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@@ -2,92 +2,130 @@
 <html>
 
 <head>
+  <meta name="viewport" content="width=device-width, initial-scale=1.0">
   <title>FRE: A Fast Method For Anomaly Detection And Segmentation</title>
   <link rel="stylesheet" href="main.css">
-  <meta name="viewport" content="width=device-width, initial-scale=1.0">
+  <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+  <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
 </head>
 
 <body>
   <article>
     <header>
+      <nav>
+        <a href="https://github.com/IntelLabs/dfm/tree/ICIP2022">ICIP 2022 Code</a> |
+        <a href="https://github.com/IntelLabs/dfm/tree/main">BMVC 2023 Code</a>
+      </nav>
       <h1>FRE: A Fast Method For Anomaly Detection And Segmentation</h1>
-      <p>Ibrahima J. Ndiour, Nilesh A. Ahuja, Utku Genc and Omesh Tickoo<br />Intel Labs</p>
+      <p class="centered">Ibrahima J. Ndiour, Nilesh A. Ahuja, Utku Genc and Omesh Tickoo<br />Intel Labs</p>
     </header>
-
-    <section class="main">
-      <h2>Abstract</h2>
-      <p>This work proposes a fast and principled method for unsupervised anomaly detection and segmentation. Our method
-        operates under the assumption of having access solely to anomaly-free training data while aiming to identify
-        anomalies of an arbitrary nature on test data.
-        We make our code available on <a href="https://github.com/intellabs/dfm">github</a>.
-      </p>
-    </section>
 
     <section class="main">
-      <h2>Contributions</h2>
-      <p>The principal contributions of the work include:
-        <ul>
-          <li>a generalized approach that utilizes a shallow linear autoencoder as a principled out-of-distribution detection method operating in the feature space produced by a pre-trained DNN.</li>
-          <li>a solid theoretical foundation for the method establishing the feature reconstruction error (FRE) as a principled measure of uncertainty.</li>
-          <li>simultaneous solving of image-level anomaly detection and pixel-level anomaly segmentation.</li>
-          <li>multiple implementation strategies addressing concerns related to memory, computational complexity, and dataset size. </li>
-          <li>extensive experimentation showing state-of-the-art quality, as well as speed, robustness and insensitivity to parameterization.</li>
-        </ul>        
-      </p>
+      <h2>Introduction</h2>
+      <p>We present a fast and principled approach to address the challenge of visual anomaly detection and segmentation. Our method operates under the assumption of having access solely to anomaly-free training data while aiming to identify anomalies of an arbitrary nature on test data. We build upon prior research and present a generalized approach that utilizes a shallow linear autoencoder to perform out-of-distribution detection on the intermediate features generated by a pre-trained deep neural network. More specifically, we compute the feature reconstruction error (FRE) and establish it as principled measure of uncertainty. We rigorously connect our technique to the theory of linear auto-associative networks to provide a solid theoretical foundation and to offer multiple practical implementation strategies. Furthermore, extending the FRE concept to convolutional layers, we derive FRE maps that provide precise pixel-level spatial localization of the anomalies within images, effectively achieving segmentation. Extensive experimentation demonstrates that our method outperforms the current state of the art. It excels in terms of speed, robustness, and remarkable insensitivity to parameterization. We make our code available on <a href="https://github.com/intellabs/dfm">github</a>.</p>
     </section>
 
     <section class="main">
-      <h2>Preliminaries: linear auto-associative networks
+      <h2>Proposed Approach
       </h2>
 
       <div class="figure-container">
-      <figure>
-        <img src="aan.png" alt="AAN" width="150px">
-        <figcaption>Auto-associative network.</figcaption>
-      </figure>
 
-      <figure>
-        <img src="subspace.png" alt="Subspace" width="150px">
-        <figcaption>FRE visualization as distance to the principal subspace.</figcaption>
-      </figure>
+        <figure>
+          <img src="fre_pipeline.png" alt="Pipeline" width="500px">
+          <figcaption>Block diagram of our proposed approach.
+          </figcaption>
+        </figure>
 
