Handling Correlated Images

Introduction

On many occasions SAR images are sampled at spacing less than their resolution. Indeed, Nyquist sampling of the detected intensity image would require spacing of half the resolution in each direction to retain all the information in the image. Figure 6.7 illustrates part of a typical SAR image obtained using the DLR ESAR system. In this mode of operation the range resolution is 4 m, the azimuth resolution is 1.8 m, the pixel spacing is 1.25 m in range and 0.68 m in azimuth. The effective number of looks (ENL) is specified to be 4.0. Based on this information we note that the image is over sampled with 3.2 samples per resolution length in range and 2.6 in azimuth. The effect of this oversampling is to make the speckle, which reflects the coherent interference between random scatterers within each resolution cell, subtend approximately 3x3 pixels.

Figure 6.7: Typical HH SAR image obtained from the DLR ESAR system.

However, as noted previously the segmentation process implicitly assumes that neighbouring pixels are not correlated. Speckle correlation will therefore impact on the quality of the segmentation reconstruction. In this case study we demonstrate the effect of such correlations on segmentation and despeckling, and show how the problem can be overcome.

Correlation and Segmentation

Previous studies have determined suitable sets of parameters for intensity segmentation. In Figure 6.5.2(a) we show a detail from Figure 6.7, together with its segmentation, using default parameters, in Figure 6.5.2(b). It is immediately apparent that the segmented image has far too many segments, which appear to follow the speckle fluctuations. This is indeed precisely the effect that correlated speckle has on segmentation. A region of high speckle intensity, subtending several pixels, is reconstructed as a single segment. The process can be understood in terms of a simple multiplication noise speckle model.

Figure 6.5.2(a) Figure 6.5.2(b)

Figure 6.5.2(c) Figure 6.5.2(d)
6.5.2(a) A detail from the original image shown in Figure 6.7 (b) The segmentation (c) The residual speckle (d) The expected residual speckle

Suppose that the observed intensity is represented as a speckle noise model such that

$$I = \sigma \times n$$

where $\sigma$ is the underlying cross-section and $n$ the multiplicative noise, which is gamma-distributed with order $L$, where $L$ is the ENL. Let us suppose that the segmentation succeeds in representing the underlying cross-section exactly. Under these conditions the variance of the intensity normalised by the cross-section is given by

$$\frac{var(I/\sigma)}{<I/\sigma>{2}} = \frac{var(n)}{<n>^{2}} = \frac{1}{L}$$

Thus, if we take the ratio of the original intensity to the derived segmentation, the result should have the properties of random speckle with the expected correlation properties and relative variance. In Figure 6.5.2(c) we show the residual speckle, derived from Figures 6.5.2(a) and 6.5.2(b). In Figure 6.5.2(d) we show a simulated speckle image having approximately the correct correlation properties. The correlation in 6.5.2(d) is assumed to correspond to a square averaging window; no attempt has been made to reproduce the correct correlation function. It is clear that 6.5.2(c) fails to show the expected speckle correlation properties, a consequence of the segmentation tending to reconstruct the individual speckle blobs. This is further borne out by the relative variance of Figure 6.5.2(c), which is 14.7, compared with the ENL of 4.0.

Speckle correlation properties

The SAR image in Figure 6.7 was of a detected (amplitude) image. In some cases the single-look-complex (SLC) image is available. In this case we can perform a detailed study of the instrument correlation properties, as distinct from any cross-section fluctuations. Consider the autocorrelation function (ACF) of the received complex field at lag $k$, which is given by

$$g_{E}(k) = \frac{<E_{0}E_{k}^{*}>}{\sqrt{<E_{0}E_{0}^*><E_{k}E_{k}^{*}>}} = \frac{<E_{0}E_{k}^{*}>}{<I>}$$

provided the field, $E$, is a stationary ergodic random process. Thus constructing the ACF of the complex field yields a correlation function with a value of 1.0 at zero lag and decaying to zero as the lag increases, provided the field has zero mean. In fact, $g_{E}(k)$ corresponds to the ACF of the instrument function so that

$$g_{E}(k) = h(0)h^{*}(k)$$

If the SLC image is detected and the ACF of the resulting intensity constructed, this will be given by

$$g_{I}(k)m= \frac{<I_{0}I_{k}>}{<I_{0}I_{k}>} = \frac{<I_{0}I_{k}>}{<I^{2}>} + 1$$

This can be split up into four separate terms. Firstly, there is the flat background of $<I^{2}>$. Secondly, there is the interference term corresponding to random scatterers within the instrument width. This would be the response if the cross-section were uniform. In this case the intensity ACF would be give by

$$g_{I}(k) \approx 1 + \frac{1}{L}\mid g_{E}(k)\mid^{2}$$

This has the same shape as the square modulus of the complex field ACF defined above. Thirdly, there is a term corresponding to incoherent imaging of the cross-section fluctuations. The fourth term represents coherent interference between random scatterers modulated by the cross-section fluctuations. This approximates to the square modulus of the complex field ACF, if the cross-section correlation function is approximately constant over the decay length of the instrument correlations. The total intensity ACF can then be approximated by

$$g_{I}(k) \approx 1 + \frac{1}{L}\mid g_{E}(k)\mid^{2} + (g_{\... ...a}(k) - 1)\left\{ 1 + \frac{1}{L}\mid g_{E}(k)\mid^{2}\right\}$$

