \omega +0.770893\right)\\ \left(\right(\cos \omega {)}^2-1.7161634\cdot \cos \omega +0.7364123\left)\right((\cos \omega {)}^2-1.887623\cdot \cos \omega +0.891187)\end{array}$$

(5)

As an example, we also give below the frequency responses of three band-pass filter components of the bank (BP1, BP4, BP7), with the intermediate BP filters having similar forms:

$$\begin{array}{r}{H}_{BP1}(\omega )=26322617.61\cdot \left(\cos \omega +0.965384\right)\left(\cos \omega +0.739977\right)\left(\cos \omega +0.683147\right)\left(\cos \omega +0.606771\right)\\ \cdot \left(\cos \omega +0.5275539\right)\left(\cos \omega +0.443055\right)\left(\cos \omega +0.354451\right)\left(\cos \omega +0.262535\right)\left(\cos \omega +0.168168\right)\\ \cdot \left(\cos \omega +0.072232\right)\left(\cos \omega -0.024376\right)\left(\cos \omega -0.120748\right)\left(\cos \omega -0.215981\right)\left(\cos \omega -0.309174\right)\\ \cdot \left(\cos \omega -0.399436\right)\left(\cos \omega -0.485886\right)\left(\cos \omega -0.567572\right)\left(\cos \omega -0.643752\right)\left(\cos \omega -0.7115434\right)\\ \cdot \left(\cos \omega -0.768665\right)\left({(\cos \omega )}^2+2.005582\cdot \cos \omega +1.005892\right)\left({(\cos \omega )}^2+1.860116\cdot \cos \omega +0.866372\right)\\ \cdot \left({(\cos \omega )}^2+1.655769\cdot \cos \omega +0.686213\right)\left({(\cos \omega )}^2-1.836976\cdot \cos \omega +0.852089\right)\\ \cdot \left({(\cos \omega )}^2-1.885554\cdot \cos \omega +0.891769\right)\left({(\cos \omega )}^2-1.998957\cdot \cos \omega +0.99906\right)\end{array}$$

(6)

$$\begin{array}{r}{H}_{BP7}(\omega )=26322617.61\cdot \left(\cos \omega +0.767204\right)\left(\cos \omega +0.71244\right)\left(\cos \omega +0.643544\right)\left(\cos \omega +0.567598\right)\left(\cos \omega +0.4858844\right)\\ \cdot \left(\cos \omega +0.399436\right)\left(\cos \omega +0.309174\right)\left(\cos \omega +0.21598123\right)\left(\cos \omega +0.120748\right)\left(\cos \omega +0.0243756\right)\\ \cdot \left(\cos \omega -0.072231567\right)\left(\cos \omega -0.168168\right)\left(\cos \omega -0.262535\right)\left(\cos \omega -0.3544512\right)\left(\cos \omega -0.443054\right)\\ \cdot \left(\cos \omega -0.527541\right)\left(\cos \omega -0.607043\right)\left(\cos \omega -0.680426\right)\left(\cos \omega -0.7555333\right)\left(\cos \omega -0.787398\right)\\ \cdot \left({(\cos \omega )}^2+2.0028175\cdot \cos \omega +1.002999\right)\left({(\cos \omega )}^2+1.884513\cdot \cos \omega +0.891301\right)\\ \cdot \left({(\cos \omega )}^2+1.834906\cdot \cos \omega +0.8499723\right)\left({(\cos \omega )}^2-1.741575\cdot \cos \omega +0.760396\right)\\ \cdot \left({(\cos \omega )}^2-1.91691\cdot \cos \omega +0.9214034\right)\left({(\cos \omega )}^2-2.027875\cdot \cos \omega +1.028599\right)\end{array}$$

(7)

$$\begin{array}{r}{H}_{BP4}(\omega )=26322617.61\cdot \left({(\cos \omega )}^2-0.473846\right)\left({(\cos \omega )}^2-0.381272\right)\left({(\cos \omega )}^2-0.29184\right)\left({(\cos \omega )}^2-0.2104413\right)\\ \cdot \left({(\cos \omega )}^2-0.140889\right)\left({(\cos \omega )}^2-0.089192\right)\left({(\cos \omega )}^2+0.050984\right)\left({(\cos \omega )}^2+0.026751\right)\\ \cdot \left({(\cos \omega )}^2+2.032066\cdot \cos \omega +1.032883\right)\left({(\cos \omega )}^2-2.032066\cdot \cos \omega +1.032883\right)\\ \cdot \left({(\cos \omega )}^2+1.923289\cdot \cos \omega +0.927906\right)\left({(\cos \omega )}^2-1.923289\cdot \cos \omega +0.927906\right)\\ \cdot \left({(\cos \omega )}^2+1.74636\cdot \cos \omega +0.7653\right)\left({(\cos \omega )}^2-1.74636\cdot \cos \omega +0.7653\right)\\ \cdot \left({(\cos \omega )}^2+1.548851\cdot \cos \omega +0.599826\right)\left({(\cos \omega )}^2-1.548851\cdot \cos \omega +0.599826\right) \end{array}$$

(8)

One brief remark is required here. What was meant by “maximally flat” is the ideal prototype filter function, with a trapezoidal shape. This term was used to make the difference between this prototype and the one used in [16], namely Gaussian. For a filter bank with less filters, we could also use the term “wide-band” filters, but in our case, with a uniform bank with nine components, the filters have rather narrow bands, so this term would be improper. As shown in Figure 2a–d, by truncating the Fourier series to a larger number of terms, we can obtain prototype filters very close to the ideal trapezoidal shape and also have a negligible ripple in the pass-band; therefore, they are practically maximally flat. Therefore, we have used this term more loosely, in a wider sense.

2.4. Design of a Non-Uniform FIR Filter Bank Prototype

In image analysis, for instance in multirate signal processing, non-uniform filter banks are also frequently used. Using the design method described in Section 2.1, a non-uniform filter bank, specifically a so-called dyadic filter bank, will be synthesized. A particular feature for such an FB is that the bandwidths of its component filters increase proportionally to their peak frequencies, such that usually the ratio between bandwidth and peak frequency remains constant; such filters are also referred to as constant-Q filter banks.