-    </div>
+      </div>
+
+      <p>At training: we pass the training data through the backbone network for a single forward pass and consider the
+        intermediate (training) features at a given layer k. Subsequently, we train a (tied) shallow linear autoencoder
+        with such features.</p>
+      <p>During inference: the autoencoder is applied to a test feature to obtain the reduced-dimension embedding, and
+        its reconstruction back into the original (feature) space. The feature reconstruction error (FRE) is then
+        calculated and used as an uncertainty score:
+      </p>
       <p>
-        <ul>
-          <li>The auto-associative network performs orthogonal projection onto the subspace spanned by the first principal eigenvectors of a covariance matrix associated with the training patterns. </li>
-          <li>How does a neural network perform when exposed to a pattern never seen previously?</li>
-          <li>In the auto-associative case, a precise quantitative answer can be given: the distortion on a new pattern is exactly given by its distance to the subspace generated by the first p eigenvectors of the data covariance matrix.</li>      
-        </ul>     
-
+        $$\begin{equation}
+        FRE(\mathbf{u}, \mathcal{T}) = \mathbf{e} \triangleq \mathbf{u}-\mathbf{\hat{u}} = \mathbf{u}-(\mathcal{T}^{inv}
+        \circ \mathcal{T})\mathbf{u}.
+        \end{equation}$$
+      </p>
+
+      <p>To derive the segmentation map, we perform channel-wise averaging on the FRE (re-arranged as a tensor) in order
+        to accumulate the FRE errors along the channel dimension. This produces a single-channel FRE anomaly map M,
+        where higher intensity regions correspond to anomalies.
+      </p>
+
+      <p>The following detection score is highly effective at discriminating between normal and anomalous samples:
+      </p>
+
+      <p>
+        $$\begin{equation*}
+        \mathbf{M}_k(i,j) = \frac{1}{C_k}\sum_{c=1}^{C_k} \mathbf{e}(c,i,j)
+        %, i\in {1, 2, \dots, W_k}, i\in {1, 2, \dots, W_k}
+        \end{equation*}$$
       </p>
     </section>
 
+
     <section class="main">
-      <h2>Proposed Approach
+      <h2>Results
       </h2>
 
+      <h3>Detection results</h3>
+
       <div class="figure-container">
+        <figure>
+          <img src="det_table1.png" alt="Detection table 1">
+          <figcaption>MVTec dataset (AUROC metric ↑).</figcaption>
+        </figure>
+      </div>
 
-      <figure>
-        <img src="fre_pipeline.png" alt="Pipeline" width="400px">
-        <figcaption>Block diagram of our proposed approach.
-        </figcaption>
-      </figure>
+      <div class="figure-container">
+        <figure>
+          <img src="det_table2.png" alt="Detection table 2">
+          <figcaption>Magnetic Tile dataset (AUROC metric ↑). </figcaption>
+        </figure>
+      </div>
 
+      <div class="figure-container">
+        <figure>
+          <img src="det_table3.png" alt="Detection table 3">
+          <figcaption>MVTec dataset for FRE across backbones (AUROC metric ↑). </figcaption>
+        </figure>
+      </div>
+
+      <h3>Segmentation results</h3>
+
+      <div class="figure-container">
+        <figure>
+          <img src="seg_table1.png" alt="Segmentation table 1" width="500px">
+          <figcaption>Pixel-AUROC and PRO metrics on MVTec dataset.
+          </figcaption>
+        </figure>
+      </div>
+
+      <div class="figure-container">
+        <figure>
+          <img src="results.png" alt="Segmentation results" width="500px">
+          <figcaption>From left to right, each set of four images comprises: original image, ground truth segmentation mask, anomaly heatmap using FRE (our method) from a single layer, anomaly heatmap using FRE from three layers.</figcaption>
+        </figure>
+      </div>
+
+      <div class="figure-container">
+        <figure>
+          <img src="seg_table2.png" alt="Segmentation table 2" width="500px">
+          <figcaption>Pixel-wise AUROC and PRO metrics on MVTec for FRE across backbones.
+          </figcaption>
+        </figure>
       </div>
 
-      <p>
-        <ul>
-          <li>The auto-associative network performs orthogonal projection onto the subspace spanned by the first principal eigenvectors of a covariance matrix associated with the training patterns. </li>
-          <li>How does a neural network perform when exposed to a pattern never seen previously?</li>
-          <li>In the auto-associative case, a precise quantitative answer can be given: the distortion on a new pattern is exactly given by its distance to the subspace generated by the first p eigenvectors of the data covariance matrix.</li>      
-        </ul>     
-
-      </p>
     </section>
-    
+
     <footer>
-      <p>November 2023</p>
+      <p class="centered">November 2023</p>
     </footer>
   </article>
 </body>

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@@ -1,21 +1,43 @@
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+    padding: 2%;
+}
+
+p {
+    text-align: justify;
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+
+p.centered { 
+    text-align: center;
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-h2 {
+footer {
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-    display: inline-block;
+h2, h3 {
+    text-align: center;
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+
+figcaption {
+    font-size: small;
+    font-style: italic;
+    text-align: center;
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+
+img {
+    max-width:100%;
+    height: auto;
+    margin: auto;
+    display: block;
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diff --git a/docs/results.png b/docs/results.png
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