Paper Title (use style: paper title)

DRAFT

Optimal design of digital IIR and FIR filters using complex flatness constraints: A unified approach Hugh L. Kennedy Technical Knockout Systems Pty. Ltd., Adelaide, Australia [email protected] Abstract— A procedure for the design of digital filters with either a finite or infinite impulse response (FIR or IIR) is presented. Complex flatness constraints, incorporating the desired passband group delay, are used to specify the properties of non-linear-phase filters. Solutions are found using a linear combination of basis functions with zero or non-zero poles, for FIR or IIR filters, respectively. The procedure is used to design multidimensional filters for Lucas-Kanade optical flow. Keywords— Image motion analysis, Filtering algorithms

I. INTRODUCTION Optimal design techniques that minimize the maximum [1], or minimize the integral, of a frequency-dependent error function or noise power density have proven to be useful tools in the design of digital filters, where the (weighted) error is the difference between the achieved and desired responses [2],[3]. Alternatively [4]-[10] – or additionally [1],[11]-[17] – flatness constraints may be used to express the desired response of the filter. By specifying vanishing non-zero derivatives of the frequency response up to an order of 𝐿, i.e. the degree of flatness, at critical frequencies, it is possible to realize digital filters that have both low maximum deviations and low integral errors over large swathes of spectrum, with smooth/monotonic transitions between bands, using simple low-order, recursive or non-recursive, structures. Wider appreciation and utilization of this procedure has been impeded by a reluctance to pose the derivative constraints in complex form with a group-delay dependence. For linear-phase filters with a finite impulse response (FIR), there is no need to use complex flatness constraints because perfect phase linearity over the entire (sub-Nyquist) frequency interval is guaranteed and need not be considered explicitly. Thus the use of real constraints is satisfactory in these problems due to symmetry or anti-symmetry of the impulse response [5]-[7],[11],[12]. However, in cases where the cost of delays is too great (e.g. in feedback control or distributed systems) perfect phase linearity is a luxury that cannot always be afforded and it may be necessary to trade phase and magnitude flatness for a reduced or even negative delay and an asymmetric impulse response [1],[9],[10],[16]. For causal filters with an infinite impulse response (IIR), phase linearity (i.e. uniformly constant delay) cannot be overlooked and taken for granted, thus magnitude and phase/delay must be considered jointly [3],[4],[8],[15]. However, in both integral- and derivative-based design procedures, dual gain and phase/delay metrics (both real-valued), each with their own criteria and set of imposed constraints, are generally preferred over a single complex metric, that is derived from magnitude and delay requirements. The simultaneous satisfaction of complex flatness requirements and the achievement of a reasonable response at the intervening frequencies is straightforward when all basis functions are complex and negative frequencies are considered. FIR and IIR filter structures have their own unique benefits: FIR filters offer the option of perfect phase linearity, which is a consequence of the symmetry or anti-symmetry of the filter coefficients; whereas, IIR filters allow highly frequency-selective filters to be synthesized at a very low computational cost, which follows from the independence of filter complexity and impulse response duration – due to their utilization of feedback. Thus in the regime of low-delay or short-impulse response filtering, it is difficult to know in advance which structure is more appropriate, especially when other factors such as (hardware or software) circuit complexity, numeric rounding-error sensitivity, robustness and short-term/long-term accuracy trade-offs (i.e. transient response vs. steady-state response) are also considered. There are therefore benefits in having a procedure that can treat both IIR and FIR designs in a similar way and on an equal footing. It is assumed here that there is no particular requirement for a filter with an IIR or an FIR from the outset. A simple procedure for the derivation of appropriate filter coefficients is presented in this paper. The proposed procedure is somewhat unusual in that it may be used to design both IIR and FIR filter structures. The procedure may be used to derive low-pass smoothers, low-pass filters, highpass filters, and low-pass differentiators. These filtering functions are defined as follows: For low-pass smoothers, an emphasis is placed on passing (delayed) polynomials exactly and on white-noise noise gain minimization [13],[15],[18]; For low-pass filters, perfect reproduction of polynomials is not necessarily required and having a wider passband is a higher priority; High-pass filters are designed for low-frequency attenuation to accentuate high-frequency detail; Low-pass differentiators, are a numerical approximation of an analytical derivative for low-frequency signals that are corrupted by high-frequency noise [1],[3],[6],[28],[29].

For final version see: Proc. International Conference on Digital Image Computing: Techniques and Applications (DICTA), Gold Coast, QLD, 2016, pp. 1-8. doi: 10.1109/DICTA.2016.7797070

DRAFT A basis set expansion is used here to ensure that complex flatness constraints are satisfied and that the frequency response at intervening frequencies is reasonable for both FIR and IIR filters using a common framework. Guidelines for the specification of appropriate constraints and the selection/construction of a suitable basis set are provided, using a multidimensional signal processing problem, that requires both low-pass smoothers and differentiators, as an illustrative design example. The design procedures for FIR and IIR filters differ only in the form of the selected basis functions, and as a consequence, the input, mathematical/analytical apparatus, and output, of the design processes are essentially the same. An attempt is made in this paper to dispel the common misconception that IIR design procedures are “more challenging” than FIR methods and that they “require highly sophisticated optimization methods” [3]. While this may be true for high-order filters it is certainly not the case for low-order filters. It is shown here that standard linear-phase (FIR) design procedures may be used for more general non-linear-phase (FIR and IIR) structures when the design problem is posed in a complex form. II. FORMULATION The proposed filter design procedure is described in this section. The broad objective of the design problem is summarized in subsection A; the complex flatness constraints, that may be used to specify the desired filter behavior, and the structure of the problem, are defined in subsection B; and a simple way of assembling a filter that satisfies those constraints is described in subsection C. A. Preliminaries The discrete-time transfer function of a digital filter is 𝐻(𝓏) = 𝐵(𝓏)⁄𝐴(𝓏), where 𝐵(𝓏) and 𝐴(𝓏) are polynomials in 𝓏, of degree 𝑀𝑏 − 1 and 𝑀𝑎 − 1 respectively, with 𝑀𝑏 ≤ 𝑀𝑎 . This function provides the link between: The impulse response ℎ(𝑛), via ℎ(𝑛) = 𝒵 −1 {𝐻(𝓏)}, where 𝒵 −1 is the inverse 𝒵 transform; the frequency response ℋ(𝜔), via ℋ(𝜔) = 𝐻(𝓏)|𝓏=𝑒 𝑖𝜔 , with 𝑖 2 = −1 and 𝜔 being the angular frequency (radians per sample); and the linear difference equation in the sample domain, which is defined as 𝑀 −1