Next, we consider for the design example a dyadic filter bank prototype in which the component filter bandwidths increase by a factor of two from low to high frequencies. A filter bank with five filters will be designed here: one LP filter, three BP filters, and one HP filter. Having previously defined the bandwidth in the conventional way, at the level $1/\sqrt{2}\cong 0.7071$, we denote the filter bandwidths as $B_i$, the central frequencies ${\omega }_{0\hspace{0.17em}i}$, and the selectivity parameters (coefficients p and s of the passband and stop-band edge frequencies) as $p_i$ and $s_i$ for each filter i of the bank. Thus, we obtain the following values: for the low-pass filter: $B_1=\pi /23=0.136591$, ${\omega }_{01}=0$, $p_1=0.033$, $s_1=0.069131$; for the first band-pass filter: $B_2=2B_1=2\pi /23=0.273182$, ${\omega }_{02}=2\pi /23$, $p_2=0.033$, $s_2=0.069131$; for the second band-pass filter: $B_3=4B_1=4\pi /23=0.546364$, ${\omega }_{03}=5\pi /23$, $p_3=0.076478$, $s_3=0.112609$; for the third band-pass filter: $B_4=8B_1=8\pi /23=1.092728$, ${\omega }_{04}=11\pi /23$, $p_4=0.163434$, $s_4=0.199565$; for the uppermost component, the high-pass filter, we have the following values:$B_5=8B_1=8\pi /23=1.092728$, ${\omega }_{05}=\pi$, $p_5=0.337346$, $s_5=0.373477$. The ideal trapezoidal dyadic prototype filter bank is shown in Figure 3a.

Regarding this non-uniform filter bank, a brief remark can be made here. Normally, in a dyadic filter bank, the uppermost HP filter has the widest bandwidth, equal to half of the frequency range, namely from $\pi /2$ to $\pi$. However, considering it as a band-pass filter with central frequency ${\omega }_{05}=\pi$, it has theoretically a double bandwidth compared to the previous BPF. Furthermore, in our case, taking an HPF with bandwidth $\pi /2$ would lead to a very narrow low-pass component, which would be too selective and therefore of very high order. Therefore, we have chosen this structure of the filter bank prototype.

The frequency responses of the dyadic FB components result through the same procedure, first finding the Fourier series expansion using (1) and (2) and then converting them into factored trigonometric polynomials, as for the uniform FB. Even if the superior filters of the bank become wider, due to the fact that the transition region is kept constant, all these filters need the same number of terms in the Fourier series for a good approximation. In fact, the resulting filters with characteristics displayed in Figure 3b are of order N = 44 (truncating the Fourier series at the first 44 terms). However, we notice that the passband ripple decreases from the low-pass to the high-pass filter. In order to make the characteristics of component filters more visible, they were represented in different colors in Figure 3a,b.

In principle, if both the uniform and non-uniform FBs contained the same number of components (for instance 9), it would make more sense to compare the results. However, because the dyadic filter has an expansion factor of two (the bandwidths of the FB components double from low to high frequencies), if we chose a dyadic filter with nine components, the lowest filters would have a very narrow bandwidth; therefore, they would be too selective. For instance, the low-pass filter (lowest component) would have a bandwidth of only pi/383 = 0.00261 pi, or 16.6 times narrower than the LP filter of the actually designed FB. Practically, the output image of this LP filter would be practically indistinguishable; it would practically extract the mean pixel value (of zero frequency) of the original image. Of course, we could conversely use a uniform filter bank with five components (the same as for the dyadic FB), but in this case, the bandwidths of the uniform FB would be too large and the result less relevant. For this reason, we have used a different number of components for the two FBs.

2.5. Circular FIR Filter Bank Obtained Using Frequency Transformation

Starting from a specified 1D prototype filter having a frequency response $H_p(\omega )$, a 2D circular filter $H({\omega }_1,{\omega }_2)$ directly results by applying to this prototype the 1D to 2D frequency transformation $\omega \to \sqrt{{\omega }_1^2+{\omega }_2^2}$:

$$H({\omega }_1,{\omega }_2)=H_p\left(\sqrt{{\omega }_1^2+{\omega }_2^2}\right)$$

(9)

The two-variable function $\cos \sqrt{{\omega }_1^2+{\omega }_2^2}$ (henceforth called circular cosine function) corresponds to the $3\times 3$ matrix

$$C=\left[\begin{array}{ccc}0.125& 0.25& 0.125\\ 0.25& -0.5& 0.25\\ 0.125& 0.25& 0.125\end{array}\right]$$

(10)

and can be approximated as follows through the 2D discrete Fourier transform of the matrix above, which is a simple particular case of the widely used McClellan transform frequently encountered in 2D FIR filter design [2,3,16]:

$$\cos \sqrt{{\omega }_1^2+{\omega }_2^2}\cong C({\omega }_1,{\omega }_2)=-0.5+0.5(\cos {\omega }_1+\cos {\omega }_2)+0.5\cos {\omega }_1\cos {\omega }_2$$

(11)

Let us now consider a zero-phase FIR filter $H_P(\omega )$, whose frequency response is given by the following trigonometric polynomial [16]:

$$H_P(\omega )=b_0+2{\displaystyle \sum _{k=1}^R{b}_k\cos k\omega }$$

(12)

3By applying the proper trigonometric identities for $\cos k\omega$ ($k=1\dots R$), the following polynomial expression in the variable $\cos \omega$ will be derived [16]:

$$H_P(\omega )=c_0+{\displaystyle \sum _{k=1}^R{c}_k(\cos \omega }{)}^k$$

(13)

where in (12) and (13), $b_0,b_k,c_0,c_k$ are the series coefficients. By applying the frequency mapping (18), the following frequency response results for the desired 2D circular filter [16]:

$$H({\omega }_1,{\omega }_2)=H_P\left(\sqrt{{\omega }_1^2+{\omega }_2^2}\right)=c_0+{\displaystyle \sum _{k=1}^R{c}_k\cdot C^k({\omega }_1,{\omega }_2)}$$

(14)

where $C({\omega }_1,{\omega }_2)=\cos \sqrt{{\omega }_1^2+{\omega }_2^2}$, defined in Equation (11).

