The Contrast Order: An Order-Based Image Quality Criterion for Nonlinear Beamformers

TL;DR

This work introduces the contrast order (CO), an order-based image quality criterion defined as $\mathrm{CO}[A,B]=\mathbb{E}[\operatorname{sign}(A-B)]$, which is invariant to strictly monotonic transformations and provides a principled, sign-preserving measure of ROI separability for nonlinear ultrasound beamformers. Accompanying CO, the effective contrast ratio (ECR) calibrates CO to Rayleigh speckle statistics via $\mathrm{ECR}[A,B]=\sqrt{\dfrac{1+\mathrm{CO}[A,B]}{1-\mathrm{CO}[A,B]}}$, yielding a familiar, distortion-insensitive metric that coincides with traditional CR under ideal Rayleigh conditions. The paper derives an unbiased, variance-bounded estimator for CO from finite ROI samples and demonstrates through simulations and PICMUS data that CO/ECR remain robust under dynamic range transformations, unlike CR/CNR which are sensitive to display mappings and nonlinear remapping. By comparing a broad set of beamformers, CO/ECR provide consistent, interpretable rankings and offer a practical framework for evaluating modern nonlinear beamformers without confounding effects from image value remapping. The approach has potential to improve cross-method comparisons and guide the design of beamforming pipelines toward genuine information gain, rather than cosmetic changes in image statistics.

Abstract

Many modern ultrasound beamformers report improved image quality when evaluated using classical criteria like the contrast ratio and contrast-to-noise ratio, which are based on summary statistics of regions of interest (ROIs). However, nonlinear beamformers and post-processing methods can substantially alter these statistics, raising concerns that the reported improvements may reflect changes in dynamic range or remapping rather than a reflection of true information gain, such as clutter suppression. New criteria like the generalized contrast-to-noise ratio (gCNR) address these concerns, but rely on noisy estimates of the underlying distribution. To address this, we introduce a new image quality criterion, called the contrast order (CO), defined as the expected value of the sign of the difference in brightness between two ROIs. The CO is invariant under all strictly monotonic transformations of the image values, as it depends only on their relative ordering, and is interpretable as the probability that one ROI is brighter than the other minus the probability that it is darker. Unlike the gCNR, the CO has a simple unbiased estimator whose variance decreases with the number of samples in each ROI. We further propose the effective contrast ratio (ECR), which calibrates the contrast order to the familiar contrast ratio such that the two coincide under ideal Rayleigh-speckle statistics. Together, the CO and ECR provide order- and sign-preserving, dynamic-range-invariant criteria for evaluating lesion contrast, offering a principled alternative to classical and newer image quality criteria when assessing modern beamformers.

Figures (7)

• Figure 1: (a) A 2D FOV $\mathcal{X}$ visualized in 3D space. (b) An image $\phi$ assigns values in $\mathcal{A}$ to each coordinate in $\mathcal{X}$. (c) The histogram of $\phi$ gives the relative occurrence of each image value in $\mathcal{A}$.
• Figure 2: The lesion detectability problem is illustrated. (a) Given the image function $\phi$ from Fig. \ref{['fig:bmode']}b, we select two regions of interest (ROIs) $\mathcal{X}_A$ and $\mathcal{X}_B$. (b) The histograms of $\phi$ in $\mathcal{X}_A$ and $\mathcal{X}_B$ are obtained as $f_A$ and $f_B$, respectively.
• Figure 3: Contrast lesions were simulated using Field II. (a) True changes in lesion contrast were simulated by changing the echogenicity of the lesion. (b) "Cosmetic" changes in lesion contrast were simulated using power compression. For both cases, the CR, ECR, CO, (signed) gCNR, and (signed) CNR are plotted. (c) The CR and ECR behave identically when measuring true changes in lesion contrast. The CO varies from -1 to +1, as does the gCNR, whereas the CNR is unbounded but follows the same trend. (d) The ECR of a $-6$ dB lesion is invariant under power compression, whereas the CR changes. Similarly, the CO and gCNR are invariant, whereas the CNR varies.
• Figure 4: Multiple beamformers were used to reconstruct grayscale targets from the PICMUS experimental contrast phantom dataset liebgott2016plane: delay-and-sum (DAS), a simple dynamic range transformation (DRT), coherence factor (CF), phase coherence factor (PCF), generalized coherence factor (GCF), Capon's minimum variance (MV), eigenspace-based minimum variance (EBMV), delay-multiply-and-sum (DMAS), short-lag spatial coherence (SLSC), a simple low-pass filter (LPF), and receive spatial compounding (SPC). The red inner circle denotes the lesion ROI, the yellow circle the lesion location, and green ring the background ROI. Note the wide variability in the image contrasts and textures.
• Figure 5: The CR, ECR, (signed) CNR, and (signed) gCNR are plotted for each beamformer for each grayscale target in Fig. \ref{['fig:picmus_images']}. Observe that the CR and CNR of the purely cosmetic DRT is different from DAS, showing their volatility. The CR and CNR, which depend on the image statistics, disagree with the trends of the ECR and gCNR, particularly for the $-6$ dB lesion.

Theorems & Definitions (9)

• Theorem 1: Invariance
• Corollary 2
• Proposition 3
• Proposition 4
• Theorem 5
• Proposition 6
• Theorem 7
• Proposition 8
• Proposition 9