In the second application, the distance from a textured surface is computed by comparing the resolution of a captured image to that of a reference image. A typical example of such problems is the estimation of the flight altitude of a plane from landscape images taken during flight. Resolution conversion in this case involves two 2 × 2 integer, non-singular decimalation and expansion matrices.
2. THIRD ORDER STATISTICS RELATIONS
Let $y(n)$ denote a rate-converted, discrete variable, $D$-dimensional process obtained from the original process $x(n)$ after fractional rate conversion:
where the rate conversion is obtained through a "downsample by matrix $L_{D \times D}$ filter by $h(n)$ - expand by matrix $M_{D \times D}$" operation and $L_{D \times D}, M_{D \times D}$ are $D \times D$ integer, non-singular, commutative and coprime matrices, [2]. Of practical interest are the cases with $det(M_{D \times D}) > det(L_{D \times D})$, meaning that $y(n)$ is observed at a resolution lower than the original (reference) resolution of $x(n)$.
Equation (1) covers the general case where resampling is possibly accompanied by rotation and azimuth changes (see [9], chap. 12). In [5] input-output cumulant expressions are established for the general case of equation (1) with full integer matrices $L_{D \times D}, M_{D \times D}$. For the purposes of the present work, we restrict ourselves to the special case of diagonal matrices $L_{D \times D} = L_{D \times D}, M_{D \times D} = M L_{D \times D}$, where $L, M$ are coprime integers. This corresponds to rescaling by the scalar factor $M/L$ alone, thus preserving the view point from which the signal is observed. This choice is made in order to simplify the mathematical notations, since here the focus is on the use of these relations for the estimation of the resampling ratio in the aforementioned applications. It is straightforward to rewrite the obtained relations, however, for the general case of general diagonal or full matrices $L_{D \times D}, M_{D \times D}$.
Equation (1) is equivalent to a resampling of the continuous signal $x_c(t)$ from which $x(n)$ was originally obtained through sampling, provided that the filter $h(n)$ employed has an ideal lowpass transfer function $H(\omega)$ with gain $L$ and cutoff frequencies $\pi/M$ over all dimensions $d = 1, ..., D$.
The third order cyclic cumulant of $y(n)$, defined as $[5]$
where $c_{3,y}(m_1, m_2; n)$ is periodic in $(n)$ with period $[0..L-1]^D$, is related to $c_{3,x}(m_1, m_2)$ through the following equation:
where $h_3(m_1, m_2) \triangleq \sum_n h(n)h(n+m_1)h(n+m_2)$ is the triple correlation of the decimation $D$-dimensional filter $h(n)$. See [3], [6] for definitions and properties of cyclic moments and cumulants.
Equation (3) can be transformed to the frequency domain, to yield relations between the cyclic bispectrum of the low resolution signals and the bispectrum of the reference resolution signals, by exploiting the coprimeness of $L, M$:
The RHS of equation (4) is a summation of frequency shifted replicas of $\tilde{C}_{3,x}(\omega_1, \omega_2)$, each replica shrunk by the scaling factor $M/L$. It is interesting to notice the resemblance that equation (4) bears to the corresponding input-output relation between the Fourier transforms of deterministic signals.
3. ESTIMATION OF THE RESOLUTION CONVERSION RATIO
The input-output relations given in the previous section allow for the computation of the resolution conversion ratio, $L/M$, provided that both $x(n)$ and $y(n)$ are available and the blurring mechanism $h(n)$ is known. The method proposed in the sequel for the computation of $L/M$ relies on matching the cumulants of the measured signal $y(n)$ to successive resolution-converted versions of the cumulants of the original signal $x(n)$. Although computationally demanding, this method is shown to converge to the true resolution conversion ratio.