Thus, through a direct substitution of $\cos \omega$ with the above-defined circular cosine function $C({\omega }_1,{\omega }_2)=\cos \sqrt{{\omega }_1^2+{\omega }_2^2}$ in the prototype $H_P(\omega )$, the 2D filter frequency response is directly derived. Then, supposing that function $H_P(\omega )$ is factored into the first-order and second-order factors in variable $\cos \omega$ and applying the above substitution in all factors of $H_P(\omega )$, the circular filter function $H({\omega }_1,{\omega }_2)$ results in the following factored form:

$$H({\omega }_1,{\omega }_2)=k\cdot {\displaystyle \prod _{i=1}^n(C+b_i)}\cdot {\displaystyle \prod _{j=1}^m(C^2+b_{1j}\cdot C+b_{2j})}$$

(15)

where we used C as a short notation for the two-variable function $C({\omega }_1,{\omega }_2)$. Because the chosen prototype is expressed as a product of elementary factors, the frequency response of the circular filter will also result in being directly factored, which may be an advantage in implementation. Therefore, the large-size kernel H corresponding to the transfer function $H({\omega }_1,{\omega }_2)$ can be written directly as a convolution of small matrices ($3\times 3$ or $5\times 5$):

$$H=k\cdot (C_1\ast \dots \ast C_i\ast \dots \ast C_n)\ast (D_1\ast \dots \ast D_j\ast \dots \ast D_m)$$

(16)

Expression (16) of kernel H corresponds to the frequency response in factored form (15). Using the matrix C of size $3\times 3$ given in (10) and considering also (15), each of the matrices $C_i$ (of size $3\times 3$) in (16) are obtained by adding the coefficient $b_i$, appearing in the first-order factors in (15), to the central element in the matrix C. Therefore, the matrix $D_j$($5\times 5$) is derived as follows:

$$D_j=C\ast C+b_{1j}\cdot C_1+b_{2j}\cdot C_0$$

(17)

In the above equation, $C_0$ is the null matrix of size $5\times 5$ with a center element of value one; $C_1(5\times 5)$ is determined by padding the matrix C (size $3\times 3$) with zeros (the symbol * stands for convolution).

2.6. Design Examples of Uniform and Non-Uniform Circular FIR Filter Banks

The frequency characteristics and their associated contour plots for all nine filters of the uniform circular filter bank are displayed in Figure 4 and Figure 5. It is easily noticed that up to 6-th band-pass filter, the filters’ characteristics are almost perfectly circular. For band-pass filters with higher central frequencies, the filter characteristics have a more obvious deviation from circularity, resembling more the shape of a rounded square. The high-pass filter, which is the component with the highest frequency (${\omega }_0=\pi$), has almost a square shape. This typical circularity distortion occurs when frequency mapping (11) is applied due to the lowest order of the McClellan approximation. Filter circularity could be improved only by using a more precise accurate approximation of the circular cosine function, but this would come at the price of higher filter complexity (kernel matrices of larger size).

Also, we plotted in Figure 5d–f the logarithmic characteristics (in decibels) for the band-pass components BP1, BP3 and BP4 of the uniform filter bank, as an example, to make the ripple more visible, especially in the stop-band.

The frequency response characteristics and corresponding contour plots of the five component filters of the non-uniform (dyadic) circular filter bank derived from the 1D prototype filters designed in Section 2.4 are displayed in Figure 6, and it can be noticed that they have a good circular symmetry.

3. Applications and Simulation Results

This section presents simulation examples of decomposition and reconstruction of a real-life test image using the two circular filter banks designed previously, namely the uniform and non-uniform CFBs, and also a quantitative evaluation of the quality of this reconstruction using specific metrics.

3.1. Image Analysis Using the Designed Circular Filter Banks

We demonstrate here an example of image analysis performed by the uniform CFB designed before. Let us consider as a test image the grayscale image in Figure 7a of size $499\times 499$, representing an aerial image of an old town; this type of image was chosen because it contains a lot of fine details of the buildings, etc. This image is filtered by applying all the nine component filters of the designed FB (1 LP filter, 7 BP filters, 1 HP filter). Thus, our test image is practically decomposed into sub-bands using the analysis filter bank designed as described before.

By filtering the original test image (a), the image that resulted at the output of the narrow LP filter is (b), and it can be noticed that it is very blurred; the fine details are no longer visible. The images derived from the first five BP filters are given in (c–g), respectively, and contain specific details corresponding to their bandwidth. Image (h), obtained at the output of the HP filter, contains the highest frequencies, which correspond to the finest details of the image.

For the filtering task carried out in MATLAB, the original test image was first converted to “double” format and its pixels were rescaled to the interval [0,1]. The images in Figure 7i–l show how the original test image is gradually reconstructed by summing up the component images into which it was decomposed by the circular filter bank.

Thus, image (i) results by summing the outputs of first two component filters (LP and BP1); image (j) is the sum of the outputs for the first three components (LP, BP1, BP2); and image (k) results by also adding the output of the component filter BP3. Finally, by summing all nine components, image (l) results, which is visually very similar to the original test image and displays very clearly all the fine details.

This suggests the possibility for the designed circular FB to be practically used as an analysis filter bank in order to decompose a given image into a set of sub-band images.

However, a consistent theoretical formulation and rigorous mathematical requirements remain to be further investigated in future work on the subject.

The decomposition of the test image into sub-band images can also be described through image energy. Thus, the energy of each resulted sub-band image can be evaluated using the well-known expression $E_k=\sqrt{{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^N{p}_{ij}^2}}$, where the image size is M × N. Also, the expression of the relative energy as a percentage can be useful:

$$E_{Rk}={\displaystyle \frac{100}{M\cdot N}}\cdot \sqrt{{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^N{p}_{ij}^2}} \left(\%\right)$$

(18)

By applying expression (18) for calculating the energies of the nine sub-band images obtained at the outputs of the designed uniform CFB, the following values are found:

ER (%) = [62.04 11.41 7.07 5.63 4.5 3.59 2.89 2.24 0.63]

(19)

which are also displayed as a barplot in Figure 8a. Summing these energy values, we find that they add up to 100%. We observe that the LP component (frequency domain around zero, with radius $\pi /16$) contains 62% of the entire image energy, while almost 86% is contained in the first four components (within a radius of $7\pi /16$) at the output of the LP filter and the first three BP filters. The relative energies of the sub-band images decrease monotonically, as shown in Figure 8a.