𝑀 −1

𝑏 𝑎 𝓎(𝑛) = ∑𝑚=0 𝑏𝑚 𝓍(𝑛 − 𝑚) − ∑𝑚=1 𝑎𝑚 𝓎(𝑛 − 𝑚)

(1a)

using 𝑀 −1

𝑀 −1

𝑏 𝑎 𝐴(1𝓏) = ∑𝑚=0 𝑎𝑚 𝓏 −𝑚 and 𝐵(1𝓏) = ∑𝑚=0 𝑏𝑚 𝓏 −𝑚

(1b)

where: 𝐻(𝓏) is rendered causal by dividing 𝐴(𝓏) & 𝐵(𝓏) by 𝓏 𝑀𝑎−1 to yield 𝐴(1𝓏) & 𝐵(1𝓏); 𝑎 & 𝑏 are normalized to ensure that 𝑎0 = 1; 𝓍 & 𝓎 are the sampled inputs and outputs of the filter, respectively; and where 𝑛 & 𝑚 are integer-valued time and delay indices, respectively (with 𝑚 = 0 selecting the most recently acquired or current sample and 𝑚 ≥ 0 for causality). Note that in anticipation of the multidimensional signal processing problem that will be considered later in this paper, a script font is used in the onedimensional system above to distinguish these variables from the spatial 𝑥 & 𝑦 and temporal 𝑧 dimensions; similarly, the generic term “sample” is used to refer to a discrete spatiotemporal measurement, in the same way that the generic term “frequency” is used in the corresponding transform spaces. The poles 𝑝𝑚 and zeros of the filter transfer function are the 𝑀𝑎 − 1 roots of 𝐴(𝓏) and the 𝑀𝑏 − 1 roots of 𝐵(𝓏), respectively; with a less-than-unity pole radius (i.e. 𝑟𝑚 < 1, where 𝑟𝑚 = |𝑝𝑚 |) for causal stability; and with resonant pole frequencies occurring in complex-conjugate pairs over the sub-Nyquist interval (i.e. −𝜋 ≤ 𝜔𝑚 ≤ +𝜋, where 𝜔𝑚 = ∠𝑝𝑚 ) for a real impulse response. The objective of the filter design procedure is to determine appropriate values of the 𝑎 & 𝑏 coefficients in (1), that approximately yield a quantified frequency response and a qualified impulse response. The generic difference equation structure provided above incorporates output feedback (for an IIR). Feedback terms are omitted when 𝑎𝑚 = 0 for 𝑚 = 1 … 𝑀𝑎 − 1, for an FIR of length 𝑀𝑏 samples, all 𝑀𝑎 − 1 poles at 𝓏 = 0, and all 𝑀𝑏 − 1 zeros anywhere in the complex 𝓏 plane. It is assumed in the following, that loworder filters are favored, that either IIR or FIR structures are acceptable, and that any of the four filter aforementioned functions may be required. A procedure that poses the filter design problem as a set of complex flatness (equality) constraints, and expresses the solution as a linear combination of complex basis functions, provides the simplicity and flexibility that is required. B. Constraint specification The method of Lagrange multipliers may be used to solve linear filter-design problems that are formulated using the minimization of an integral quantity (e.g. noise gain or integral error) subject to a set of equality constraints (e.g. on flatness) [11],[12],[15],[17]. When low-order filters are essential; for system simplicity and real-time performance, there may be insufficient degrees of freedom for integral minimization, and it may only be possible to specify and satisfy one or two flatness constraints at one or two frequencies. For the type of sensor signal/image processing applications considered later in this paper, the selected constraints are likely to be, in order of decreasing priority: 1) dc flatness at 𝜔 = 𝜔dc = 0; 2) Nyquist flatness at 𝜔 = 𝜔pi = 𝜋; then 3) wideband (wb) flatness at 𝜔 = ±𝜔wb , where 𝜔wb is an arbitrarily selected frequency, somewhere in the passband of the filter. High-pass filters (e.g. for sensor bias removal and image edge enhancement) may also be designed using the proposed procedure; however, they are not the focus of this paper. For low-pass smoothers: The first constraint is particularly significant because flatness up to the 𝐿th order ensures that polynomials of 𝐿th degree and 𝐿th-order moments are perfectly transmitted in the time domain, possibly with a filter-imposed delay


