*This is the second part of **"Anomaly Detection: UFlow"**, in which we introduced UFlow, our proposed method for anomaly detection in images based on Normalizing Flows.*

In the first part of Anomaly Detection:UFlow, we went through the method and explained how to generate the first output, an anomaly map composed of an anomaly score for each image pixel.

In this part, we explain how to compute the second output of the method, an **automatic segmentation of the anomalies**. It can be seen as classifying each pixel as anomalous or normal. In this approach, an estimate of the **Number of False Alarms (NFA)** is associated to each anomaly candidate, and detections are obtained by thresholding the NFA [8]. The proposed method has no parameters to tune and uses level sets organized into a tree structure to segment the anomalies automatically.

Before discussing this method, we should first introduce the ** A Contrario** framework and the

*Number of False Alarms*.

## NFA: Number of False Alarms

The a contrario framework [8] is a multiple-hypothesis testing methodology that has been successfully applied to derive unsupervised statistical detection thresholds in a wide variety of detection problems, such as alignments for line segment detection [44], clustering [45], image forgery detection [46], and even anomaly detection [47–49], to name a few.

It is based on the **non-accidentalness principle** [51]. Given that we do not usually know what the anomalies look like, it focuses on modeling normality by defining a null hypothesis, also called background model. Relevant structures are detected as **large deviations from this model** by evaluating how likely it is that an observed structure or event *E* would happen under the null hypothesis.

**One of its most useful characteristics is that it provides an easily interpretable detection threshold based on controlling the NFA, defined as**

Here *N_T* denotes the number of events tested, and *Pr_H0(E)* is the probability of the event *E* being a realization of the background model *H0*. Besides providing a robust detection threshold, as it will be shown, the NFA value itself has a clear statistical meaning: **it is an estimate of the expected number of times, among the tests that are performed, that such tested event could be generated by the background model** [8]. In other words, an event E is** ε-meaningful** if its expected number of occurrences is less than ε under the normality assumption. Consequently, **ε provides an upper bound on the expected number of false detections we can obtain on anomaly-free images.** A low NFA value means that the observed pattern is too unlikely to be generated by the background model and, therefore, indicates a meaningful anomaly.

As the events *E* corresponding to anomalies can have arbitrary sizes and shapes, we must test all possible configurations of connected sets of pixels within every image. But this is computationally inefficient and sometimes even impossible in a reasonable amount of time. Therefore, we propose to **use the anomaly map to select the regions to be tested from the set of connected components of its level sets**. These components are naturally ordered by inclusion, providing a convenient hierarchical representation that can be organized in a tree structure.

## Detecting anomalies from the level sets of the anomaly map

In image processing, level sets are the basis for a wide variety of morphological filters, and, as they are closely related to object boundaries, they have been used for edge detection [52, 53], registration [54, 55], image quantization [56], and segmentation [57], among others. All the information of a gray-valued image *u(i,j)* is contained in a set of binary images obtained by thresholding at different levels. Since no information is lost, level sets provide a complete representation: any image can be easily reconstructed from the whole family of its (upper) level sets [58]:

Level sets naturally define a hierarchical representation, ordered by their geometrical inclusion. In the case of the upper level sets, each connected component defined by a certain level is included in another connected component defined by a lower level. Therefore, they can be naturally embedded in a tree representation.

Specifically, for each scale we propose to use the upper level sets of an image defined from the NF embeddings as

where *C *is the number of channels of the embedding.

**Figure 1** shows a toy example that illustrates the procedure. Figure 1a represents the image u, with values *u(i,j)* denoted by numbers inside each pixel. Figure 1b shows the raveled indexes that serve as labels for the tree nodes shown in Figure 1c. The tree is built so that the nodes are given by the connected components of the level set *L_{λ,c}*, and the edges represent the inclusion, connecting level set components from two different levels.

The tree of connected components allows us to **significantly reduce the number of image regions we effectively test**. Instead of computing the NFA of all possible N_T connected regions of any shape and size, which would be intractable, we will limit this computation to the connected components of the upper level sets in the tree. **By construction, these regions are expected to be good candidates for anomalies.**

To compute the NFA of a connected component L_{λ,c} in the tree, we need to evaluate the **Probability of False Alarm (PFA)**, i.e., the probability of occurrence of a region like the observed L_{λ,c} under the background model of normality. In the absence of anomaly, the embedding produced by the U-Flow model is

i.i.d. Therefore, variables *u(i,j)* are identically distributed as a Chi-Squared distribution of order *C*. Then, as the minimum pixel value in the connected component *L_{λ,c}* is *λ*, it follows that its probability of false alarms is given by

Once we have computed the log(PFA) values of all connected components in the tree, to obtain the final regions, we iteratively perform prune and merge procedures until convergence (until the tree does not change anymore). See the paper for more details.

### log(NFA) computation

The scales are merged by upsampling the resulting *log(PFA)* heatmaps at different scales *l*, up to the original input size *(H, W)*, and keeping the minimum value at each pixel position *(i, j)*:

Finally, the log(NFA) value is given by:

with

## Results

Providing an **operation point** is crucial in almost any industrial application. Most recent deep-learning industrial anomaly detection methods in the literature focus on generating anomaly maps and evaluating them using the AUROC metric; they do not provide detection thresholds or anomaly segmentation masks.

Here, we report the results of anomaly segmentation based on an unsupervised threshold obtained by setting NFA = 1 (log(NFA) = 0). As explained in the paper, this threshold means that, in theory, we authorize, on average, one false anomaly detection per image.

As the state-of-the-art methods to which we compare do not provide detection thresholds, we adopt two strategies: (i) we compute an oracle-like threshold that maximizes the mIoU for the testing set, and (ii) we use a fair strategy that only uses training data to find the threshold. In the latter, the threshold is set to allow at most one false positive in each training image, as it would be analogous to setting NFA = 1 false alarm. As seen in the table below, our automatic thresholding strategy significantly outperforms all others, even when compared with their oracle-like threshold.

### Visual results

### Robustness

Additionally, it is important to note that our results obtained with the **oracle-like threshold and the automatic threshold are very close**. This demonstrates the validity of the proposed statistical derivation and the inter-scale independence achieved by the proposed architecture. To further illustrate this point and to show the robustness of the anomaly detection method with respect to the threshold on the NFA, the figure below depicts the *mIoU* for all the MVTec-AD categories as a function of − log(NFA). These graphs show that **the unsupervised threshold log(NFA) = 0 is always near the optimal point**, which is the oracle threshold, corresponding to the maximum mIoU reported in Table 1. Not only are the oracle and the automatic thresholds close, but the variation in mIoU is very low in a wide range of possible thresholds, showing that **this detection strategy is robust and parameter-free in practice.**

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