A similar image decomposition is performed with the non-uniform (dyadic) circular filter bank, as shown in Figure 9. From the same original test image in Figure 7a, five sub-band images will result, as specified. The images recovered by summing the first two and three components are given in Figure 9f,g respectively, while (h) is the recovered image by summing all five component sub-band images. Using again Equation (18), the energies of the corresponding five sub-band images obtained at the outputs of the designed dyadic CFB are found as

ERd (%) = [64.06 11.81 10.27 8.99 4.87]

(20)

and are also displayed as a barplot in Figure 8b, showing a monotonic decrease.

Thus, the decomposition of the test image into sub-band images can also be described through image energy. For the uniform filter bank, the fact that the energies of the sub-band images decrease monotonically, the energy corresponding to the LP filter being the largest, explains the fact that the original image can be gradually reconstructed by successively adding the sub-band images from low to high frequencies, and that the higher components have the least effect on the recovered image. Also, the fact that the relative energies given by (18), with values shown in (19), add up to 100% indicates that the frequency domains of the image are correctly separated by this analysis filter bank.

As a particular feature, in the designed rotationally invariant filter banks, the image spectrum is separated into concentric regions with increasing peak frequencies. Due to this rotational invariance, the corresponding energy in each sub-band remains more or less constant. For this reason, circular filter banks with different shapes of the component filters (maximally flat, Gaussian, etc.) are very useful in specific feature extraction and classification tasks in image processing. An additional advantage of the proposed FB is that the filters’ transfer functions are real-valued (zero-phase), so they do not introduce any phase distortions in the filtered images.

In our simulations of the 2D filtering, we have used the natural approach when applying a convolution kernel to a digital image, namely the so-called free boundary conditions, in which pixel locations that lie outside the image boundaries are interpreted as having a value of zero. As can be noticed from the resulting filtered images, there are no visible artifacts or any other distortions occurring due to boundary conditions.

3.2. Quantitative Evaluation of Image Reconstruction Using RMSE and PSNR

While the provided examples show the capabilities of the designed circular filter banks in image analysis and reconstruction, and the recovered image obtained by summing all the sub-band component images is visually and subjectively quite close to the original image, it would be useful to evaluate their performance in a quantitative manner as well using some objective metrics commonly applied in image processing. The most suitable measures in our case would be the root mean squared error (RMSE) and, derived from it, the peak signal-to-noise ratio (PSNR), which is the ratio between the maximum value or power of a signal and the power of noise that affects the signal.

Because signals or images usually have a very large dynamic range, the PSNR is most commonly expressed as a logarithmic quantity in decibels (dB).

Given a grayscale (monochrome) image unaffected by noise I and its noisy or otherwise distorted version J, RMSE is defined mathematically as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{M\times N}}\cdot {\displaystyle \sum _{i=0}^{M-1}{\displaystyle \sum _{j=0}^{N-1}{\left[I(i,j)-J(i,j)\right]}^2}}}$$

(21)

Based on the root mean squared error, the PSNR (in decibels) is given by the following expression:

$$PSNR=20\cdot {\log }_{10}\left(Max(I)/RMSE\right)$$

(22)

In the above formula, $Max(I)$ is the maximum possible pixel value of the image. In the most common situation, when pixels are represented on 8 bits, this value is 255. Of course, in our case of image reconstruction from sub-band components that have resulted from analysis using the designed CFB, the recovered image is not affected by noise; however, it slightly differs from the original image due to imperfect reconstruction.

In calculating the RMSE and PSNR, the noisy image $J(i,j)$ in (21) is assimilated to the reconstructed image, which is somewhat distorted compared to the ideal original image $I(i,j)$.

The reconstruction error can be shown by displaying the difference between the original image and the image reconstructed by adding all the nine sub-band component images in the case of the uniform CFB, as shown in Figure 10a. For the non-uniform CFB, the difference between the original image and the image reconstructed by adding all the five sub-band images is given in Figure 10b. Visually, in both cases, the error is relatively small, with the maximum and minimum difference between pixel values being 31.63 and −44.89 for the uniform CFB and 28.94 and −41.68 for the non-uniform CFB, respectively.

The calculated values for the two metrics, the RMSE and PSNR, are as follows: for the uniform CFB, RMSE = 17.8 and PSNR = 23.12 dB, while for the non-uniform CFB, RMSE = 15.21 and PSNR = 24.49 dB. The slightly better results for the uniform CFB may be due to the smaller circularity distortions of its component filters, and the smaller number of components also implies less overlapping between adjacent regions of the frequency plane and a therefore more precise partitioning or image energy.

4. Efficient Implementation of the 2D Anisotropic FIR Filters Using Polyphase and Block Filtering Approach

Next, an efficient implementation with a low arithmetic complexity and a high degree of parallelism is developed at the system level for the designed 2D directional FIR filters, employing a polyphase structure which achieves a 2D filtering task with a relatively large convolution kernel (of size 65 × 65). In order to implement the convolution operation with a kernel of this size, a block filtering technique [27] combined with polyphase decomposition will be used.