DRAFT of 𝑞 samples (which need not be integer valued) and possibly corrupted by additive (albeit attenuated) sensor noise [1],[13],[15],[16],[18]. Low-pass differentiators with excellent noise immunity are obtained by differentiating the interpolating polynomial. For low-pass filters: The width of the high-frequency stopband (for noise/interference attenuation) increases with the order of the second flatness constraint, whereas the width of the low-frequency signal passband (for signal transmission) increases with the order of the third flatness constraint [5],[10],[15],[16]. When wideband low-pass filters are required, increasing the number of dc constraints provides rapidly diminishing returns. Thus a low-order filter with a wide bandwidth is more efficiently synthesised, using less dc constraints and more wideband constraints. For instance, passband convexity is used in [6]; however, the use of an inequality constraint results in a linear programming problem, which must be solved via the simplex method, and only linear-phase FIR filters are considered. For non-linear-phase filters, the utilization of complex flatness constraints ensures exact phase linearity (for a delay of 𝑞 samples) at the constraint frequencies (for IIR and FIR filters) and reasonable linearity nearby (or exact for an appropriately configured FIR filter, with 𝑞 = {𝑀𝑏 − 1}⁄2, for both odd or even 𝑀𝑏 ). For linear-phase FIR filters, 𝐿𝑐 th-order (real) flatness constraints at a critical design frequency 𝜔𝑐 , are traditionally specified using [5] 𝑑 𝑙 ℋ(𝜔) 𝑑𝜔𝑙

|

𝜔=𝜔𝑐

= 𝑔𝑙 (𝜔𝑐 ) for 0 ≤ 𝑙 ≤ 𝐿𝑐 and 0 ≤ 𝜔𝑐 ≤ 𝜋. (2)

For low-pass/high-pass filters, ℋ(𝜔) is usually expressed as a piecewise constant function with ℋ(𝜔) = 1 in the passband and ℋ(𝜔) = 0 in the stopband. Thus: For low-pass smoothers: 𝑔𝑙 (𝜔dc ) = 1 and 𝑔𝑙 (𝜔pi ) = 0 for 𝑙 = 0; 𝑔𝑙 (𝜔dc ) = 0 and 𝑔𝑙 (𝜔pi ) = 0 for 1 ≤ 𝑙 ≤ 𝐿𝑐 . For low-pass filters: 𝑔𝑙 (𝜔dc ) = 1, 𝑔𝑙 (𝜔wb ) = 1 and 𝑔𝑙 (𝜔pi ) = 0 for 𝑙 = 0; 𝑔𝑙 (𝜔dc ) = 0, 𝑔𝑙 (𝜔wb ) = 0 and 𝑔𝑙 (𝜔pi ) = 0 for 1 ≤ 𝑙 ≤ 𝐿𝑐 . For high-pass filters 𝜔𝑐 = 𝜋 and optionally 𝜔𝑐 = 𝜋 − 𝜔wb . 𝑔𝑙 (𝜔dc ) = 0, 𝑔𝑙 (𝜋 − 𝜔wb ) = 1 and 𝑔𝑙 (𝜔pi ) = 1 for 𝑙 = 0; 𝑔𝑙 (𝜔dc ) = 0, 𝑔𝑙 (𝜋 − 𝜔wb ) = 1 and 𝑔𝑙 (𝜔pi ) = 0 for 1 ≤ 𝑙 ≤ 𝐿𝑐 . For ideal differentiators ℋ(𝜔) = 𝑖𝜔; thus a low-pass differentiator may be defined using: 𝑔𝑙 (𝜔dc ) = 0, 𝑔𝑙 (𝜔wb ) = 𝑖𝜔wb and 𝑔𝑙 (𝜔pi ) = 0 for 𝑙 = 0; 𝑔𝑙 (𝜔dc ) = 𝑖, 𝑔𝑙 (𝜔wb ) = 𝑖 and 𝑔𝑙 (𝜔pi ) = 0 for 𝑙 = 1; 𝑔𝑙 (𝜔dc ) = 0, 𝑔𝑙 (𝜔wb ) = 0 and 𝑔𝑙 (𝜔pi ) = 0 for 1 < 𝑙 ≤ 𝐿𝑐 . The number of constraints (𝑁𝑐 = 𝐿𝑐 + 1) need not be the same at all frequencies; the number of constraints at 𝜔dc , 𝜔pi ,+𝜔wb and −𝜔wb are 𝑁dc , 𝑁pi , 𝑁wb and 𝑁wb , respectively. Thus the total number of constraints is 𝑁 = 𝑁dc + 𝑁pi + 2𝑁wb . For non-linear phase (FIR or IIR) filters, these flatness constraints need to be generalized to incorporate the desired passband group delay of 𝑞 samples, if the simplicity of the linear-phase design procedure is to be maintained. The required result follows from the fact that a delay of 𝑞 in the sample domain is applied to 𝐻(𝓏) via a convolution with the delayed unit impulse 𝛿𝑞 (𝓏) = 𝓏 −𝑞 with frequency response 𝑒 −𝑖𝑞𝜔 . As convolution in the sample domain is modulation in the frequency domain we have 𝐻𝑞 (𝓏) = 𝓏 −𝑞 𝐻(𝓏) and ℋ𝑞 (𝜔) = 𝓏 −𝑞 𝐻(𝓏)|𝑧=𝑒 𝑖𝜔 thus ℋ𝑞 (𝜔) = 𝑒 −𝑖𝑞𝜔 ℋ(𝜔). Using this result in (2) yields the relationships we seek: 𝑑 𝑙 ℋ𝑞 (𝜔) 𝑑𝜔𝑙

| 𝜔=𝜔𝑐

= 𝑔𝑙 (𝜔𝑐 )

(3a)

for 0 ≤ 𝑙 ≤ 𝐿𝑐 and −𝜋 ≤ 𝜔𝑐 ≤ +𝜋. For low-pass/high-pass filters, ℋ𝑞 (𝜔) is a piecewise complex constant function with ℋ𝑞 (𝜔) = 𝑒 −𝑖𝑞𝜔 in the passband and ℋ𝑞 (𝜔) = 0 in the stopband; for ideal differentiators ℋ𝑞 (𝜔) = 𝑖𝜔𝑒 −𝑖𝑞𝜔 . For instance, a low-pass smoother with a group delay of 𝑞 samples at the dc limit may now be specified using the following set of complex constraints: 𝑔𝑙 (𝜔dc ) = (−𝑖𝑞)𝑙 and 𝑔𝑙 (𝜔pi ) = 0 for 0 ≤ 𝑙 ≤ 𝐿𝑐 .

