Bispectrum analysis energy feature maps of ultrasound RF signals

The high-order spectral analysis method analyzes the spectral characteristics of a signal by introducing high-order statistics, such as third-order moments and fourth-order moments, reflecting the nonlinear characteristics and phase correlation of the signal. Assuming x_bs(n_bs) represents a certain RF signals sequence, where n_bs=1, …,256 is the number of samples. Bispectrum of RF signals x_bs(n_bs) is defined as third-order cumulant C₃ Fourier transform BS, where the third-order cumulant C₃ is (20):

Among them, k_bs and l_bs are time delays, so the bispectrum BS of the RF signals is:

Among them, f₁ and f₂ represent the horizontal and vertical frequency axes.

Figure 1 shows the local bispectrum analysis maps of the central region of six breast tumors. They exhibit differences in distribution and have regularity. Some patients’ maps, like Figures 1A, B display centralized patterns, whereas others, such as Figures 1C-F show scattered patterns. The scattered mode indicates a wide frequency distribution of echo signals. Both scattered and centralized maps contain various rich distribution patterns. For instance, Figures 1C, D showcase obvious second harmonic components. In other maps, although there may not be evident second harmonics, significant distribution differences exist for other frequency components around the center frequency. These differences in frequency component distribution within local bispectrum analysis maps are closely linked to the internal microstructure of breast tumors, potentially aiding in breast cancer detection.

Construction of bispectrum analysis energy features (A-F) are local bispectrum analysis maps of the central region of six breast tumors. (G) is a complete bispectrum analysis map of RF signal segment x_bs(n_bs). A0 region of (H) represents the diagram of the complete bispectrum analysis, and the non-overlapping region S0 is marked. The S0 region in (I) is divided into three equal parts starting from the origin along the direction of the f₂ axis, obtaining the S1, S2, and S3 regions, respectively. The bispectrum analysis energies of the non-overlapping regions of the S1, S2, S3, and S0 regions are calculated respectively to form four new bispectrum analysis energy features of RF signal segment x_bs(n_bs).

Based on the distribution characteristics mentioned above, we conducted frequency subdivision in the bispectrum analysis map to design four new features for each RF signal segment x_bs(n_bs), representing four different frequency components. Figure 1G illustrates the complete bispectrum analysis map with overlaps. The A0 region in Figure 1H represents the complete bispectrum analysis diagram, with a non-overlapping region S0 labeled. In Figure 1I, the diagram is divided into A1, A2, and A3 regions, with equal divisions in the horizontal direction. The overlap between S0 and A1 forms S1 (a low-frequency non-overlapping region), and the overlap between S0 and A3 forms S3 (a high-frequency non-overlapping region). S0 and A2 overlap to form S2, which encompasses a portion of both low-frequency and high-frequency components. The S0 region includes all frequency components of the S1, S2, and S3 regions. By calculating the energy of S1, S2, S3, and S0 regions, we obtain four new bispectrum analysis energy features for each RF signal segment xbs(nbs). Subsequently, we extract these four new bispectrum analysis energy features from all RF signal segments in the key frame for each patient, resulting in each patient’s four bispectrum analysis energy feature maps: BS_S1, BS_S2, BS_S3, and BS_A.

Figures 2A-D display the bispectrum analysis energy feature maps BS_S1, BS_S2, BS_S3, and BS_A of a benign patient, respectively. Similarly, Figures 3A-D show the bispectrum analysis energy feature maps BS_S1, BS_S2, BS_S3, and BS_A of a malignant patient, respectively. Although visually similar, a closer examination of image details revealed subtle differences among the four feature maps for each patient. To quantitatively analyze these differences and correlations, gray histograms, mutual information (MI), and root mean square error (RMSE) were utilized. For a benign patient, Figures 2E, G depicted the statistical results of gray histograms, MI, and RMSE for the four bispectrum analysis energy feature maps. The histogram provided a visual representation of pixel value distribution, with BS_A being mostly covered by the histograms of BS_S1 (blue highlighted areas), BS_S2 (rose highlighted areas), and BS_S3 (red highlighted areas). The feature maps after frequency subdivision exhibited regular pixel distribution characteristics. The RMSE in Figure 2G indicated the similarity between feature maps, with higher values suggesting greater distinctiveness and necessity for classification. The average RMSE between BS_S1 and the other three feature maps was 3.451, whereas the average RMSE between BS_S3 and the other three feature maps was 2.388. Both BS_S1 and BS_S3 had higher average RMSE compared with BS_S2 (1.972) and BS_A (2.067), highlighting their representativeness and importance. MI in Figure 2F measured the strength of the relationship between random variables. The average MI between BS_S1 and the other three feature maps was the highest at 0.981, followed by BS_S3 at 0.979. This indicated that BS_S1 and BS_S3 contained the majority of information from the other feature maps.

Bispectrum analysis energy feature maps of a benign patient and their differences analysis. (A–D) are four types of bispectrum analysis energy feature maps of a benign patient with breast tumor. (E) The overlapping histogram results of four bispectrum analysis energy feature maps of BS_S1, BS_S2, BS_S3, and BS_A. (F) The mutual information (MI) between each two of the four new bispectrum analysis energy feature maps. (G) The root mean square error (RMSE) between each two of the four new bispectrum analysis energy feature maps.

Bispectrum analysis energy feature maps of a malignant patient and analysis of their differences. (A-D) are four types of bispectrum analysis energy feature maps of a malignant patient with breast tumor. (E) The overlapping histogram results of four bispectrum analysis energy feature maps of BS_S1, BS_S2, BS_S3, and BS_A. (F) The mutual information (MI) between each two of the four new bispectrum analysis energy feature maps. (G) The root mean square error (RMSE) between each two of the four new bispectrum analysis energy feature maps.

In summary, the quantitative analysis results demonstrated that BS_S1 and BS_S3 provided greater information content and differences compared with BS_A and BS_S2. Similar observations were made for malignant patients, where BS_S1 represented the low-frequency energy and BS_S3 represented the high-frequency energy. Frequency subdivision contributed to enhanced information richness and value of bispectrum analysis energy feature maps, enabling a more comprehensive analysis of microscopic and macroscopic tissue information.