Factors affecting the normality of channel outputs of channelized model observers: an investigation using realistic myocardial perfusion SPECT images

Abstract

The channelized Hotelling observer (CHO) uses the first- and second-order statistics of channel outputs under both hypotheses to compute test statistics used in binary classification tasks. If these input data deviate from a multivariate normal (MVN) distribution, the classification performance will be suboptimal compared to an ideal observer operating on the same channel outputs. We conducted a comprehensive investigation to rigorously study the validity of the MVN assumption under various kinds of background and signal variability in a realistic population of phantoms. The study was performed in the context of myocardial perfusion SPECT imaging; anatomical, uptake (intensity), and signal variability were simulated. Quantitative measures and graphical approaches applied to the outputs of each channel were used to investigate the amount and type of deviation from normality. For some types of background and signal variations, the channel outputs, under both hypotheses, were non-normal (i.e., skewed or multimodal). This indicates that, for realistic medical images in cases where there is signal or background variability, the normality of the channel outputs should be evaluated before applying a CHO. Finally, the different degrees of departure from normality of the various channels are explained in terms of violations of the central limit theorem.

Keywords: channelized Hotelling observer; image quality; model observers.

Fig. 1

Sixty-four bin histogram plots of the reconstructed counts per unit volume (counts/cm³) in the different organs.

Fig. 2

The images are noise-free short-axis postprocessed images for different defects and phantoms. The arrows indicate the defects’ position, which were generated with a severity of 100% to aid visualization. The images shown in (a)–(d) are from the male phantom with the smallest value for all three anatomical parameters. Images (a) and (b) show anterolateral defects with extents of 25% and 5%, respectively. Images (c) and (d) show inferior defects with extents of 25% and 5%, respectively. The images shown in (e) and (f) have an anterolateral defect with an extent of 25%, where (e) is from the male phantom with the largest value for all three anatomical parameters and (f) is from the female phantom with the smallest value for all three anatomical parameters. The images shown in (g) and (h) have an inferior defect with an extent of 25%, where (g) is from the male phantom with the largest value for all three anatomical parameters and (h) is from the female phantom with the smallest value for all three anatomical parameters.

Fig. 3

The six channels used in this work. The leftmost column represents the lowest frequency channel (channel 1). The channel’s start frequency and width increase from left to right. The rows are (a) the frequency domain channels, (b) the spatial domain channels, and (c) the horizontal profiles through the origin of the spatial channels as indicated by the line in the leftmost image in (b), where the horizontal axis is the pixel number and the vertical axis is the pixel value.

Fig. 4

Histogram plots of the channel outputs when neither signal nor anatomical variability was included. The horizontal axis is the channel output intensity and the vertical axis is the frequency of occurrence. The columns represent the outputs from the six channels as defined in Fig. 3. The rows are (a) without uptake variability and (b) with uptake variability. Sixty-four histogram bins were used.

Fig. 5

Q–Q plots comparing the distributions of standardized channel outputs and the theoretical standard normal distribution when neither signal nor anatomical variability was included. The horizontal axis represents the quantiles of standard normal distribution; the vertical axis is the quantiles of the standardized channel outputs. The columns represent the outputs from the six channels defined in Fig. 3. Rows (a) and (b) are without uptake variability, where plots in (a) represent defect-absent and (b) defect-present data. Rows (c) and (d) are with uptake variability, where plots in (c) represent defect-absent and (d) defect-present data.

Fig. 6

Histogram plots of the channel outputs with anatomical variability and without signal variability. The axes are as described in Fig. 4. The columns represent the outputs from the six channels defined in Fig. 3. The rows are histogram plots for the cases of (a) the two male phantoms with different sizes, (b) two different genders, and (c) all 54 phantoms. The histograms in rows (a) and (b) used 16 bins while those in row (c) used 64 due to the larger number of feature vectors.

Fig. 7

Q–Q plots comparing the distributions of standardized channel outputs with the theoretical standard normal distribution with anatomical variability and without signal variability. The axes are as described in Fig. 5. The columns represent the outputs from the six channels defined in Fig. 3. Rows (a) and (b) are for the case of the two male phantoms with different sizes, where plots in (a) represent defect-absent and (b) defect-present data. Rows (c) and (d) are for the case of the two phantoms with different genders, where plots in (c) represent defect-absent and (d) defect-present data.

