Two-dimensional window design

Windowing is a process where an index limited sequence has its maximum energy concentrated in a finite frequency interval. This can be extended to an N-dimension where the N-D window has the limited support and maximum concentration of energy in a separable or non-separable N-D passband. The design of an N-dimensional window particularly a 2-D window finds applications in various fields such as spectral estimation of multidimensional signals, design of circularly symmetric and quadrantally symmetric non-recursive 2D filters, design of optimal convolution functions, image enhancement so as to reduce the effects of data-dependent processing artifacts, optical apodization and antenna array design.

Two dimensional window

Consider a two-dimensional window function (or window array) $w(n_{1},n_{2})$ with its Fourier transform denoted by $W(w_{1},w_{2})$. Let $i(n_{1},n_{2})$ and $I(w_{1},w_{2})$ denote the impulse and frequency response of an ideal filter and $h(n_{1},n_{2})$ and $H(w_{1},w_{2})$ denote the impulse and frequency response of a filter approximating the ideal filter, then we can approximate $I(w_{1},w_{2})$ by $h(n_{1},n_{2})$. Since $i(n_{1},n_{2})$ has an infinite extent it can be approximated as a finite impulse response by multiplying with a window function as shown below

$h(n_{1},n_{2})=i(n_{1},n_{2})w(n_{1},n_{2})$
and in the Fourier domain

$H(w_{1},w_{2})={\frac {1}{(2\pi )^{2}}}[I(w_{1},w_{2})**W(w_{1},w_{2})]$

The problem is to choose a window function with an appropriate shape such that $H(w_{1},w_{2})$ is close to $I(w_{1},w_{2})$ and in any region surrounding a discontinuity of $I(w_{1},w_{2})$, $H(w_{1},w_{2})$ shouldn't contain excessive ripples due to the windowing.

2-D Window function from 1-D function

There are four approaches for generating 2-D windows using a one-dimensional window as a prototype. .

Approach I

One of the methods of extending the 1-D window design to a 2-D design is by using circular symmetry, i.e., by rotating the 1-D windows. A function is said to possess circular symmetry if it can be written as a function of its radius, independent of $\theta$ i.e. $f(r,\theta )=f(r).$
If w(n) denotes a good 1-D even symmetric window then the corresponding 2-D window function is
$w(n_{1},n_{2})=w({\sqrt {n_{1}^{2}+n_{2}^{2}}})$ for $|{\sqrt {n_{1}^{2}+n_{2}^{2}}}|\leqslant a$ (where $a$ is a constant)
and
$w(n_{1},n_{2})=0$ for $|{\sqrt {n_{1}^{2}+n_{2}^{2}}}|>a$

The transformation of the Fourier transform of the window function in rectangular co-ordinates to polar co-ordinates results in a Fourier-Bessel transform expression which is called as Hankel transform. Hence the Hankel transform is used to compute the Fourier transform of the 2-D window functions.

If this approach is used to find the 2-D window from the 1-D window function then their Fourier transforms have the relation

${\frac {1}{2\pi }}H(w_{1},w_{2})**W(w_{1},w_{2})=H(w)*W(w)$

where:

$H(w)=\left\{{\begin{matrix}1,&w\geq 0\\0,&w<0\\\end{matrix}}\right.$ is a 1-D step function

and

$H(w_{1},w_{2})=\left\{{\begin{matrix}1,&w_{1}\geq 0\ and\ all\ w_{2}\\0,&w_{1}<0\ and\ all\ w_{2}\\\end{matrix}}\right.$ is a 2-D step function.
In order to calculate the percentage of mainlobe constituted by the sidelobe, the volume under the sidelobes is calculated unlike in 1-D where the area under the sidelobes is used.
In order to understand this approach, consider 1-D Kaiser window whose window function is given by

$w[n]=\left\{{\begin{matrix}{\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N-1}}-1\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},&0\leq n\leq N-1\\\\0&{\mbox{otherwise}}\\\end{matrix}}\right.$

then the corresponding 2-D function can be derived as

$w(n_{1},n_{2})=\left\{{\begin{matrix}{\frac {I_{0}\left(\alpha {\sqrt {1-{\frac {\sqrt {n_{1}^{2}+n_{2}^{2}}}{a^{2}}}}}\right)}{I_{0}(\alpha )}},&|r|\leqslant a\\0&{\mbox{otherwise}}\\\end{matrix}}\right.$

where:

• $r={\sqrt {n_{1}^{2}+n_{2}^{2}}}$
• N is the length of the 1-D sequence,
• I₀ is the zeroth-order modified Bessel function of the first kind,
• α is an arbitrary, non-negative real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design.

This is the most widely used approach to design the 2-D windows. But the disadvantage of this approach is that when the 1-D window is not defined for values of ${\sqrt {n_{1}^{2}+n_{2}^{2}}}$ then interpolation is required. Also, the frequency characteristics of the 1-D window are not preserved in 2-D cases by this rotation method.

