Methods of computing impulse-response weights of 1- and 2-D matched filters and LMS filters for suppressing clutter in an electrooptic sensor’s output are developed and illustrated with examples. The methods are applicable to signals from scanning or staring sensors viewing point or extended sources against variable backgrounds, provided signal shape and orientation are known. The matched-filter design technique is based on isotropic power-spectral clutter models whose parameters also must be known. Images of sensor output are assumed to provide the requisite information about signals and backgrounds. The LMS design technique is based on deterministic polynomial clutter models. An LMS filter estimates signal amplitude explicitly and local clutter parameters implicitly by performing a least-squares fit of a signal-plus-clutter model to the sensor output at every point of the scene. Thus clutter parameters need not be known for LMS design, although qualitative knowledge of the background may facilitate choice of the clutter model.

Larry B. Stotts and Lawrence E. Hoff Appl. Opt. 53(22) 5042-5052 (2014)

References

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The meaning and use of these arrays are discussed in reference 1. They are not vectors or matrices except where the symbol t for “transpose” occurs. In that case they are regarded as row vectors or as matrices for evaluating the expression involved. The arrays operate by convolution, indicated by an asterisk *. The first step of convolution is array reversal. Convolving an operator array with an array of samples estimates a derivative at the sample points. Operator arrays must always fully overlap sample arrays. This table gives operator arrays for Δy = Δx = 1 sample interval. 2-D arrays for unequal x and y sampling intervals must be constructed as described in reference 1.
The first array is from forward and backward difference approximations of the derivative, the second array from a central difference approximation. The two approximations are the same for even order derivatives. Central arrays estimate derivatives at central weights, forward and backward arrays at first and last weights, respectively. Since the 2-D arrays involve only even operators, they are effectively derived from central difference formulae and thus estimate derivatives at central weights.

a.b.c.d may have any values except those which make the derivative components vanish. The central symmetry of the signals ensures that Σ s_{j}y_{j}^{m}=Σ y_{j}^{m}=0 for odd m. These are the conditions for Eq. (17) to reduce to Eqs. (19,20).
(D’s) = (d^{n}s dy^{n}) at sample point i or j. Reference 1 describes estimation of the derivatives. If the derivative components vanish, the coefficient matrix of Eq. (19) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.
The remaining weights are centrally symmetric with the preceding ones.

TABLE 3:

2-D, First-order, LMS Filters for Certain Symmetric Signals

a,b,c,d may have any values except those which make the derivative components vanish. The symmetry of the signals is such that Σs_{j}x_{j}=Σs_{j}y_{j}=Σx_{j}=Σy_{j}=Σx_{j}y_{j}=0. These are the conditions for Eq. (34b) to be a solution of Eq. (30).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
In these examples the relative weights are given also by (Ks_{j} − Σs_{j}). The conditions for this are (1) a linear clutter function (B^{00}+B^{10}x_{j}+B^{01}y_{j}) and (2) the signal symmetry specified above.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.

TABLE 4:

2-D, Third-order, LMS Filters for Certain Symmetric Signals

a,b,c,d may have any values except those which make the derivative components vanish. The symmetry of the signals is such that Σ s_{j}x_{j}^{p}y_{j}^{r}=Σ x_{j}^{p}y_{j}^{r}=0 with p, or r, or both odd. These are the conditions for Eq. (30) to reduce to Eqs. (32a,b,c,d).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.

TABLE 5:

Alternative Third-order LMS Filters for Two Signals of Table 4

a,b,c may have any values except those which make the derivative components vanish. The signal symmetry is such that Σ s_{j}x_{j}^{p}y_{j}^{r}=Σ x_{j}^{p}y_{j}^{r}=0 with p, or r, or both odd. These are the conditions for Eq. (30) to reduce to Eqs. (32a,b,c,d).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives evaluated as near as possible to the signal center, and that the filter vanishes with the derivative components. In practice the zero weights are not included in the impulse response because they do not affect the filter output. They are included here to show that the calculations actually give zero weights at the indicated points.