As a preliminary step, employing sub-expression sharing methods, an algorithm which performs 2D filtering with a kernel of size 6 × 6 was developed, as presented below. To achieve this task, the kernel of the 2D filter obtained from the design and the input image to be filtered is decimated by factors 6 and 11, respectively; after this, a polyphase filtering technique is used. Through this approach, four partial output component images are calculated in parallel, namely $Y_0$, $Y_1$, $Y_2$, $Y_3$, given by Equations (23)–(30):

$$Y_0=A_0\times diag\left(A_0^T{H}^T\right)\times M_0\times X_{2D}$$

(23)

In the expression above, the matrices have the following form:

$$A_0=\left[\begin{array}{ccccc}{B}_0& B_0& B_0& B_0& B_0\\ O_1& B_0& O_1& O_1& O_1\\ O_1& O_1& B_0& O_1& O_1\\ O_1& O_1& O_1& B_0& O_1\\ O_1& O_1& O_1& O_1& B_0\\ O_1& O_1& O_1& O_1& O_1\end{array}\right]\hspace{1em}{M}_0=\left[\begin{array}{ccccccccccc}{B}_2& -B_2& -B_2& -B_2& -B_2& -B_2& O_2& O_2& O_2& O_2& O_2\\ O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2\end{array}\right]$$

(24)

Similarly, the matrices $Y_1$, $Y_2$, $Y_3$ are given by the following formula:

$$Y_1=A_1\times diag\left(A_1^T{H}^T\right)\times M_1\times X_{2D}$$

(25)

where

$$A_1=\left[\begin{array}{ccccc}{B}_0& O_1& O_1& O_1& O_1\\ O_1& B_0& B_0& B_0& B_0\\ O_1& O_1& B_0& O_1& O_1\\ O_1& O_1& O_1& B_0& O_1\\ O_1& O_1& O_1& O_1& B_0\\ B_0& O_1& O_1& O_1& O_1\end{array}\right]\hspace{1em}{M}_1=\left[\begin{array}{ccccccccccc}{O}_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2\\ O_2& -B_2& B_2& -B_2& -B_2& -B_2& -B_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2\end{array}\right]$$

(26)

$$Y_2=A_2\times diag\left(A_2^T{H}^T\right)\times M_2\times X_{2D}$$

(27)

where

$$A_2=\left[\begin{array}{ccccc}{O}_1& O_1& O_1& O_1& O_1\\ B_0& O_1& O_1& O_1& O_1\\ O_1& B_0& B_0& B_0& B_0\\ O_1& O_1& B_0& O_1& O_1\\ O_1& O_1& O_1& B_0& O_1\\ B_0& O_1& O_1& O_1& B_0\end{array}\right]\hspace{1em}{M}_2=\left[\begin{array}{ccccccccccc}{O}_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2\\ O_2& O_2& -B_2& -B_2& B_2& -B_2& -B_2& -B_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2\end{array}\right]$$

(28)

$$Y_3=A_3\times diag\left(A_3^T{H}^T\right)\times M_3\times X_{2D}$$

(29)

where

$$A_3=\left[\begin{array}{cccccc}{O}_1& O_1& O_1& O_1& O_1& O_1\\ O_1& O_1& O_1& O_1& O_1& O_1\\ O_1& O_1& O_1& O_1& O_1& O_1\\ B_0& B_0& B_0& O_1& O_1& O_1\\ O_1& B_0& O_1& B_0& B_0& O_1\\ O_1& O_1& B_0& O_1& B_0& B_0\end{array}\right]\hspace{1em}{M}_3=\left[\begin{array}{ccccccccccc}{O}_2& O_2& O_2& -B_2& -B_2& -B_2& B_2& -B_2& -B_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2& O_2\\ O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2& O_2\\ O_2& O_2& O_2& O_2& -B_2& -B_2& -B_2& -B_2& B_2& -B_2& O_2\\ O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& O_2& B_2& O_2\\ O_2& O_2& O_2& O_2& O_2& -B_2& -B_2& -B_2& -B_2& -B_2& B_2\end{array}\right]$$

(30)

In expressions (24), (26), (28), and (30), O1 is a null matrix of size 6 × 21 and O2 is a null matrix of size 21 × 11, and the block matrices have the following form:

$$B_0=\left[\begin{array}{ccccccccccccccccccccc}1& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 1& 0& 0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 0& 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 0& 1& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 0& 1& 0& 1& 1\end{array}\right]$$

(31)

$$B_2=\left[\begin{array}{ccccccccccc}1& -1& -1& -1& -1& -1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& -1& 1& -1& -1& -1& -1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& -1& -1& 1& -1& -1& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& -1& -1& -1& 1& -1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -1& -1& -1& -1& 1& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& -1& -1& -1& -1& -1& 1\end{array}\right]$$

(32)

Next, summing up the intermediate results $Y_0$, $Y_1$, $Y_2$, $Y_3$ derived from (23) to (30), we obtain the output vector $Y$ containing 36 samples of the filtered image:

$$\begin{array}{l}Y=Y_0+Y_1+Y_2+Y_3\\ ={\left[\begin{array}{cccccccccccccccccc}{Y}_{00}& \dots & Y_{05}& Y_{10}& \dots & Y_{15}& Y_{20}& \dots & Y_{25}& Y_{30}& \dots & Y_{35}& Y_{40}& \dots & Y_{45}& Y_{50}& \dots & Y_{55}\end{array}\right]}^T\end{array}$$

(33)

The vector $H$ occurring in Equations (23), (25), (27) and (29) has the following form:

$$H=\left[\begin{array}{cccccccccccccccccc}{h}_{00}& \dots & h_{05}& h_{10}& \dots & h_{15}& h_{20}& \dots & h_{25}& h_{30}& \dots & h_{35}& h_{40}& \dots & h_{45}& h_{50}& \dots & h_{55}\end{array}\right]$$

(34)

while the input vector $X_{2D}$ is as follows:

$$X_{2D}=\left[\begin{array}{cccccccccccccccc}{x}_{0,0}& x_{0,1}& \dots & x_{0,6}& x_{1,0}& x_{1,1}& \dots & x_{1,6}& \dots & x_{10,0}& x_{10,1}& x_{10,2}& x_{10,.3}& x_{10,4}& x_{10,5}& x_{10,6}\end{array}\right]$$

(35)

The main purpose of the developed polyphase filtering algorithm for 2D FIR filters was to reduce the arithmetic complexity and, at the same time, accelerate the calculation using a parallel filtering architecture. It is widely known that performing a direct 2D convolution implies much redundancy in computation owing to overlapping blocks of input data; by discarding a large volume of such unrequired, redundant calculations, a significant reduction in the computational complexity of the system will be obtained.

