## 3 Results and Discussion  
The performance of the fractal convolution filter was evaluated by the ImageJ software [30] with 40 digital mammograms. All the processing steps were implemented as a simple macro inside the program. The mathematical and logical operators were provided by the host software package. Mammograms were provided by the Department of Radiology of the Puerta de Hierro University Hospital after careful data anonymization and ethical consent, and from the database called the Breast Cancer Digital Repository (BCDR), created by a consortium formed between the University of Oporto and the Centro Extremeno de Tecnologias Avanzadas (CETA) [31]. The mammograms with dense background tissues and those showing the presence of microcalcifications were previously selected by a radiologist. For illustrative purposes, three representative cases were selected for the following figures. The first is an example of a dense mammogram. The second shows localized dense tissues. The last depicts dense tissue placed above microcalcifications. Figure 5 offers a typical example of a dense mammogram with microcalcifications. Enhanced images were achieved by convolving the original  
Figure 4: Detail of the Lena image with 8 \(\times\) zoom using the nearest neighbor and fractal convolution, respectively, and their histograms. Figure 3: The original Lena image and directional edge filtering using fractal masks (512 \(\times\) 512, 8 bit). mammogram with a set of 5 \(\times\) 5 fractals rotated in angular increments of 45, as previously explained. The result is shown on the right of Figure 5.  
After determining the ROI in the enhanced mammogram (Figure 6), it is zoomed by a factor of 8\(\times\). As seen in Figure 7a, the original zoomed image is pixilated, but mask convolution eliminates these artifacts. The result is shown in Figure 7b. Finally, mathematical operators (logarithm and exponential) were applied to copies of the processed ROI and their results were combined by the XOR operator. As shown in Figure 7c, the shape and dimensions of the microcalcification are well-enhanced. In Figure 8, the largest microcalcification was enlarged to detail its shape. The result of this segmentation process can be used for automatic classification, pattern matching, and machine learning postprocessing steps. The processing steps are shown in Figure 8a-c.  
Figure 5: Original and enhanced mammograms. Figure 6: Region of Interest (ROI) selection. Figure 9 shows a mammogram with localized-dense tissues. Albeit somewhat difficult, small calcifications can be noted on the left of the original image. They can be more clearly observed on the right of the enhanced image, and its clustered pattern organization is also noticeable. The possible malignant region is located and indicated in Figure 10. Another example of a detailed lesion is represented in Figure 11. A study based on the application of the presented algorithms allows the lesion to be considerably enhanced. Another example of a complex mammogram is presented in Figure 12, where dense tissue is located above the microcalcifications that are very difficult to visualize. The enhanced image is shown on the right of Figure 12. The final result reveals a significant number of microcalcifications. Figure 8: Detail of the largest microcalcification in Figure 7. (**a**) Isolated microcalcification from Figure 7a, (**b**) Fractal convolved and contrast enhanced microcalcification, (**c**) XOR segmented microcalcification. Figure 3:0 Detail of the ROI selected in Figure 3:1 (**a**) Original zoomed ROI, (**b**) Fractal convolved and contrast enhanced ROI, (**c**) XOR segmented ROI. Figure 3:2 The original and enhanced mammograms. As in the above-mentioned cases, analysis is focused on the ROI so that each microcalcification can be characterized.