Thus the instrumental component of the intensity ACF near zero lag has the form

$$\frac{1}{L}\mid g_{E}(0)\mid^{2}g_{\sigma}(0) \approx \frac{1}{L}\left(1+ \frac{1}{\nu}\right)$$

where $\nu$ is the order parameter of the cross-section fluctuations, whereas the total value is

$$\frac{1}{L}\left( 1 + \frac{1}{\nu}\right) + \frac{1}{\nu}$$

Let us now demonstrate the complex field correlation properties using SLC data from the same SAR radar as the intensity image above. In this mode, the DLR ESAR has resolution of 4 m in slant range and 0.72 m in azimuth. The sample spacing is 2.5 m in slant range and 0.34 m in azimuth so that this data is over-sampled by 1.6 in slant range and 2.1 in azimuth. Figure 6.8 shows the derived ACF of the complex field. The ENL is 1.0. From Figure 6.8 we note that the width at half height is 1.4 pixels (range) and 1.9 pixels (azimuth), which is consistent with the defined correlations.

Now we construct ACF of the detected intensity image, shown in Figure 6.9. In line with the discussion above, we break the intensity ACF into terms corresponding to those proportional to the correlation of the instrument function, the second and fourth terms above, the flat background (already subtracted) and the third term arising from the incoherent imaging of the cross-section variations. Since the intensity derived from the SLC data has $L \approx 1$ the instrumental profile term can be separated from the total, as shown by the red lines in Figure 6.9. It is apparent that the intensity ACF is consistent with the measured instrumental ACF in Figure 6.8 and the expression above.

Figure 6.8: ACF of the complex field for the SLC image. Azimuth response (full line), slant range response (dashed line)

Figure 6.9: Intensity ACF for the SLC data. Azimuth response (full lines), range (dashed). Terms proportional to instrument response (red); cross-section ACF (green).

The next step is to derive the effective resampling values. The instrumental part of the intensity ACF has a maximum value of 2.2 at zero lag. A fraction of 4field ACF, in Figure 3. This yields resampling values of 3 and 2 for azimuth and range lines respectively.

This section has demonstrated how the speckle correlations, introduced by the imaging instrument function, appear in the complex field and intensity ACFs. The next stage is to use the information from these ACFs to remove the effects of correlation from the SAR intensity images.

Resampling to remove speckle correlations.

Let us initially consider how to remove speckle correlations when SLC data is available. This is not so in many instances. Therefore it is necessary also to show how the same effect can be achieved when only intensity data is available.

Consider first the instrument ACF shown in Figure 6.8 above. The effect of correlations is insignificant for lag values of 3 pixels in range and 4 in azimuth. If the intensity image were resampled by taking every 3rd pixel in range and 4th in azimuth we would expect the samples to be essentially uncorrelated. This somewhat crude recipe is easy to apply, and has the desired effect. Unfortunately, it is found to lose some of the information contained in the images for two reasons. Firstly, the actual instrumental correlation lengths are expected to be 1.6 and 2.1 pixels respectively so that further separation of samples is losing uncorrelated information and degrades the resolution. Secondly, if we take averages of the intensity over the resampling interval, the speckle in the resultant image is reduced. The reduction is not equal to the number of pixels averaged, of course, because the speckle is correlated. However, this averaging slightly mitigates against the loss of information resulting from degraded resolution. In practice, we find that taking samples at a spacing corresponding to about 20% correlation along each axis achieves the desired effect. In this example, therefore, the resampling spacing should be 2.1 and 2.9 pixels respectively. Thus resampling corresponds to an average and shrink operation on the intensity over 2 and 3 pixels in range and azimuth respectively.

Let us now investigate how this applies to the SLC imagery.

Figure 6.5.3(a) Figure 6.5.3(b)

Figure 6.5.3(c) Figure 6.5.3(d)

Figure 6.5.3(e) Figure 6.5.3(f)
Figure 6.5.3
Comparison of original and resampled segmentation. (a) original image; (b) segmentation of original image; (c) resampled image; (d) segmentation of resampled image; (e) ratio image for original data; (f) ratio image after resampling.