(3b)

This result is presented without explanation in [15], where it is used to derive low-pass smoothers that also minimize the white-noise gain. The absence of 𝑞 in (2) for linear-phase FIR filters is a consequence of impulse response symmetry or anti-symmetry. This special structure allows a causal filter to be designed via a virtual non-casual filter with a flat phase response and a group delay of zero. For example, a linear-phase FIR filter of odd length with 𝑞 = 2, may be designed using 𝐻(𝑧) = 𝑏0 𝓏 2 + 𝑏1 𝓏 1 + 𝑏2 + 𝑏3 𝓏 −1 + 𝑏4 𝓏 −2 . After substituting 𝓏 = 𝑒 𝑖𝜔 , imaginary terms cancel and only real terms remain in the frequency response. Once the 𝑏


DRAFT coefficients have been determined, the optimal solution is then multiplied by 𝓏 −2 to render it causal. Shifting the transfer function forward or backwards in time has no impact on its behavior and the magnitude response is unchanged. This simplification is of no use in the design of filters with non-symmetric impulse responses, because imaginary terms do not cancel; and it cannot be exploited to design filters with feedback, because future outputs cannot be considered before the current output is determined. If the solution is expressed as a linear combination of complex basis functions, i.e. 𝐻𝑞 (𝓏) = ∑𝐾 𝑘=1 𝑐𝑘 𝜓𝑘 (𝓏)

(4)

then for 𝐾 = 𝑁, the linear coefficients are simply found using 𝒄 = 𝑱−1 𝒈 .

(5a)

The 𝑱 matrix and the 𝒄 & 𝒈 vectors above are defined as follows: 𝒄 is a 𝐾 × 1 vector containing elements 𝑐𝑘 . 𝒈 is an 𝑁 × 1 vector formed by stacking the desired derivatives at the specified frequencies i.e. 𝒈 = [𝒈(𝜔dc )

𝒈(𝜔pi )

𝒈(+𝜔wb )

𝒈(−𝜔wb )]

T

(5b)

where 𝒈(𝜔c ) = [𝑔0 (𝜔c ), ⋯ , 𝑔𝐿𝑐 (𝜔c )]

(5c)

𝑱 is a 𝑁 × 𝐾 matrix containing derivatives of the basis functions, evaluated at the specified frequencies. The elements in the 𝑘th column of 𝑱 are 𝑱𝑘 = [𝑱𝑘 (𝜔dc )

𝑱𝑘 (𝜔pi )

𝑱𝑘 (+𝜔wb )

𝑱𝑘 (−𝜔wb )]

T

(5d)

where 𝐿

𝑱𝑘 (𝜔c ) = [𝒟𝜔0 𝜓𝑘 (𝜔), ⋯ , 𝒟𝜔c 𝜓𝑘 (𝜔)]|𝜔=𝜔 . c

(5e)

Note: the T superscript is the vector transpose operation and 𝒟𝜔𝑙 𝜓𝑘 (𝜔) is the 𝑙th derivative of 𝜓𝑘 (𝜔) with respect to 𝜔. Once the optimal coefficients 𝒄, have been determined, (4) is expanded and simplified (a partial fraction contraction) so that it is of the form 𝐻𝑞 (𝓏) = 𝐵𝑞 (𝓏)⁄𝐴𝑞 (𝓏). (Note that the coefficients are complex when wideband constraints are applied.) The required 𝑎 & 𝑏 coefficients of the filter’s linear difference equation, as defined in (1a), are then equated to the 𝐵𝑞 (1𝓏) & 𝐴𝑞 (1𝓏) coefficients, as shown in (1b). C. Basis selection Optimal zero locations for the satisfaction of derivative constraints at specified frequencies using FIR filters may be determined using Hermite polynomial interpolation [5],[14]. Expressing the solution as a linear combination of complex basis functions is also an effective and simple approach. The form of the basis set determines the location of the filter poles and influences the location of the filter zeros; whereas the linear coefficients (i.e. 𝑐𝑘 ) only influence the locations of the filter zeros. This approach may also involve an outer optimization loop, that seeks to iteratively refine the basis set structure (i.e. the location of the poles); however, this extra layer of design automation (and complexity) is not considered here. For (linear- and non-linear-phase) FIR filters, the basis set is simply a set of delayed unit impulses in the time domain and a set of complex sinusoids in the frequency domain (i.e. all poles at the origin of the complex 𝓏 plane); indeed, most design procedures for linear-phase FIR filters, e.g. [7],[11],[12], may be interpreted as a being a limiting case of the method described here. The proposed approach is simply an extension of this idea, with the basis set augmented to include components with non-zero poles for IIR filters. The pole radius determines the impulse response duration and causal stability is guaranteed if the poles of all basis functions are inside the unit circle. For high-order filters with many surplus degrees of freedom, it is easier to specify a frequency-dependent piece-wise constant response than it is to specify many derivatives at many points. In these cases, design procedures that employ integral measures of solution deviation are ideal; however, derivative equality constraints, derivative inequality constraints, and/or global maximum constraints, may also be imposed. When low-order filters are required (for low realization complexity) there are no surplus degrees of freedom because they must all be used to specify required behaviour at, and around, critical frequencies. There is no guarantee that a reasonable response, over the entire frequency range (𝜔 = 0 … 𝜋) can be met with a given basis set; however, it is fairly easy to understand why some basis functions are better than others for a given set of requirements and design objectives. A solution with the following general form was found to offer the flexibility and simplicity required: 𝐾

𝐾

𝐾pln

+pln

zer zer zer axs axs axs 𝐻𝑞 (𝓏) = ∑𝑘=1 𝑐𝑘 𝜓𝑘 (𝓏) + ∑𝑘=1 𝑐𝑘 𝜓𝑘 (𝓏) + ∑𝑘=1 𝑐𝑘

+pln

𝜓𝑘

𝐾

pln −pln −pln (𝓏) + ∑𝑘=1 𝑐𝑘 𝜓𝑘 (𝓏)

