Linear Filter Kernel Estimation Based on Digital Camera Sensor Noise

We study linear filter kernel estimation from processed digital images under the assumption that the image’s source camera is known. By leveraging easy-to-obtain camera-specific sensor noise fingerprints as a proxy, we have identified the linear crosscorrelation between a pre-computed camera fingerprint estimate and a noise residual extracted from the filtered query image as a viable domain to perform filter estimation. The result is a simple yet accurate filter kernel estimation technique that is relatively independent of image content and that does not rely on hand-crafted parameter settings. Experimental results obtained from both uncompressed and JPEG compressed images suggests performances on par with highly developed iterative constrained minimization techniques. Introduction Inferring the processing history of digital images is a core problem of digital image forensics [1, 2]. A rich body of literature reflects the plethora of different operations an image may undergo after being captured by an acquisition device, focussing on aspects of manipulation detection, localization, and parameter estimation. Purely technical solutions prevail, leaving the question as to what operations are to be considered legitimate or malicious to human interpretation [3]. Recent advances in data-driven techniques advocate the idea of general-purpose image forensics, where mostly very high-dimensional feature spaces are chosen to detect, identify, or localize a variety of operations in a unified framework [4–8]. The generality of these techniques comes at the cost of a relatively coarse granularity with respect to intra-class variations within the same type of processing (e. g., different filter sizes or strengths). Even with practicable solutions to overcome issues of scalability in multi-classification [4, 5, 8], it seems infeasible to assume that a single model will ever be informative enough to discern between a large number of parameter settings across multiple processing types. Targeted techniques with the ability to estimate parameters of specific processing operations, such as previous JPEG quantization tables [9, 10], the shape of non-linear intensity mappings [11], or parameters of affine transformations [12–14] will thus remain to play an important role in the realm of image forensics. Along these lines, our focus here is on estimating the coefficients of an unknown linear filter kernel a query image may have been subjected to. In contrast to the analysis of non-linear filtering, where a large number of works have focussed on the specifics of median filtering [15, 16, among many others], the interest in linear filtering has appeared relatively scattered over the forensics community [17–20]. At the same time, however, blind deconvolution and point spread function (PSF) estimation are widely established fields in image processing of course [21–23], with applications including image restoration and enhancement, or the removal of motion blur. The major challenge in this stream of research is that estimating a filter kernel is an ill-posed problem in the absence of the original unfiltered image. The key is to incorporate informative prior models about the original image into the estimation process. In classical blind deconvolution, such prior models will typically pertain to a generic digital image, effectively acting as a proxy of the captured scene. For forensic purposes, where the focus is clearly on processing applied after image capturing, this problem can be approached in a much more narrow sense by taking knowledge about image acquisition into account, e. g., through reasonable assumptions that the original image underwent a demosaicing procedure [17], or that the original image was stored in JPEG format [19]. In the remainder of this paper, we demonstrate how knowledge of the acquisition camera’s sensor noise fingerprint [24] can facilitate efficient and effective solutions to the linear filter kernel estimation problem. Similar to digital camera identification in the presence of geometric distortion [25–27], the premise here is that the camera fingerprint, as part of the image, will undergo the same processing the image undergoes. Under the assumption of linear filtering, the effects of this processing can be measured directly in the linear cross-correlation between the camera’s “clean” fingerprint and the fingerprint estimate obtained from the processed image. This way, the (assumedly) known camera fingerprint imposes a strong prior model of the unfiltered signal in the estimation process. Before we detail our approach below, the following sections give a brief overview of the general filter kernel estimation problem and some background on digital camera sensor noise forensics. We then continue with a description of a simple filter kern estimation technique and present experimental results. Model and Notation We adopt a generic linear shift-invariant (LSI) model y[i, j] = ∑ u,v h[u,v] · x[i−u, j− v]+n[i, j] (1) in which the observed image y[i, j] results from the linear twodimensional convolution of the “clean” image x[i, j] with a filter kernel h[u,v], plus some measurement noise n[i, j]. The goal of filter kernel estimation is to determine the coefficients h[u,v] of the unknown filter from the image y[i, j], where it is often instrumental to assume a (without loss of generality) square kernel support of S× S, S = 2S0 + 1, i. e., −S0 ≤ u,v ≤ S0, and ∑u,v h[u,v] = 1. It is convenient to write Equation (1) as a system of linear equations,