Fig. 8

As Fig. 7 for the case of 54 phantoms, where plots in (a) represent defect-absent and (b) defect-present data.

Fig. 9

Histogram plots of the channel outputs with signal location variability and without anatomical variability. The extent and severity of the defects were both equal to 25%. The axes are as described in Fig. 4. The columns represent the outputs from the six channels defined in Fig. 3. The rows are from (a) the anterolateral, (b) inferior, and (c) the mixture of anterolateral and inferior defects. Sixty-four histogram bins were used.

Fig. 10

Histogram plots of the channel outputs with signal severity variability and without anatomical variability. The extent of the defects was 25% and they were located in the anterolateral wall. The axes are as described in Fig. 4. The columns represent the outputs from the six channels defined in Fig. 3. The rows are from (a) the mixture of defect severities of 10% and 25% and (b) 25% and 50%. Sixty-four histogram bins were used.

Fig. 11

Q–Q plots comparing the distributions of standardized channel outputs with the theoretical standard normal distribution with signal severity variability and without anatomical variabilities. The extent of the defects was 25% and they were located in the anterolateral wall. The axes are as described in Fig. 5. The columns represent the outputs from the six channels defined in Fig. 3. The rows are for defect-present data, having a mixture of defect severities of (a) 10% and 25% and (b) 25% and 50%.

Fig. 12

Histogram plots of the channel outputs with signal extent variability and without anatomical variability. The defects’ extents were 5% and 25%. They were located at the anterolateral wall. The axes are as described in Fig. 4. The columns represent the outputs from the six channels defined in Fig. 3. The rows represent the (a) 25% and (b) 50% severity cases. Sixty-four histogram bins were used.

Fig. 13

Q–Q plots comparing the distributions of standardized channel outputs with the theoretical standard normal distribution with signal extent variability and without anatomical variability. The defects’ extents were 5% and 25%. They were located at the anterolateral wall. The axes are as described in Fig. 5. The columns represent the outputs from the six channels defined in Fig. 3. The rows are for the defect-present data having severities of (a) 25% and (b) 50%.

Fig. 14

A schematic illustrating the mixture of two unimodal distributions. The rows are from the case of: (a) $|m_{\mathrm{diff}}|>s_{\text{sum}}$: the resulting distribution is thus bimodal and (b) $|m_{\mathrm{diff}}|<s_{\text{sum}}$: the resulting distribution is thus unimodal.

Fig. 15

(a) A noise-free short-axis image of a male phantom with small body, heart, and subcutaneous adipose thickness. The arrows indicate the four pixel locations used to compute the histograms. The four histograms from locations 1 to 4 are shown from left to right. Plots in (b) and (c) represent the histograms of the pixels (b) before windowing and (c) after windowing. The variations in pixel values are due to noise and uptake variations.

Fig. 16

Histogram plots of the channel outputs with uptake variability for different cutoffs. The axes are as described in Fig. 4. The columns represent the outputs from the six channels defined in Fig. 3. The rows are from cutoffs of (a) 0.08 and (b) $0.24\text{cycles}/\text{pixel}$. Sixty-four histogram bins were used.

Fig. 17

Q–Q plots comparing the distributions of standardized channel outputs with the theoretical standard normal distribution for different cutoffs. The axes are as described in Fig. 5. The columns represent the outputs from the six channels defined in Fig. 3. The rows are for defect-present data with uptake variability having cutoffs of (a) 0.08 and (b) $0.24\text{cycles}/\text{pixel}$.

Fig. 18

Q–Q plots comparing the distributions of standardized equally weighted channel outputs with the theoretical standard normal distribution for defect-absent class. The axes are as described in Fig. 5. The left and the right columns represent the cutoffs 0.08 and $0.24\text{cycles}/\text{pixel}$, respectively. The rows are from the cases of: (a) and (b) no uptake variability before windowing, (c) and (d) no uptake variability after windowing, (e) and (f) with uptake variability before windowing, and (g) and (h) with uptake variability after windowing.