Evaluación y dimensionado de la instalación

Errors from two different sources can also be very closely-coupled in the super- resolution problem. In this example, we show that errors in some of the geometric parameters θ(k) _{can to a small extent be mitigated by a small increase in the size of} the blur kernel.

We take images from the synthetic Frog dataset of Figure 3.17, which is con- structed in exactly the same way as the Keble dataset, but using an entirely different scene as the ground truth high-resolution image. Again, we take 16 low-resolution images at a zoom factor of 4, a PSF width of 0.4 low-resolution pixels, and various levels of i.i.d. Gaussian noise. Because the ground-truth image is much smoother than the Keble image, with fewer high-frequency details, it is in general easier to super-resolve, and leads to reconstructions with much lower RMS error than the Keble image dataset.

Errors are applied to the θ parameters governing the horizontal and vertical shifts of each image, with standard deviations of 0, 0.01, 0.02, 0.03, 0.04 and 0.05 low-resolution pixels. For each of these six registrations of the input data, a set of

(c) closeup (b) closeup

(a) closeup

(c) σ

err=10% of intensity range (b) σ

err=2% of intensity range (a) No photometric error

Figure 3.16: Reconstructing the Keble dataset with photometric error: im- ages. The upper row shows the full super-resolution image, and the lower row shows a close-up of the main window, where the different levels of detail are very notice- able. (a) Reconstruction with ground truth photometric parameters: the sources of error here are merely the additive noise on the low-resolution images and the loss of high frequency detail due to subsampling. (b) The best reconstruction possible with a 2% error on the photometric shift parameters, λ(₂k). (c) The best reconstruction possible with a 10% error. Notice that the edges are still well-localized, but finer details are smoothed out due to the increase in prior strength.

Ground truth Frog image image 5 image 4 image 3 image 2 σ = 5 image 1 σ = 1 σ = 0

Figure 3.17: The synthetic Frog dataset. At the top is the ground truth Frog image. Below is an array of low-resolution images generated from this ground truth image. Like the Keble dataset in Figure 3.4, all these low-resolution images have a Gaussian point-spread function of std 0.4 low-resolution pixels, a zoom factor of 4, an 8DoF projective transform, and a linear photometric model. Each column represents a different registration vector, and each row represents a different level of the Gaussian i.i.d. noise added to the low-resolution images, measured in grey levels, assuming an 8-bit image (i.e. 256 grey levels). Because large areas of the frog image are smooth, this dataset represents less of a challenge to reconstruct than the Keble image, since smoothness-based image priors describe the distribution of its pixels more effectively than they describe the Keble image.

0.35 0.4 0.45 0.5 11

12 10

psf standard dev.

RMSE with respect to ground truth

Error wrt PSF size, for various degrees of mis−registration

σ_err = 0.00 σ_err = 0.01 σ_err = 0.02 σ_err = 0.03 σ_err = 0.04 σ_err = 0.05

Figure 3.18: Reconstructing the Frog image with small errors in geometric registration and point-spread function size. The six colours represent six levels of additive random noise added to the shift parameters in the geometric registration. The curves represent the optimal error as the PSF parameter γ was varied about its ground truth value of 0.4. The larger the registration error, the bigger the error in γ is in order to optimize the result.

super-resolution images is recovered as the PSF standard deviation, γ is varied, and for each setting ofγ, the prior strength ratio giving the best reconstruction is found. Figure 3.18 shows how the best error for each of the six registration cases varies with the point-spread function size. When the geometric registration is known accurately, the minimum falls at the true value of γ. However, as the geometric registration parameters drift more, the point at which the lowest error is found for any given geometric registration increases. This can be explained intuitively because the uncertainty in the registration tends to spread the assumed influence of each high-resolution pixel over a larger area of the collection of low-resolution images.

(d) diff wrt best x

(c) result for σ

err

=0.05

(b) diff wrt best x

(a) result for σ

err

=0.01

Figure 3.19: Reconstructing the Frog dataset with smell levels of geometric error. (a) The best result with σerr = 0.01, corresponding to the second-lowest

black dot in Figure 3.18. (b) The difference between this reconstruction and the best reconstruction given no geometric error (corresponding to the lowest black dot in Figure 3.18), amplified by a factor of four. (c) The result with σerr = 0.05

(uppermost black dot of Figure 3.18). (d) The difference between this reconstruction and the best reconstruction, amplified by a factor of four.

Figure 3.19 shows two high-resolution images reconstructed with geometric reg- istration error standard deviations of 0.01 and 0.05 low-resolution pixels, and also shows the difference between these and the best reconstruction for this dataset using the true registration. Localization problems and the effects of a stronger image prior are clearly visible along the strong image edges, particularly around the frog’s eye, especially where the standard deviation is as high as one twentieth of a low-resolution pixel.