From this point on, the 2D algorithm discussed before for a simpler situation will be extended from a small kernel of size 6 × 6 to a kernel of larger size (31 × 31). For this purpose, in order to obtain an efficient parallel computing structure, we will further employ a block filtering combined with a polyphase decomposition technique.

In this polyphase approach, we will perform a decimation by factor 6 of the filter kernel. Nevertheless, before decimation, the kernel must be enlarged up to a dimension multiple of six, in our case 66 × 66, by bordering it with zeros. At the same time, we also applied a decimation by a factor of nine to the input image, and thus an input image of size 63 × 63 was derived.

Using the block processing and polyphase decomposition techniques previously described, the following efficient algorithm has been developed for the implementation of the designed 2D FIR filter bank.

Thus, for the special case in which the kernel matrix is 18 × 18 and the input matrix is size 33 × 33, the vectors $H_{00}^T$, $H_{01}^T$,$\dots$,$H_{05}^T$, $H_{10}^T$, $H_{11}^T$,$\dots$,$H_{15}^T$, $H_{20}^T$, $H_{21}^T$,$\dots$,$H_{25}^T$, $H_{30}^T$, $H_{31}^T$,$\dots$,$H_{35}^T$, $H_{40}^T$, $H_{41}^T$,$\dots$,$H_{45}^T$, $H_{50}^T$, $H_{51}^T$,$\dots$,$H_{55}^T$ will have the generic form $H_{ij}$ as expressed below (for $i=0,\dots ,4$, $j=0,\dots ,4$):

$$H_{ij}=\left[\begin{array}{ccccccccc}{h}_{0+i,0+j}& h_{0+i,6+j}& h_{0+i,12+j}& h_{6+i,0+j}& h_{6+i,6+j}& h_{6+i,12+j}& h_{12+i,0+j}& h_{12+i,6+j}& h_{12+i,12+j}\end{array}\right]$$

(36)

For instance, the vectors $H_{00}$, $H_{12}$, $H_{33}$ resulted from the general Formula (36) will be as follows:

$$\begin{array}{l}{H}_{00}=\left[\begin{array}{ccccccccc}{h}_{0,0}& h_{0,6}& h_{0,12}& h_{6,0}& h_{6,6}& h_{6,12}& h_{12,0}& h_{12,6}& h_{12,12}\end{array}\right]\hspace{1em}(i=0,\hspace{0.17em}j=0)\\ H_{12}=\left[\begin{array}{ccccccccc}{h}_{1,2}& h_{1,8}& h_{1,14}& h_{7,2}& h_{7,8}& h_{7,14}& h_{13,2}& h_{13,8}& h_{13,14}\end{array}\right]\hspace{1em}(i=1,\hspace{0.17em}j=2)\\ H_{33}=\left[\begin{array}{ccccccccc}{h}_{3,3}& h_{3,9}& h_{3,15}& h_{9,3}& h_{9,9}& h_{9,15}& h_{15,3}& h_{15,9}& h_{15,15}\end{array}\right]\hspace{1em}(i=3,\hspace{0.17em}j=3)\end{array}$$

(37)

For a simpler demonstration of the proposed filtering procedure, it was first shown in a particular case of lesser complexity, in which the filter kernel is of size 18 × 18, while the size of the input image is 33 × 33. Starting from this simple case, the results can be easily extended for the kernel of the circular FIR filter designed before, of size 65 × 65, extended to size 66 × 66 (by bordering with zeros), in order to perform the decimation with a factor value of six.

In the next step, the simpler algorithm described before for a 2D filter with a 6 × 6 kernel and 11 × 11 input matrix can be easily extended by performing a decimation of the filter kernel by factor 6 and a decimation of the input matrix by factor 11. In this way, by decimating with factor 6 instead of the original kernel of size 18 × 18, 36 matrices of size 3 × 3 will be the result. As an example, for matrix $H_{01}^T$, through decimating factor 6, the following block matrix of size 3 × 3 will result in the following:

$${H^{\prime }}_{01}=\left[\begin{array}{ccc}{h}_{0,1}& h_{0,7}& h_{0,13}\\ h_{6,1}& h_{6,7}& h_{6,13}\\ h_{12,1}& h_{12,7}& h_{12,13}\end{array}\right]$$

(38)

In the following step, by concatenating the rows of ${H^{\prime }}_{01}$, the matrix $H_{01}^T$ results from Equation (36). We will also substitute the vector $X_{2D}$ with matrix $X_{2D}$ using general expression (35). The vectors $X_{00}$, $X_{01}$, $X_{02}$, $X_{03}$, …, $X_{66}$, which compose the matrix $X_{2D}$ and are related to the input image, are defined by the general formula expressed as follows:

$$X_{ij}=\left[\begin{array}{ccccccccc}{x}_{22+i,22+j}& x_{22+i,11+j}& x_{22+i,0+j}& x_{11+i,22+j}& x_{11+i,11+j}& x_{11+i,0+j}& x_{0+i,22+j}& x_{0+i,11+j}& x_{0+i,0+j}\end{array}\right]$$

(39)

For example, the vectors $X_{03}$, $X_{31}$, $X_{66}$ resulting from expression (39) are as follows:

$$\begin{array}{l}{X}_{03}=\left[\begin{array}{ccccccccc}{x}_{22,25}& x_{22,14}& x_{22,3}& x_{11,25}& x_{11,14}& x_{11,3}& x_{0,25}& x_{0,14}& x_{0,3}\end{array}\right]\hspace{1em}(i=0, j=3)\\ X_{31}=\left[\begin{array}{ccccccccc}{x}_{25,23}& x_{25,13}& x_{25,1}& x_{14,23}& x_{14,13}& x_{14,1}& x_{3,23}& x_{3,12}& x_{3,1}\end{array}\right]\hspace{1em}(i=3, j=1)\\ X_{66}=\left[\begin{array}{ccccccccc}{x}_{28,28}& x_{28,17}& x_{28,6}& x_{17,28}& x_{17,17}& x_{17,6}& x_{6,28}& x_{6,17}& x_{6,6}\end{array}\right]\hspace{1em}(i=6, j=6)\end{array}$$

(40)