In Figure 6.5.3(a) we show a detail from the intensity derived from the SLC image. The segmented version, using default parameters, illustrated in Figure 6.5.3(b), again shows the effects of speckle correlations. Many of the regions in this image are clearly resulting from original speckle blobs. Both these images have been rescaled for a square display. Next we apply the resampling derived above; applying average and shrink to the intensity of 2 pixels in range (x) and 3 in azimuth (y). The resampled intensity is shown in Figure 6.5.3(c) with the corresponding segmentation in 6.5.3(d). This segmentation has the expected form with large areas of uniform cross-section and very few speckle-sized segments. The ratio image for the original data ( 6.5.3a) divided by the segmentation ( 6.5.3b) is shown in Figure 6.5.3(e). The ``speckle'' blobs are clearly not of the correct size for the expected correlations, indicating that much of the speckle fluctuation is appearing in the segmented version. The residual is merely the contributions which are uncorrelated on the scale of a pixel. This is confirmed by calculating the ENL, which is 1.70 instead of the theoretical value of 1.0. Figure 6.5.3(f) shows the corresponding ratio after the original intensity data was resampled. Speckle correlations are now on a much larger scale, as expected, and the calculated ENL is 1.03. Thus, this resampling scheme has indeed removed the effect of speckle correlations to a large extent.

Thus we conclude this section by noting that the effects of speckle correlations in the intensity data can be removed by estimating the required resampling from the complex field or intensity ACFs. The former is more accurate; the latter can lead to complications since it includes the ACF of the cross-section.

Resampling applied to the original intensity image

Let us conclude this study by applying the methods discussed in the previous section to the original intensity image. The first step is to construct the intensity ACF of the data, which is shown in Figure 6.10. The situation is complicated by the fact that the cross-section component of the ACF, at lags greater than 4 pixels has an appreciable slope. From the theoretical expression for the intensity ACF we see that the component of the ACF proportional to the instrument function has the form near zero

Figure 6.10: Plots of the intensity ACF of the data in Figure 1. Azimuth ACF (full line), range ACF (dashed line).

lag of where $\nu$ is the order parameter of the cross-section fluctuations. The original data has $L \approx 4$.

The origin of Figure 6.10 then indicates that $\nu = 4$ and that the instrument function component of the intensity ACF has a magnitude of 0.31. Thus the correct lag values for resampling are at an ACF value of about 0.26, i.e. lags of 2.9 (azimuth) and 3.3 (range). The nearest integer resampling number is 3 along each axis. It should be noted that it is difficult to separate the instrumental and cross-section components of the ACF in this fashion. Fortunately we only need to approximate the resampling values to the nearest pixel. A reasonable approximation can be obtained by observing that the gradient of both ACFs is reduced significantly at around a lag value of 3, indicating that the speckle part of the ACF has decayed away at about this value. When the averaging and resampling by 3x3 is applied to the original image the image shown in Figure 6.5.4(a) is obtained, while Figure 6.5.4(b) shows the corresponding segmentation. It is obvious that the segmented form of the resampled image contains large uniform regions, as one would expect. This is in stark contrast to the original version in Figure 6.5.2(b). There are no speckle blob sized segments in Figure 6.5.4(b), which should be compared with the very large number visible in 6.5.2(b). Figure 6.5.4(c) shows the ratio of the original intensity to the reconstructed cross-section after resampling. This has the expected scale of correlations in the speckle blobs, similar to those in Figure 6.5.2(d) and completely different from the ratio image obtained from the original data, shown in Figure 6.5.2(c) and repeated here for comparison in Figure 6.5.4(d). Finally, the ENL derived from the ratio image is 4.00, consistent with the defined value. This consistency indicates that the segmentation derived in this way can be trusted.

Figure 6.5.4(a) Figure 6.5.4(b)

Figure 6.5.4(c) Figure 6.5.4(d)
Segmentation of the resampled original image from Figure 6.5.2. (a) Image resampled by 3x3; (b) segmentation of (a); (c) ratio of (a) to (b); (d) corresponding ratio for original data (from Figure 6.5.2(c)).

Segmentation: Conclusions

In summary, this study has demonstrated the following aspects of applying segmentation to correlated data:

• If segmentation is applied directly to SAR intensity images that have been sampled at spacing less than the speckle correlation width, which is determined by the resolution of the system, the resulting segmentation is dominated by speckle-sized segments.

• The ratio of the original intensity to the segmented version does not correspond to the expected speckle both in apparent ENL and the speckle correlation properties. Only residual correlations on a scale of one pixel remain.

• If the SLC image is available, the speckle correlations can be evaluated directly from the complex field ACF. The resampling factors then correspond to the lag where the field ACF falls to 20
• Similar information can be derived from the intensity ACF, though more indirectly. The component due to the instrumental response can be separated from that due to the underlying cross-section fluctuations. The resampling factors then correspond to the lags where the instrumental part of the ACF falls to 4
• Since the resample factor needs only be accurate to the nearest pixel, it is often adequate to identify where the slope of the intensity ACF changes significantly. This marks the point where the instrumental component has largely decayed away.
• Both methods have been applied to the resampling and segmentation of data from both SLC imagery and multi-look intensity imagery. In both cases the residual speckle component is consistent with the expected ENL and the correlation properties similar to those expected.
• These techniques offer the means to segment intensity data with any speckle correlation properties in an optimum fashion; optimising both the retention of information by speckle reduction through averaging the intensity and correlation removal by resampling.