Approach II

Another way of deriving the 2-D window is from the outer product of two 1-D windows, i.e., $w(n_{1},n_{2})=w_{1}(n_{1})w_{2}(n_{2}).$ The property of separability is exploited in this approach. The window formed has a square region of support and is separable in the two variables. In order to understand this approach, consider 1-D Kaiser window then the corresponding 2-D function is given by

$w(n_{1},n_{2})=\left\{{\begin{matrix}{\frac {I_{0}\left(\alpha {\sqrt {1-{\frac {n_{1}}{a^{2}}}}}\right)I_{0}\left(\alpha {\sqrt {1-{\frac {n_{2}}{a^{2}}}}}\right)}{I_{0}^{2}(\alpha )}},&|n_{1}|\leqslant a,|n_{2}|\leqslant a\\0&{\mbox{otherwise}}\\\end{matrix}}\right.$

The Fourier transform of $w(n_{1},n_{2})$ is the outer product of the Fourier transforms of $w_{1}(n_{1})\ and\ w_{2}(n_{2})$. Hence $W(w_{1},w_{2})=W_{1}(w_{1})W_{2}(w_{2})$.

2-D Window Functions

Using the above approaches, the 2-D window functions for few of the 1-D windows are as shown below. When Hankel transform is used to find the frequency response of the window function, it is difficult to represent it in a closed form. Except for rectangular window and Bartlett window, the other window functions are represented in their original integral form. The two dimensional window function is represented as $w(r)$ with a region of support given by $|r|<a$ where the window is set to unity at origin and $w(r)=0$ for $|r|>a.$ Using the Hankel transform, the frequency response of the window function is given by

$W(f)=\int _{0}^{\infty }\!rw(r)J_{0}(fr)\,dr$.

where $J_{0}$ is Bessel function identity.

Rectangular Window

Figure1: 2-D circularly symmetric window surface plot

Figure2: 2-D circularly symmetric window contour plot

The two dimensional version of a circularly symmetric rectangular window is as given below

$w(r)=\left\{{\begin{array}{ll}1,&|r|\leqslant a\\0,&|r|>a\\\end{array}}\right.$

The window is cylindrical with the height equal to one and the base equal to 2a. The vertical cross-section of this window is a 1-D rectangular window.
The frequency response of the window after substituting the window function as defined above, using the Hankel transform, is as shown below
$W(f)=\int _{0}^{\infty }\!rJ_{0}(fr)\,dr$

Bartlett Window

The two dimensional mathematical representation of a Bartlett window is as shown below

$w(r)=\left\{{\begin{array}{ll}1-{\frac {|r|}{a}},&|r|\leqslant a\\0,&|r|>a\\\end{array}}\right.$

The window is cone-shaped with its height equal to 1 and the base is a circle with its radius 2a. The vertical cross-section of this window is a 1-D triangle window.
The Fourier transform of the window using the Hankel transform is as shown below
$W(f)=\int _{0}^{\infty }\!r(1-{\frac {|r|}{a}})J_{0}(fr)\,dr$

Kaiser Window

The 2-D Kaiser window is represented by

$w(r)=\left\{{\begin{array}{ll}{\frac {I_{0}\left(\alpha {\sqrt {1-(({\frac {r}{a}})^{2}}}\right)}{I_{0}(\alpha )}},&|r|\leqslant a\\0,&{\mbox{otherwise}}\\\end{array}}\right.$

The cross-section of the 2-D window gives the response of a 1-D Kaiser Window function.
The Fourier transform of the window using the Hankel transform is as shown below
$W(f)=\int _{0}^{\infty }\!r({\frac {I_{0}\left(\alpha {\sqrt {1-(({\frac {r}{a}})^{2}}}\right)}{I_{0}(\alpha )}})J_{0}(fr)\,dr$

Tseng window

A 1-D Tseng window can be extended to a two dimensional window by following the concept of rotation as done before. This window finds applications in antenna array design for the detection of AM signals. The reason for mentioning this window here is to emphasize that while extending the window to 2-D there might arise a situation where the corresponding 1-D window will not have a value.

The relation $W(K_{1},K_{2})=W({\sqrt {K_{1}^{2}+K_{2}^{2}}})$ is used for $K_{1}=-N,-N+1,...,N-1$ and $K_{2}=-N,-N+1,...,N-1.$

$K_{1}$ and $K_{2}$ are normalized independent variables with $K_{1}=f_{1}T$ and $K_{2}=f_{2}T$ where $T$ is the sampling time interval.

While designing a 2-D window there are two features that have to be considered for the rotation. Firstly, the 1-D window is only defined for integer values of $K$ but ${\sqrt {K_{1}^{2}+K_{2}^{2}}}$ value isn't an integer in general. To overcome this, the method of interpolation can be used to define values for $W(K_{1},K_{2})$ for any arbitrary $W({\sqrt {K_{1}^{2}+K_{2}^{2}}}).$ Secondly, the 2-D FFT must be applicable to the 2-D Tseng window. Thus in order to facilitate the above two features, the window function is given by $W(K_{1}+K_{2})=W({\sqrt {K_{1}^{2}+K_{2}^{2}}})+W(I+\alpha )$ ≃ $W(I+\alpha _{0})$ where $I$ is the largest integer which does not exceed ${\sqrt {K_{1}^{2}+K_{2}^{2}}},$ $0$≥$\alpha$>$1$ and $\alpha _{0}$ is the closest value to $\alpha .$