0028 DS are selected by the connectivity of the coherent image pairs, not by the number of effective pairs¶

Status¶

Accepted

Date¶

2026-10-06

Context¶

Decision 0024 asks to select by the weighted temporal coherence only together with the effective number of image pairs eff_n_pairs, because a high weighted value that rests on few pairs is not reliable. But the minimum of eff_n_pairs has no meaning of its own and scales with the number of image pairs P = N (N - 1) / 2 (about 0.45 P gave the same quality as the plain temporal coherence >= 0.8 on the 17 images of the sample data and on 92 images of Campi Flegrei, 2026-10-06). What makes a weighted temporal coherence unreliable is that the coherent pairs do not link all images: the phase between two groups of images that no coherent pair connects is not constrained by the data, whatever the weighted value is.

The graph with the images as nodes and the image pairs with a positive weight as edges is useless as it is: a pair of pure noise has |coh|^2 > 1/n_looks with probability 0.37, so the graph of every point, noise included, is connected (sample 99.6 %, Campi Flegrei 100 %).

Decision¶

ds_temp_coh(coh, ph, image_pairs=None, block_size=128, n_looks=None, alpha=1e-3) returns with n_looks also n_components, int16, shape (n_points,): the number of connected components of the graph of the images whose edges are the image pairs with |coh_k|^2 > 1 - p^(1/(n_looks-1)). For |coh| of a pair without coherence the probability of |coh|^2 > x is (1-x)^(n_looks-1), so an incoherent pair is an edge with probability p. p is chosen so that the random graph on N images with edge probability p is connected with probability alpha (Gilbert's recursion for the probability that G(N, p) is connected): a point without any coherent pair has n_components 1 with probability alpha, whatever N is. 1e-6 <= alpha <= 0.5 (below, the recursion loses its digits). Points with n_looks <= 1 get N.
t_coh, t_coh_w and eff_n_pairs are unchanged (t_coh bit for bit); eff_n_pairs stays an output but is not used for the selection. The weighted API function emperical_co_emi_temp_coh_pc returns n_components as fifth output; the commands ds-temp-coh and emperical-co-emi-temp-coh-pc get the output n_components / n_components_dir and the input alpha.
The examples select the DS with n_components == 1 and t_coh_w >= 0.9 (alpha 1e-3).
Limits of the GPU kernel: block_size x nimages at most 393216 (4 bytes of shared memory per image and warp, 48 KB per block; 3072 images at the default block size); the CPU version has no limit.

Consequences¶

Breaking: the weighted outputs of ds_temp_coh and emperical_co_emi_temp_coh_pc have one more element; code that unpacks them fails.
Measured (A100, 100 000 points x 4186 image pairs, random coherence, 92 images): ds_temp_coh without n_looks 3.9 ms before and 4.3 ms now, with n_looks 4.2 ms before and 15.2 ms now; the phase linking step of the sample (733 000 points) takes 21 s with and without the change. Per point 2 more bytes of output.
On the sample data (17 images, 732 727 candidates) 70.1 % of the candidates are connected; connected and t_coh_w >= 0.9 keeps 58.4 % with a mean cosine of the phase difference of the sequential interferograms to reference points 11-20 pixels away of 0.739 and 9.7 % of the points below 0.5, the plain t_coh >= 0.8 53.2 %, 0.753 and 6.9 %: a little looser. On an 800 x 800 block of Campi Flegrei (92 images) 74.0 % with 0.927 against 69.7 % with 0.933.
Pure noise simulations (17 and 92 images, 4000 trials) give the connected fraction alpha within the sampling error although the edges of the speckle noise are not independent (tests check 17 images).
alpha only decides how a noise point is treated, not the quality of partly coherent points: for these the weighted temporal coherence decides, with its empirical threshold.

Do not¶

Do not use the pairs with a positive weight as edges: every point is connected then.
Do not select by the weighted temporal coherence without n_components (this replaces the last rule of decision 0024, which asked for eff_n_pairs).
Do not make the minimum of eff_n_pairs a selection parameter again: it depends on the number of images.