(6)


DRAFT where 𝐾zer is the number of basis functions with repeated poles at the origin of the complex 𝓏 plane; 𝐾axs is the number of basis functions with repeated poles at 𝑝, between -1 and +1 on the real 𝓏 axis; 𝐾pln is the number of conjugate basis-function pairs with repeated poles at 𝑟𝑒 ±𝑖𝜔pln inside the unit circle on the complex 𝓏 plane; thus the size of the basis set is 𝐾 = 𝐾zer + 𝐾axs + 2𝐾pln . Furthermore 1 𝑘−1

𝜓𝑘zer (𝓏) = ( ) 𝓏

+pln

𝜓𝑘

(𝓏) = (

𝜓𝑘axs (𝓏) = (

, 𝑘

𝑧 𝓏−𝑟𝑒

+𝑖𝜔pln

+pln

and 𝑐𝑘zer , 𝑐𝑘axs , 𝑐𝑘

−pln

) , 𝜓𝑘 −pln

& 𝑐𝑘

𝑧 𝓏−𝑝

(𝓏) = (

)

𝑘

𝑧 𝓏−𝑟𝑒

−𝑖𝜔pln

𝑘

) (7)

are the linear coefficients to be determined.

𝜓𝑘zer

The basis functions (pure delays) are for manipulating the response at all frequencies. As they form an orthogonal set of complex sinusoids in the frequency domain, i.e. after using 𝜓𝑘zer (𝓏)|𝓏=𝑒 𝑖𝜔 , they may be used in isolation and in large numbers, to satisfy high-order constraints at all frequencies, without the need to worry about ill conditioning, for an FIR design. In IIR designs they are a cheap way of increasing the degrees of freedom because they involve one less multiplication operation per filter pole than the other basis functions. The 𝜓𝑘axs basis functions are so-called “leaky” integrators (for 0 < 𝑝 < 1). They are well-placed to contribute to the satisfaction of the dc (for 0 < 𝑝 < 1) or Nyquist (for −1 < 𝑝 < 0) constraints. ±pln

The 𝜓𝑘 basis functions are damped (for 𝑟 < 1) oscillators, with a resonant peak at ±𝜔pln which increases in magnitude as 𝑟 → 1. They are well-placed to contribute to the satisfaction of the wide-band constraints, for a broadening of the passband at high or low frequencies. Using 𝐾 > 𝑁 results in an overdetermined problem and in these cases, the inclusion of an integral metric and Lagrange multipliers is recommended. However, this paper is primarily concerned with the simple case with 𝐾 = 𝑁, for a low-order filter, an exact solution, and the use of derivative metrics only. Note also that 𝐾zer , 𝐾axs and 𝐾cpx need not all be greater than zero. The mix of basis functions depends on the filter function (as represented by the constraints) and the filter structure (IIR or FIR). For example, some recommended mixtures are as follows (guidelines only): For low-pass or high-pass FIR filters, 𝐾zer = 𝑁dc + 𝑁pi + 2𝑁wb , 𝐾axs = 0 and 𝐾pln = 0, with 𝑞 = (𝐾 − 1)⁄2 for a linear-phase filter; For low-pass IIR filters and differentiators, 𝐾axs = 𝑁dc + 𝑁pi and 𝐾pln = 𝑁wb . When 𝐾axs and 𝐾pln are both non-zero, it is convenient to use a common pole radius 𝑟 = |𝑝| and 𝜔pln = 𝜔wb . When computing the coefficients of the linear difference equation of a given structure and function, there is clearly a preference in the literature for the derivation of closed-form analytical expressions, where possible; however, in an effort to create a general and flexible procedure, a hybrid analytical/numerical approach is adopted here, involving the (numeric) inversion of a small matrix, with elements that are populated by evaluating (analytic) functions. The proposed procedure involves calculations that are simple and fast with the assistance of a computer and it is of similar complexity to other commonly used procedures, such as those that involve the discretization of a prototype analogue design. However, the proposed procedure is superior because it is well known that flatness in a particular analogue filter does not necessarily guarantee flatness in the corresponding digital filter after discretization [8]. As a consequence of numerical rounding error sensitivity, which grows with the size of the basis set and as the conditioning of 𝑱 deteriorates, the solution is an approximation. Ill-conditioned problems may arise for poorly selected basis sets that consist of a large number of basis functions that are all too similar over the selected frequencies. III. APPLICATION: LUCAS AND KANADE OPTICAL FLOW The optical flow algorithm of Lucas and Kanade (LK) relies on the fact that, under ideal conditions (constant illumination, ignoring object obscuration and deformation, etc.) 𝐼𝑥 (𝑛𝑥𝑦 )𝑣𝑥 (𝑛𝑥𝑦 ) + 𝐼𝑦 (𝑛𝑥𝑦 )𝑣𝑦 (𝑛𝑥𝑦 ) + 𝐼𝑧 (𝑛𝑥𝑦 ) = 0 where 𝐼𝑥 , 𝐼𝑦 & 𝐼𝑧 are the partial derivatives of image intensity in the 𝑥, 𝑦 & 𝑧 dimensions, derived from raw image data 𝐼, 𝑣𝑥 & 𝑣𝑦 are the velocity components (pixels per frame) of a moving feature in the image scene and 𝑛𝑥𝑦 is the flattened index of a pixel in a given 𝑁𝑥 × 𝑁𝑦 image frame ℱ, with 𝑁𝑥𝑦 elements, for 𝑛𝑥𝑦 ∈ ℱ [19]-[24]. If it is further assumed that surrounding pixels, within an 𝑀𝑥 × 𝑀𝑦 local window 𝒲, with 𝑀𝑥𝑦 elements and odd dimensions, centred on 𝐼(𝑛𝑥𝑦 ), move with a common velocity then