Tables (5)

TABLE 1

– Derivative Operators and Array Approximations^{a}

Operator

Array Symbol and Approximation

(d/dy)

[w^{1}] = [1 − 1] or
$\frac{1}{2}{\left[1\phantom{\rule{0.2em}{0ex}}0-1\right]}^{}$^{b}

(d^{2}/dy^{2})

[w^{2}] = [1 −2 1]

(d^{3}/dy^{3})

[w^{3}] =[1 −3 3 −1] or
$\frac{1}{2}{\left[1-2\phantom{\rule{0.2em}{0ex}}0\phantom{\rule{0.2em}{0ex}}2-1\right]}^{}$^{b}

(d^{4}/dy^{4})

[w^{4}]= [1 −4 6 −4 1]

(d^{5}/dy^{5})

[w^{5}] = [1 −5 10 −10 5 −1] or
$\frac{1}{2}{\left[1-4\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}0-5\phantom{\rule{0.2em}{0ex}}4-1\right]}^{}$^{b}

The meaning and use of these arrays are discussed in reference 1. They are not vectors or matrices except where the symbol t for “transpose” occurs. In that case they are regarded as row vectors or as matrices for evaluating the expression involved. The arrays operate by convolution, indicated by an asterisk *. The first step of convolution is array reversal. Convolving an operator array with an array of samples estimates a derivative at the sample points. Operator arrays must always fully overlap sample arrays. This table gives operator arrays for Δy = Δx = 1 sample interval. 2-D arrays for unequal x and y sampling intervals must be constructed as described in reference 1.
The first array is from forward and backward difference approximations of the derivative, the second array from a central difference approximation. The two approximations are the same for even order derivatives. Central arrays estimate derivatives at central weights, forward and backward arrays at first and last weights, respectively. Since the 2-D arrays involve only even operators, they are effectively derived from central difference formulae and thus estimate derivatives at central weights.

a.b.c.d may have any values except those which make the derivative components vanish. The central symmetry of the signals ensures that Σ s_{j}y_{j}^{m}=Σ y_{j}^{m}=0 for odd m. These are the conditions for Eq. (17) to reduce to Eqs. (19,20).
(D’s) = (d^{n}s dy^{n}) at sample point i or j. Reference 1 describes estimation of the derivatives. If the derivative components vanish, the coefficient matrix of Eq. (19) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.
The remaining weights are centrally symmetric with the preceding ones.

TABLE 3:

2-D, First-order, LMS Filters for Certain Symmetric Signals

a,b,c,d may have any values except those which make the derivative components vanish. The symmetry of the signals is such that Σs_{j}x_{j}=Σs_{j}y_{j}=Σx_{j}=Σy_{j}=Σx_{j}y_{j}=0. These are the conditions for Eq. (34b) to be a solution of Eq. (30).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
In these examples the relative weights are given also by (Ks_{j} − Σs_{j}). The conditions for this are (1) a linear clutter function (B^{00}+B^{10}x_{j}+B^{01}y_{j}) and (2) the signal symmetry specified above.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.

TABLE 4:

2-D, Third-order, LMS Filters for Certain Symmetric Signals

a,b,c,d may have any values except those which make the derivative components vanish. The symmetry of the signals is such that Σ s_{j}x_{j}^{p}y_{j}^{r}=Σ x_{j}^{p}y_{j}^{r}=0 with p, or r, or both odd. These are the conditions for Eq. (30) to reduce to Eqs. (32a,b,c,d).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives estimated as near as possible to the signal center, and that the filter vanishes with the derivative components.

TABLE 5:

Alternative Third-order LMS Filters for Two Signals of Table 4

a,b,c may have any values except those which make the derivative components vanish. The signal symmetry is such that Σ s_{j}x_{j}^{p}y_{j}^{r}=Σ x_{j}^{p}y_{j}^{r}=0 with p, or r, or both odd. These are the conditions for Eq. (30) to reduce to Eqs. (32a,b,c,d).
(D_{x}^{p}D_{y}^{r}s)_{i,j}=∂^{(p+r)}s/∂x^{p}∂y^{r} at sample point i or j. The derivatives are estimated with Δx = Δy = 1 sample interval, as described in reference 1. If the derivative components vanish, the coefficient matrix of Eq. (32a) is singular, and the equation cannot be solved for a signal-amplitude filter.
The relative filter weights are in braces. Ordinarily the derivative factor is an unseen part of the filter normalization (gain) constant. The derivative factor is shown here to emphasize that the weights are built up from the model-signal’s derivatives evaluated as near as possible to the signal center, and that the filter vanishes with the derivative components. In practice the zero weights are not included in the impulse response because they do not affect the filter output. They are included here to show that the calculations actually give zero weights at the indicated points.