# Blind Deconvolution with Scale Ambiguity
## 1 Introduction  
Signal processing is a hot research topic in the field of electronics and information, which has been researched everywhere in today's digital era, especially for cultural, military, health and scientific research domains. As a 2D signal, image plays an important role for people to obtain information and the study of images has attracted much attention. Single image deblurring is a classical problem in image processing communities. A few image deblurring methods [1; 2; 3; 4; 5; 6; 7] may be the most representatively used to handle deblurring problems. The work of [PERSON] et al. [8] provided an overview of a series of deblurring methods [9; 10; 11; 12; 13; 14]. When the blur is spatially invariant, two unknowns, i.e., a blur kernel (a.k.a. point spread function, PSF) and a latent image, are expected to be recovered from a single blurred input. The convolution operator is the most commonly used to describe the blur process:  
\[f=u*k+\varepsilon, \tag{1}\]  
where \(f\), \(u\), \(k\) and \(\varepsilon\) represent the blurred image, the latent image, the blur kernel, and inevitable additive Gaussian noise, respectively. \(*\) denotes the convolution operator. Blind image deblurring is a well-known ill-posed problem because there are infinite pairs of \(u\) and \(k\) which can satisfy (1). To make the problem well-posed, plenty of methods focus on makingassumptions on the latent image, the blur kernel, or both [5; 10; 12; 15; 16; 17; 18]. The sparsity of image gradient [11; 19; 20] has been studied as one kind of image prior in image deblurring. Unfortunately, [PERSON] et al. [19] analyzed that within the maximum a posterior (MAP) framework, the deblurring methods based on the sparsity of image gradient tend to favor blurred images over clear images. Consequently, some other natural image priors that favor clear images over blurred ones have also been introduced. [PERSON] et al. [10] designed a new normalized sparsity prior to estimate blur kernels. [PERSON] and [PERSON] [13] presented a blind deblurring method, which uses internal patch recurrence as a cue for kernel estimation. [PERSON] et al. [5] proposed an effective image deblurring method based on the dark channel prior. This algorithm enforces the sparsity of the dark channel of latent images for kernel estimation. Based on the analysis of [PERSON] et al. [19] that sharp edges usually work well in MAP framework, [1; 9] proposed to introduce a heuristic edge selection step during kernel estimation process. Nevertheless, these methods are likely to fail when strong edges are not available in the observed images. While the aforementioned methods [5; 10; 11; 13] mainly focus on developing effective image priors for blind image debluring. Very Few deblurring methods [17; 21] pay attention on the relative scale ambiguity between the latent image and the blur kernel. Mathematically, there is an inherent scale ambiguity in the convolutional image degradation model (1). That is, if \((u,k)\) is a pair of solutions (we ignore the noise), then the blurred image \(f\) can undoubtedly be generated by \((su,k/s)\), where \(s>0\) and \(s\
eq 1\) is a scale factor. Almost all existing deblurring methods remove this ambiguity by imposing \(L^{1}\) normalization on the blur kernel to fix the scale. It is a common constraint to make the sum of all the kernel weights euql to one. Nonetheless, little theoretical analysis would strictly prove whether this choice is beneficial to kernel estimation or not. This indicates that, any other norm could be used to eliminate the scale ambiguity. As evidenced by the recent work, [PERSON] et al. [17] used the Frobenius norm to fix the scale ambiguity, which enables isotropic total variation (TV) regularized deconvolution formulation to achieve the state-of-the-art results.