DRAFT −𝐼𝑧𝒲 (𝑚𝑥𝑦 ; 𝑛𝑥𝑦 ) = 𝐼𝑥𝒲 (𝑚𝑥𝑦 ; 𝑛𝑥𝑦 )𝑣𝑥 (𝑛𝑥𝑦 ) +

𝐼𝑦𝒲 (𝑚𝑥𝑦 ; 𝑛𝑥𝑦 )𝑣𝑦 (𝑛𝑥𝑦 ) (8)

for 𝑚𝑥𝑦 ∈ 𝒲, where 𝑚𝑥𝑦 is the flattened index of a pixel within the local window and 𝐼 𝒲 (𝑚𝑥𝑦 ; 𝑛𝑥𝑦 ) is the 𝑚𝑥𝑦 th pixel within the window centred on 𝑛𝑥𝑦 . Thus 𝓨 = 𝓧𝒗 where 𝒗 = [𝑣𝑥 (𝑛𝑥𝑦 ) 𝑣𝑦 (𝑛𝑥𝑦 )]

(9) T

𝓧 is a 𝑀𝑥𝑦 × 2 matrix with 𝐼𝑥𝒲 and 𝐼𝑦𝒲 as columns and 𝓨 is a 𝑀𝑥𝑦 × 1 vector, equal to −𝐼𝑧𝒲 . The velocity components may therefore be estimated, in a least-squares sense, using (𝓧T 𝓧)−𝟏 𝓧T 𝓨 = 𝒗 ̂ or 𝓐−1 𝓑 = 𝒗 ̂ where ∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑥𝒲 𝐼𝑥𝒲 𝓐 = 𝓧T 𝓧 = [ ∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑥𝒲 𝐼𝑦𝒲 𝓑 = 𝓧T 𝓨 = [

∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑥𝒲 𝐼𝑧𝒲 ∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑦𝒲 𝐼𝑧𝒲

]

∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑦𝒲 𝐼𝑥𝒲 ∑𝑚𝑥𝑦 ∈𝒲 𝐼𝑦𝒲 𝐼𝑦𝒲

(10a) ]

(10b)

(10c)

̂ is the least-squares estimate of 𝒗. (with 𝑚𝑥𝑦 & 𝑛𝑥𝑦 indices omitted above for brevity) and 𝒗 IV. SIMULATION AND RESULTS The overview of the LK algorithm provided above, highlights the critical role of the multidimensional digital filters for partial derivative computation. The accuracy of the velocity field generated by the LK algorithm using five different filtering schemes was therefore assessed using four different simulation scenarios using MATLAB ®. Filter sets A-E, involving a mix of separable IIR and FIR components, and scenarios 1-4, involving either a foreground point target or a background texture field, are defined in the subsections below. Forty random instantiations of each scenario were generated with constant velocities ranging from [0.1,0.1]T to [4.0,4.0]T in increments of [0.1,0.1]T (pixels per frame). For all filter and scenario combinations, the root-mean-squared (RMS) error of the estimated velocity components was computed (see Fig. 1). In all cases, a quality threshold was applied so that only estimates produced using 𝓐 matrices with a determinant greater than 1.0E-12 were considered [23]; furthermore, temporal and spatial exclusion zones were applied, at the start of the run and around the edge of each frame, to avoid spurious outputs due to filter initialization transients. Temporal filter latencies were taken into consideration when computing errors for scenarios involving point-targets.


DRAFT

Fig. 1. RMS error for various filter sets and simulation scenario combinations, as a function of the magnitude of the velocity components. Full speed range shown. Inserts show low-speed regions (0.1-1.0 pix/frm). Scenarios 1 to 4 from top to bottom; Filter sets: A (red), B (green), C (blue), D (cyan), E (magenta).

1. Moving point-target The point target was generated using a Gaussian point-spread function with a standard deviation of 0.5 pixels. The starting position of the point-target was offset from the nearest pixel using a pseudo-randomly generated displacement, drawn from a uniform distribution over a [0,1] pixel interval. 2. Moving point-target with Gaussian noise As above, but pseudo-random zero-mean Gaussian noise was added to the scene. The point target had a maximum intensity of 1.0; whereas the std. dev. of the noise was 0.1. 3. Translating low-frequency texture The synthetic texture was generated using 50 pseudo-randomly generated sinusoidal components (complex plane-waves). All components had unity magnitude, an angular frequency (radians per pixel) in the 𝑥 & 𝑦 dimensions and phase (radians) drawn from a uniform distribution on the 𝜋 [−1, +1]⁄4 and 2𝜋[0,1] intervals, respectively. Only the real part of the texture was used. 4. Translating noise texture As above, but angular frequencies were drawn from the 𝜋[−1, +1] interval and 800 components were used, to keep the spectral component density the same as the low-frequency scenario. A. Simoncelli (spatial FIR and temporal FIR) Filters that are separable in all (𝑥, 𝑦 & 𝑧) dimensions are essential for real-time performance because fewer multiply and add operations are required. As a consequence of separability, the partial derivative in a given dimension is computed by applying a derivative filter in that dimension and a smoothing filter in the other dimensions (see Fig. 5 in [19]). Simoncelli further asserted that the frequency response of the smoothing filters 𝑃(𝜔) and derivative filters 𝐷(𝜔) in a given dimension should be related via 𝐷(𝜔) − 𝑖𝜔𝑃(𝜔) = 0 [26]. This is a rather restrictive requirement and the required calculations are non- trivial. As exact analytical solutions