The vectors $X_{00}$, $X_{01}$, $X_{02}$, $X_{03}$, …, $X_{11,\hspace{0.17em}11}$ have been derived as shown next. With the aim of easier understanding, our demonstration takes into account the simpler case in which the input image is of size 33 × 33, being decimated by factor 11. Thus, instead of the input matrix being size 33 × 33, 121 matrices of size 3 × 3 will result. Considering, for example, the particular case of $X_{10}$, performing decimation by factor 11, we obtain the following 3 × 3 block matrix:

$${X^{\prime }}_{10}=\left[\begin{array}{ccc}{x}_{1,0}& x_{1,11}& x_{1,22}\\ x_{12,0}& x_{12,11}& x_{12,22}\\ x_{23,0}& x_{23,11}& x_{23,22}\end{array}\right]$$

(41)

Subsequently, the rows of ${X^{\prime }}_{01}$ are concatenated, and then the resulting vector is reversed and the vector $X_{1,0}$ is determined from the following general expression:

$$X_{1,0}=\left[\begin{array}{ccccccccc}{x}_{23,22}& x_{23,11}& x_{23,0}& x_{12,22}& x_{12,11}& x_{12,0}& x_{1,22}& x_{1,11}& x_{1,0}\end{array}\right]$$

(42)

Although the algorithm description was limited to a particular case of a 33 × 33 input matrix, to follow its operation more easily, it is straightforward to use it also in a more general situation. As a relevant example, a 2D FIR filtering with a kernel of size 6 × 6, applied to a 11 × 11 pixel input image, was decomposed into a number of 189 1D inner products using the block matrix equations given as follows:

$$Y_0=\left[\begin{array}{ccccc}{D}_0& D_0& D_0& D_0& D_0\\ O_3& D_0& O_3& O_3& O_3\\ O_3& O_3& D_0& O_3& O_3\\ O_3& O_3& O_3& D_0& O_3\\ O_3& O_3& O_3& O_3& D_0\\ O_3& O_3& O_3& O_3& O_3\end{array}\right]\times diag\left(\left[\begin{array}{cccccc}{O}_4& O_4& O_4& O_4& O_4& D_1\\ O_4& O_4& O_4& O_4& D_1& D_1\\ O_4& O_4& O_4& D_1& O_4& D_1\\ O_4& O_4& D_1& O_4& O_4& D_1\\ O_4& D_1& O_4& O_4& O_4& D_1\end{array}\right]H^T\right)\times \left[\begin{array}{ccccccccccc}{D}_2& -D_2& -D_2& -D_2& -D_2& -D_2& O_5& O_5& O_5& O_5& O_5\\ O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5\end{array}\right]\times X_{2D}$$

(43)

$$Y_1=\left[\begin{array}{ccccc}{D}_0& O_3& O_3& O_3& O_3\\ O_3& D_0& D_0& D_0& D_0\\ O_3& O_3& D_0& O_3& O_3\\ O_3& O_3& O_3& D_0& O_3\\ O_3& O_3& O_3& O_3& D_0\\ D_0& O_3& O_3& O_3& O_3\end{array}\right]\times diag\left(\left[\begin{array}{cccccc}{D}_1& O_4& O_4& O_4& O_4& D_1\\ O_4& O_4& O_4& O_4& D_1& O_4\\ O_4& O_4& O_4& D_1& D_1& O_4\\ O_4& O_4& D_1& O_4& D_1& O_4\\ O_4& D_1& O_4& O_4& D_1& O_4\end{array}\right]H^T\right)\times \left[\begin{array}{ccccccccccc}{O}_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5\\ O_5& -D_2& D_2& -D_2& -D_2& -D_2& -D_2& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5\end{array}\right]\times X_{2D}$$

(44)

$$Y_2=\left[\begin{array}{ccccc}{O}_3& O_3& O_3& O_3& O_3\\ D_0& O_3& O_3& O_3& O_3\\ O_3& D_0& D_0& D_0& D_0\\ O_3& O_3& D_0& O_3& O_3\\ O_3& O_3& O_3& D_0& O_3\\ D_0& O_3& O_3& O_3& D_0\end{array}\right]\times diag\left(\left[\begin{array}{cccccc}{D}_1& O_4& O_4& O_4& D_1& O_4\\ O_4& O_4& O_4& D_1& O_4& O_4\\ O_4& O_4& D_1& D_1& O_4& O_4\\ O_4& D_1& O_4& D_1& O_4& O_4\\ D_1& O_4& O_4& D_1& O_4& O_4\end{array}\right]H^T\right)\times \left[\begin{array}{ccccccccccc}{O}_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5\\ O_5& O_5& -D_2& -D_2& D_2& -D_2& -D_2& -D_2& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5\end{array}\right]\times X_{2D}$$

(45)

$$Y_3=\left[\begin{array}{cccccc}{O}_3& O_3& O_3& O_3& O_3& O_3\\ O_3& O_3& O_3& O_3& O_3& O_3\\ O_3& O_3& O_3& O_3& O_3& O_3\\ D_0& D_0& D_0& O_3& O_3& O_3\\ O_3& D_0& O_3& D_0& D_0& O_3\\ O_3& O_3& D_0& O_3& D_0& D_0\end{array}\right]\times diag\left(\left[\begin{array}{cccccc}{O}_4& O_4& D_1& O_4& O_4& O_4\\ O_4& D_1& D_1& O_4& O_4& O_4\\ D_1& O_4& D_1& O_4& O_4& O_4\\ O_4& D_1& O_4& O_4& O_4& O_4\\ D_1& D_1& O_4& O_4& O_4& O_4\\ D_1& O_4& O_4& O_4& O_4& O_4\end{array}\right]H^T\right)\times \left[\begin{array}{ccccccccccc}{O}_5& O_5& O_5& -D_2& -D_2& -D_2& D_2& -D_2& -D_2& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5& O_5\\ O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5& O_5\\ O_5& O_5& O_5& O_5& -D_2& -D_2& -D_2& -D_2& D_2& -D_2& O_5\\ O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5& O_5& D_2& O_5\\ O_5& O_5& O_5& O_5& O_5& -D_2& -D_2& -D_2& -D_2& -D_2& D_2\end{array}\right]\times X_{2D}$$

