# Grapher Patch

-graph-store identifies the Graph Store managed by the HTTP service. In order to dispatch requests to manage named or default graphs by embedding them in the query component of the Graph Store URL, the URL will need to be known a priori.

## Grapher patch

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As mentioned earlier, "multipart/form-data" can be dispatched to implementations of this protocol. When used with POST this operation MUST be understood as a request that the origin server perform an RDF merge of the graphs - that the documents submitted with the multipart form are a serialization of - into the RDF graph content identified by the request or encoded IRI. In such a case, if the Content-Type is not provided, implementations MAY attempt to determine it from the file's extension rather than respond with 400 Bad Request.

SPARQL 1.1 Update can be used as a patch document. In particular, SPARQL 1.1 Update requests that manage the graph associated with the RDF graph content identified (directly or indirectly) in the request can be used as the RDF payload of a HTTP PATCH request to modify it. If a SPARQL 1.1 Update request is used as the RDF payload for a PATCH request that makes changes to more than one graph or the graph it modifies is not the one indicated, it would be prudent for the server to respond with a 422 Unprocessable Entity status.

Total variation (TV) based models are very popular in image denoising but suffer from some drawbacks. For example, local TV methods often cannot preserve edges and textures well when they face excessive smoothing. Non-local TV methods constitute an alternative, but their computational cost is huge. To overcome these issues, we propose an image denoising method named non-local patch graph total variation (NPGTV). Its main originality stands for the graph total variation method, which combines the total variation with graph signal processing. Schematically, we first construct a K-nearest graph from the original image using a non-local patch-based method. Then the model is solved with the Douglas-Rachford Splitting algorithm. By doing so, the image details can be well preserved while being denoised. Experiments conducted on several standard natural images illustrate the effectiveness of our method when compared to some other state-of-the-art denoising methods like classical total variation, non-local means filter (NLM), non-local graph based transform (NLGBT), adaptive graph-based total variation (AGTV).

Recently, image denoising has also been studied from the point of view of graph signal processing (GSP) [9]. Kheradmand and Milanfar [3] proposed a general graph-based regularization framework which was later modified by Pang et al. [10] who used patch gradients instead of patch intensities to define the patch self-similarity. Mahmood et al. [11] proposed an adaptive graph total variation (AGTV) for tomographic reconstruction. Though these methods provide good experimental performance, some problems remain: (i) there is neither theoretical justification nor intuitive interpretation of the relationship between the graph structure derived from the image and the image denoising performance; (ii) these algorithms involve complicated mathematical construction and large calculations.

In addition, one of the key issues in performing graph signal processing (i.e. denoising in our case) concerns the selection of edge weights [12]. Indeed, these weights have a significant effect on the amount of noise removal. In [13], Smolka et al. compute the weights using a Gibbs distribution of the intensities for the adjacent pixels. Black et al. [14] derived these weights via robust statistics. A more common but less robust approach exploits a Gaussian kernel function where the weights are calculated only from two isolated pixels based on their intensities and location information [15]. A more reliable idea is to consider the pixel neighborhood due to the high degree of redundancy in natural images [6]. In such a way Buades et al. [4] proposed to use a windowed non-local means filter to characterize one pixel instead of only using the pixel itself. Some graph-signal based image denoising methods also borrow the image patch thought to construct the graph, the most typical scheme being AGTV. However, they only take the image patch intensity into consideration and ignore the location information of the patch. Thus, image spatial information has not been utilized.

To overcome the above problems, we propose a non-local patch graph total variation (NPGTV) as a novel method for natural image denoising. Our method can be seen as an improved version of AGTV by considering the pixel coordinate as an ingredient to construct the K nearest neighbor graph (KNN graph). More clearly, both the image patch intensity and patch location information are taken into account. By doing so, image details can be preserved at a greatest extent. In addition, in this paper, we also analyze the impact of the patch size and of the K value of the KNN graph on the denoising performance. It is important to notice that AGTV reconstructs and utilizes graph total variation, repeatedly, our method merely constructs a non-local patch graph and use GTV model once. In this way, time cost is significantly reduced. As we will see in the sequel, our proposed method can achieve better performance compared to some recent and efficient non-local based denoising methods and total variation based denoising methods.

Although the AGTV algorithm can perform well on the tomographic data denoising, there still exist some shortages that hinder it from being applicable to the natural image denoising problem. First, only image intensity is taken into consideration during the graph construction while ignoring the patch location information. Second, the patch graph needs to be constructed repeatedly in the whole algorithm, which will lead to the tedious hyperparameter tuning problem. How to overcome both deficiencies and construct an effective GTV-based image denoising algorithm becomes our main motivation to propose the NPGTV algorithm.

In this section, we describe the proposed NPGTV algorithm accordingly to three steps: (i) representation of an image as a weighted undirected graph; (ii) establishment and solving of the total variation model; (iii) description of our complete NPGTV proposal. This choice stands for the main steps of our algorithm depicted in Fig 1. First, as shown in Fig 1(a) and 1(b), a group of non-local image patches are extracted from a noisy image. By next, a KNN graph is derived from these patches as illustrated in Fig 1(c). Then the NPGTV model is established. Finally, the denoised image is achieved by performing the Douglas-Rachford splitting algorithm on to this model (Fig 1(d)).

One of the core steps of our method is to construct from an image a graph G for GTV regularization. To do so, we build a weighted undirected graph G = (V, E) to describe an image by considering its pixels as elements of V. The set E contains the corresponding edge information. The edge em,n only exists if the node vm and vn are connected. One naïve way consists in connecting each pixel to its neighbors. One can thus obtain a 4-connect graph or an 8-connect graph. Another strategy, like in AGTV [11], connects image patch center at each pixel through the K nearest neighbor (KNN) algorithm. However, both methods suffer from various drawbacks. The former, being a local model, will tend to alter image details when denoising. The latter is a non-local model that merely uses patch intensity to calculate the distance between two image patches without considering central pixel coordinate. To go beyond these disadvantages, we propose a graph construction method that combines patch intensity with pixel coordinates. The method consists of four steps:

From the above, it is easy to find that the patch size s, the K value of KNN algorithm and the spatial constraint parameter λ will have an impact on the denoising result. All three parameters are relevant to the noise level. Generally speaking, these parameters will take large values under high noise levels, and vice versa. But they cannot be too high. For the first two parameters, the larger the patch size and K, the smoother the image will be. If the patch size and K take too large values, some image details will be removed. Besides, too large patch size will lead to image edge blur (just as illustrated in Fig 3). Too large K values will also increase the calculation cost slowing down our method. Thus in the experiments presented in our paper, we set a patch size of 99 pixels under high noise level (larger than 15db) and 55 pixels under weak noise level (less than15db). The value of K is fixed to 5. Such parameters have been shown to be robust while details and fine structure can be better preserved.

The patch size is 20, K is 9 and λ is 0.05. Although the noise is removed, some image details are also eliminated such as building outline and grass on the ground. Besides, the image is blurred. A virtual outline appears around the image edges.

As for the spatial constraint parameter λ, the location information of the patch is normalized into the range of gray level of the image. By doing so, patch intensity and location are more relevant when constructing the graph with the KNN algorithm. Notice that, when λ = 0, is degraded into vi. As a consequence, the similarity between two patches is measured based on the distance of their intensity level. When λ takes a high value, patch location will become the main ingredient to determine the similarity between two patches. Taking such value may not work well since it is extremely sensitive to minor transformations, both in geometry (shifts and rotations) and in imaging conditions (lighting or noise) [23]. Therefore, the selection of the value of λ heavily depends on the range of gray level of the image. In our experiment, the gray

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