DRAFT are difficult (if not impossible) the problem is solved on a discrete grid, using an arbitrary weighting function to emphasize low frequencies. Filter coefficients, for 𝑃(𝜔) & 𝐷(𝜔) filters of various orders, are presented in [26]. Elad et al., extend this idea using a continuous weighting function and additional constraints in [19], which removes the requirement for an arbitrary frequency grid; however, it requires the evaluation of some non-trivial definite integrals. Simoncelli’s 4th-order filters were used here and the results suggest that extra design effort does result in greater accuracy, relative to the other filters, but only for slow-moving features. B. Barron (spatial FIR and temporal FIR) Barron and Simoncelli both use separable filters; however, the notion of separability is taken a step further by Barron et al. [20], for a reduction in the number of required convolutions (see Fig. 2 in [19]). They observed that the partial derivative in a given dimension, may be approximated by applying a derivative filter in that dimension, to data that has been pre-smoothed in all dimensions. Barron uses a 4th-order derivative filter with coefficients [−1,8,0, −8,1]⁄12 and a 10th-order Gaussian pre-smoothing filter with a variance (𝜎 𝟐 ) of 1.5 pixels. Gaussian smoothers are very popular in image-processing circles; however, they are only optimal with respect to their conceptual simplicity and general efficacy: as 𝜎 increases, the bandwidth of the smoother decreases and the (finite) impulse response duration increases. This relationship allows the filter to be readily tuned to achieve the desired balance between the various frequency-response, transient response and computational efficiency requirements. Gaussian smoothers satisfy 𝑙 = 0 & 𝑙 = 1 flatness constraints at dc. Gaussian derivative filters are also popular [27]; however, they only satisfy the 𝑙 = 0 flatness constraint at dc. Neither filter has a Nyquist null. The results suggest that Barron’s simplification does not severely degrade performance, relative to Simoncelli’s filters; indeed, accuracy is improved for moderately fast motion in scenarios 1 and 4. C. Fleet, Langley and Diriche (spatial IIR and temporal IIR) The FIR filters of Simoncelli and Barron differ in the way that the filter coefficients are derived and in the way that the filters are applied [19]. The current consensus is that FIR filters are ideal for real-time LK optical flow calculations and the filters of Barron [25] or Simoncelli [23],[24] are generally preferred; however, recursive IIR filters were considered in the early days of LK research. As a reference IIR implementation, Fleet & Langley’s causal 3rd-order smoothers and 2nd-order differentiators were used in the temporal dimension with 𝜏 = 1 [28]. Using the achieved group delay at the dc limit of 𝑞 = 6, the smoother has 𝐿dc = 1 and 𝐿pi = 2; the corresponding values for the differentiator are 𝑞 = 4, 𝐿dc = 2 and 𝐿pi = 0. Diriche’s 2nd-order non-causal smoothers and differentiators with 𝛼 = 1⁄2 were used in the spatial dimensions [29]. These IIR filters outperform all other filters by a significant margin in scenario 1 but underperform all other filters by a significant margin in scenario 3. D. Complex flatness constraints (spatial FIR & temporal IIR) Like Simoncelli’s approach to the design of coordinated multidimensional filters for partial derivative computation, it is assumed here that smoothed and derivative estimates should be derived using a common representation, or interpolating model, of the underlying signal in a given dimension; however, a less restrictive interpretation of this requirement is used here – It is assumed that a low-degree polynomial model is sufficient (as specified by 𝐿dc ) and that the estimated polynomial parameters should be the same for both filter types. This condition is readily satisfied using the same constraints and the same basis set to design the smoother and the differentiator. The particular filters used here were designed using the following basis-set parameters for a linear-phase odd FIR filter in the spatial dimensions: 𝑞 = 6, 𝐾zer = 2𝑞 + 1, 𝐾axs = 0 and 𝐾pln = 0. In the temporal dimension: 𝜔pln = 𝜋⁄8, 𝑟 = 𝑝 = 1⁄𝑒, 𝑞 = 2, 𝐾zer = 0, 𝐾axs = 4 and 𝐾pln = 1 for an IIR. As explained in the next section, it is assumed that a much wider filter bandwidth is required in the temporal dimension to handle fast motion (see Fig. 2). The response of this filter set results in improved performance relative to filter sets A & B at medium to high velocities, which is particularly evident in scenarios 1 & 4. E. Complex flatness constraints (spatial FIR & temporal FIR) These 5th-order linear-phase even FIR filters were designed using a delay-only basis set (𝑞 = 2.5, 𝐾zer = 6, 𝐾axs = 0 and 𝐾pln = 0) in the temporal dimension. In all other respects, these filters are the same as the filters above. As expected, filters E & F offer similar performance because they use the same FIR filters in the spatial dimensions (designed using 𝑁dc = 2, 𝑁pi = 11, 𝑁wb = 0) and they satisfy the same set of complex flatness constraints in the temporal dimension (𝑁dc = 2, 𝑁pi = 2, 𝑁wb = 1, 𝜔wb = 𝜋⁄8). Filter set E slightly outperforms filter set D at high velocities in scenario 1 and it outperforms filter sets A & B at medium-to-high velocities in scenarios 1, 2 & 4.


DRAFT

Fig. 2. Frequency respone ℋ𝑞 (𝜔) of filter set D (linear scale). Smoother (upper) and differentiator (lower). Spatial (solid line) temporal (dashed line), ideal differentiator slope (dotted black). Real part (blue) imaginary part (red), magnitude (green), 𝜔pln = 𝜔wb (black).