(46)

Finally, the derived output vector is given by the following expression:

$$\begin{array}{l}Y=Y_0+Y_1+Y_2+Y_3\\ ={\left[\begin{array}{cccccccccccccccccc}{Y}_{00}& \dots & Y_{05}& Y_{10}& \dots & Y_{15}& Y_{20}& \dots & Y_{25}& Y_{30}& \dots & Y_{35}& Y_{40}& \dots & Y_{45}& Y_{50}& \dots & Y_{55}\end{array}\right]}^T\end{array}$$

(47)

In Equations (43)–(46), the block matrices are, respectively, $D_0=B_0\otimes U_{11}$, with the vector $U_{11}=\left[\begin{array}{ccccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right]$ and $D_1=B_1\otimes I_{11}$, where $I_{11}$ is the $11\times 11$ identity matrix (with ones on the main diagonal and zeros elsewhere); we also have $D_2=B_2\otimes I_{11}$. The matrices $O_3$, $O_4$, and $O_5$ are null matrices of size 6 × 189, 189 × 54, and 189 × 99, respectively.

5. Discussion

The proposed synthesis procedure for 2D rotationally invariant filter banks is fully analytical without resorting to any numerical optimization algorithms. It is well known that analytical design techniques yield parametric filters, with closed-form and adjustable frequency responses. As far as the authors are aware, this type of analytical design for 2D FIR circular filter banks has not been approached before by other researchers. Referring to related works, other analytical methods for the design of 2D circular filters of IIR type, including circular filter banks, have been approached before [15,16,17]. However, a rigorous performance comparison with other circularly symmetric filters found in the literature would be very difficult to make. For instance, the design approaches for circular filters given in [10,11,12,13,14] are very different from our design and lead to other types of circular filters with different characteristics and applications, so they are rather difficult to compare with our proposed technique.

The trapezoidal-shaped prototype was chosen here because it can be conveniently approximated analytically by a Fourier series truncated to a number of terms according to the imposed accuracy and having less ringing (ripple) than the ideal square-shaped LP filter. The transition region has an adjustable steepness, and the filter is nearly maximally flat with a given small ripple. Moreover, this filter is zero-phase since its transfer function is real; therefore, the component frequencies are not phase-shifted, so the filtered images will not have phase distortions.

The shape distortions of the filters’ characteristics are mainly due to circularity distortions for the upper filters of the bank (as show the contour plots, the circular shape tends more to a rounded square); this is slightly visible from the fifth band-pass filter and more pronounced for the 6-th and 7-th BP filters and the HP filter. However, even if the marginal filters are not perfectly circular, the components of the designed filter bank relatively compactly cover the frequency plane, the contour plots being “concentric” and overlapping, intersecting at the imposed, conventional value of 0.7071.

In regard to implementation, the proposed technique is an extension of the method used in [16] but has a higher degree of parallelism and significantly diminishes the arithmetic complexity. To highlight its efficiency, the developed filtering procedure can be compared with the direct convolution between an image and the large filter kernel in regard to the computational requirements. Thus, the filtering operation of an $M\times N$ image, performed by an FIR filter with a kernel size $m\times n$, implies a direct convolution between a $m\times n$ matrix and a $M\times N$ matrix. For each one of the total $M\times N$ image pixels, $m\times n$ multiplications are performed, such that the 2D filtering complexity is of the order $O(MNmn)$. Also, the total number of additions would be $(N+n^2)(M+m^2)$.

In the simpler case that has been used to exemplify our implementation, the filter kernel has a size of 18 × 18, while the image is 33 × 33; thus, we would have in the case of direct convolution 352,836 multiplications with 127,449 additions. In our approach, we are using only 189 inner products with 3 × 3 multiplications and 3 × 3 additions for each, which is 189 × 3 × 3 = 1701 multiplications and 1701 additions, plus 4320 additions in the post-processing stage.

The proposed system-level implementation structure based on block filtering and polyphase decomposition techniques can be achieved with different values of decimation factors for both the image to be filtered and the filter kernel; thus, a filter implementation with various degrees of parallelism can be derived. In this work, we performed a decimation by factor 6 of the filter kernel and a decimation by factor 9 to the input image. For comparison, in the previous paper [16], we used decimation factor 4 for the filter kernel and factor 5 for the input image, respectively. Therefore, in the current paper, we obtain a higher degree of parallelism, with the drawback that the resulting matrices have a larger size. A systematic generalization of this implementation is probably difficult to be achieved. This implementation is practically independent of the prototype filter shape, which is an advantage. In fact, it can be applied to any 2D FIR filter with a given kernel. Depending on its size and the chosen decimation factor, a bordering with zeros may be required, as shown. For 2D filters of higher order (corresponding to more selective prototypes), the filter kernel has a larger size, and the implementation is obviously more complex.

6. Conclusions

The developed synthesis procedure for 2D rotationally invariant filter banks is fully analytic and does not use numerical optimization algorithms. As explained before, by applying a specific frequency mapping to a chosen prototype with maximally flat characteristics and a given bandwidth and transition region, we directly obtain the factored frequency responses for the component filters of the 2D CFB for any desired specifications (number of filters in the FB, bandwidth, overlap, transition steepness). Through a simulation example, it is shown that both the uniform and non-uniform CFBs can be practically used as an image analysis FB by decomposing the given image into a set of sub-band component images, from which the original image can be easily and precisely reconstructed. The efficient proposed implementation based on the polyphase and block filtering approach gives a filter structure with a high degree of parallelism and reduced arithmetic complexity. Based on the accurate reconstruction of the original test image from its sub-band components, as the simulation results show, in future work, we intend to investigate if such rotationally invariant filter banks (both uniform and non-uniform) could be systematically used in the sub-band coding of images. Another aspect would be to improve the filter bank circularity, reducing its marginal shape distortions, by finding a more accurate frequency mapping, possibly of higher order. In terms of implementation, we will also explore how to optimally choose the decimation factors for the filter kernel and input image in order to minimize the number of arithmetic operations and possibly find other efficient implementation structures which ensure low computational complexity.