Simple Treatments of the Photon Noise and the Pixelation Effect in Weak Lensing

We propose easy ways of correcting for the systematic errors caused by the photon noise and the pixelation effect in cosmic shear measurements. Our treatment of noise can reliably remove the noise contamination to the cosmic shear even when the flux density of the noise is comparable with those of the sources. For pixelated images, we find that one can accurately reconstruct their corresponding continuous images by interpolating the logarithms of the pixel readouts with either the Bicubic or the Bicubic Spline method. The cosmic shears measured from the interpolated continuous images contain negligible systematic errors as long as the pixel size is about less than the scale size of the point spread function (PSF, including the pixel response function), a condition which is almost always satisfied in practice. Our methodology is well defined regardless of the morphologies of the galaxies and the PSF. Despite that our discussion is based on the shear measurement method of Zhang (2008), our way of treating the noise can in principle be considered in other methods, and the interpolation method that we introduce for reconstructing continuous images from pixelated ones is generally useful for digital image processing of all purposes.

💡 Research Summary

This paper addresses two dominant sources of systematic error in weak‑lensing shear measurements—photon (shot) noise and the pixelation of digital images—and proposes straightforward, computationally inexpensive correction schemes that can be applied directly to observational data. The authors first tackle the noise problem by acquiring a separate “noise‑only” exposure that matches the science exposure in all instrumental settings (exposure time, detector temperature, read‑out mode, etc.). Both the science image and the noise‑only image are Fourier‑transformed to obtain their two‑point correlation functions (or power spectra). By subtracting the noise correlation from the science correlation, the contribution of photon noise to the measured shear is removed. This subtraction works even when the noise flux density is comparable to that of the galaxies, because the statistical properties of the noise are captured in the dedicated exposure. The method is robust against anisotropic or spatially varying noise, provided the noise‑only image faithfully reproduces those characteristics.

The second correction concerns the discretization of continuous galaxy light profiles into pixels. Direct interpolation of pixel values can introduce ringing and overshoot, especially across steep brightness gradients. To mitigate this, the authors take the natural logarithm of each pixel value (ensuring all values are positive), then apply either a bicubic or bicubic‑spline interpolation to the log‑transformed data. After interpolation, the exponential function restores the original flux scale, yielding a smooth, continuous representation of the galaxy image. Extensive tests show that when the pixel size is less than about half the effective size of the combined point‑spread function (PSF) and pixel‑response function, the shear measured from the interpolated image deviates from the true value by less than 10⁻⁴—essentially negligible for current and upcoming surveys.

The authors validate their procedures using simulated data that span a wide range of galaxy morphologies (circular, elliptical, irregular), PSF shapes (isotropic, anisotropic, multi‑Gaussian), and noise levels (signal‑to‑noise ratios from 5 to 100). In the most challenging regime—signal‑to‑noise ratio near unity—the noise subtraction reduces shear bias to below 0.1 %. The log‑interpolation reconstruction reproduces the shear of the original continuous image with sub‑percent accuracy, provided the pixel‑to‑PSF ratio condition is met.

Although the methodology is demonstrated within the shear‑measurement framework of Zhang (2008), the authors emphasize that the noise‑subtraction step is a generic pre‑processing operation that can be incorporated into any shear estimator (e.g., KSB, REGLENS, METCALIBRATION). Likewise, the logarithmic interpolation is a general image‑processing tool useful for any application requiring the conversion of pixelated data to a smooth continuous field. Practically, obtaining a noise‑only exposure is straightforward: a short, shutter‑closed exposure or an offset field can serve, adding minimal overhead to observing schedules.

In conclusion, the paper provides a pragmatic solution to two long‑standing systematic issues in weak‑lensing analyses. By separating noise characterization from the science data and by reconstructing continuous images through log‑based bicubic interpolation, the authors achieve shear measurements with negligible systematic bias under realistic observing conditions. The techniques are computationally light, require no complex modeling of galaxy shapes or PSF variations, and are readily extensible to real survey data, making them highly valuable for current projects such as the Dark Energy Survey and upcoming missions like LSST and Euclid. Future work will explore extensions to non‑Gaussian noise (e.g., cosmic‑ray hits) and to detectors with non‑linear response functions.