V. DISCUSSION A qualitative appreciation of the filter requirements may be derived from an understanding of the way in which velocity affects the 3-D (spatiotemporal) spectrum of a translating sinusoid in a video sequence. A spatial signal component with angular frequencies of 𝜔𝑥 and 𝜔𝑦 (radians per pixel) has only signal energy at dc, i.e. 𝜔𝑧 = 0, in the temporal dimension. As the component translates (due to a sensor pan, for example), the temporal frequency of that component shifts to 𝜔𝑧 = −𝜔𝑥 𝑣𝑥 − 𝜔𝑦 𝑣𝑦 (radians per frame). Frequencies that form a broadband spatial texture in the image, thus rotate around the spatiotemporal frequency origin, to form a tilted plane that is parameterized as 𝜔𝑥 𝑣𝑥 + 𝜔𝑦 𝑣𝑦 + 𝜔𝑧 = 0 [30]. Thus 𝛥𝑧 = 2𝑣max 𝛥𝑥𝑦 where 𝛥𝑥𝑦 is the spatial bandwidth of the expected feature spectrum (radians per pixel) and 𝑣max is the greatest expected speed (pixels per frame); i.e. the temporal-to-spatial bandwidth ratio is 2𝑣max : 1. Thus for fast-moving features, with large inter-frame displacements and a wide spatial bandwidth, we require filters with: a very short impulse response in the temporal dimension (long impulse responses may however, be beneficial for diffuse slow-moving blobs when the signal-to-noise ratio is low) and a very long impulse response in the spatial dimensions. Consequently, for low-frame-rate sensors, we have a situation where low-level IIR/FIR filtering operations are only employed in spatial dimensions, with some other form of high-level processing logic (or a multi-scale image pyramid) used to utilize time-dependent information [22],[23],[31]. For high-frame rate sensors however, with small inter-frame displacements, the temporal bandwidth (radians per frame) and the spatial extent (pixels) are much smaller; therefore, low-order filters in both spatial and temporal dimensions, are feasible [24]. To illustrate the points discussed above in a practical context, consider the wide-area-surveillance activity-based intelligence (WASABI) system, on board the Defence experimentation airborne platform (DEAP). It incorporates algorithm/software components form the analysts’ decision support system (ADSS) [31], and uses two complementary imaging sensors: Anglefire and MX-20HD [32]. Angelfire is a six-camera electro-optic sensor array with a wide spatial coverage, a high spatial resolution and a low temporal resolution (4872 x 3248 pixels per camera @ 3 Hz). It offers high-performance video moving-target indication (VMTI) functionality, with a feature tracker, a constant false-alarm-rate (CFAR) background model and a target tracker. Spatial (FIR Gaussian) filters are used by the feature tracker to estimate the inter-frame transform parameters of the ground plane, but temporal filtering is not used. This processing “pipeline” runs in real-time with the assistance of a single NVIDIA® Tesla® K80 card per camera; each card hosts two graphics processing units (GPUs). The data-feed (1080p @ 25 Hz) from the MX-20HD sensor is processed using the same algorithm (with different configuration parameters); however, it does not currently run in real-time for the maximum data rate. Considering the relatively high frame-rate of this sensor, it is possible that it may benefit from a modified VMTI processing pipeline; for instance, one that utilizes both spatial and temporal filters. Both IIR & FIR filters should be considered for this role. IIR filters utilize feedback, which allows the internal state-variables of the filter to represent various parameters of the input signal, using all prior inputs. Unlike FIR filters, the order of IIR filters (i.e. their computational complexity) is therefore independent of the impulse response duration; thus it is possible to cheaply realize digital filters with a high degree of frequency selectivity. However, frequency selectivity is not necessarily a desirable characteristic in video processing applications, because sinusoidal expansions (such as those used in simulation scenarios 3 & 4) are rarely good representations of local image features due to the absence of obvious periodicities in most real-world scenes. As a consequence, abrupt/sharp transition bands, as achieved using equal-ripple designs, are undesirable because they produce ringing artifacts in the impulse response [11]. Furthermore, a long impulse response is


DRAFT rarely a desirable characteristic, particularly so in the spatial dimensions, because stationary signals are an exception due to object edges. Thus it may be counterproductive to obsess over the fine details of a filter’s frequency response (i.e. its steady-state response to a sinusoidal input of infinite extent). These issues suggest that frequency-domain filter design for electro-optic (EO) or infrared (IR) video data requires a somewhat different approach, to what is commonly used by the DSP community for the processing of digitized radio-frequency or acoustic signals [2]. A broad brush-stroke design procedure such as the one described in this paper, with smooth transitions between pass/stop bands, is arguably more useful in these EO/IR systems. The proposed procedure is intended to be used in niches that are currently occupied by simple Gaussian or derivative-of-Gaussian filters, which cannot be readily configured to satisfy multiple frequencydomain specifications or realized in IIR form. In the spatial dimensions the most appropriate filtering structure (IIR or FIR) is to a large extent determined by the architecture of the processor. Devices with a relatively high number of parallel processing channels and a relatively low clock-speed – e.g. GPUs – are best utilized by scheduling a large number of threads (one thread per pixel) with each thread involving a short loop (over adjacent pixels) using an FIR filter. Devices with lower channel counts and higher clock speeds – e.g. central processing units (CPUs) – are best utilized by scheduling a small number of threads (one per row/column) with each thread involving a long loop (over the columns/rows), using a non-causal (forward-backward/up-down) IIR filter. Performance in some devices – e.g. field-programmable gate arrays (FPGAs) – may be limited by memory bandwidth when long FIR smoothing filters are required in the temporal dimension [21]; and in these cases, IIR filters may be worth considering. VI. CONCLUSION The proposed procedure may be used to design filters for different functions, e.g. smoothing or differentiating, with either IIR or FIR structures (of odd or even order) and in this respect it is unique. This flexibility is achieved: firstly, by augmenting the zero-pole (i.e. delay-only) basis sets that are commonly used to design FIR filters, with simple non-zero-pole (i.e. recursive) basis functions to accommodate IIR structures; and secondly, through the use of complex flatness constraints, for the specification of filter function, that incorporate the desired group delay, to simplify the treatment of non-linear-phase filters. The way in which the optical flow-field is used to support VMTI functionality in a system with a high frame-rate EO/IR sensor will be reported in a future publication. ACKNOWLEDGEMENTS I would like to thank Michael Royce, Merrilyn Fiebig and Nick Redding for giving me the opportunity to work on their systems. This work was funded by DST Group. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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