Ljiljana Platiša, Bart Goossens, Ewout Vansteenkiste, Subok Park, Brandon D. Gallas, Aldo Badano, and Wilfried Philips, "Channelized Hotelling observers for the assessment of volumetric imaging data sets," J. Opt. Soc. Am. A 28, 1145-1163 (2011)

Current clinical practice is rapidly moving in the direction of volumetric imaging.
For two-dimensional (2D) images, task-based medical image quality is often assessed
using numerical model observers. For three- dimensional (3D) images, however, these
models have been little explored so far. In this work, first, two novel designs of a
multislice channelized Hotelling observer (CHO) are proposed for the task of
detecting 3D signals in 3D images. The novel designs are then compared and evaluated
in a simulation study with five different CHO designs: a single-slice model, three
multislice models, and a volumetric model. Four different random background
statistics are considered, both Gaussian (noncorrelated and correlated Gaussian
noise) and non-Gaussian (lumpy and clustered lumpy backgrounds). Overall, the results
show that the volumetric model outperforms the others, while the disparity between
the models decreases for greater complexity of the detection task. Among the
multislice models, the second proposed CHO could most closely approach the volumetric
model, whereas the first new CHO seems to be least affected by the number of training
samples.

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The following notation applies: M, number of voxels in the
image; ${M}_{\mathrm{FOV}}$, number of voxels in the FOV (LB, CLB); ${\sigma}_{s}$, spread parameter of the 3D Gaussian signal; ${a}_{s}$, magnitude of the 3D Gaussian signal; ${\sigma}_{b}$, standard deviation of the 3D Gaussian kernel (CNB) or spread
parameter of the 3D Gaussian lump (LB); ${a}_{b}$, peak intensity level in the background image; $\overline{K}$, mean number of lumps in the FOV (LB, CLB); ${L}_{x}$, ${L}_{y}$, and ${L}_{z}$, characteristic lengths of the asymmetrical lumps in the
x, y, and z directions,
respectively (CLB).

Table 2

Parameters of the LG Channels^{
a
}

Background
Category

Signal Size

${a}_{u}$

${P}_{2\mathrm{D}}$

${P}_{3\mathrm{D}}$

WNB

${\sigma}_{s}=8$

12

3

4

CNB

${\sigma}_{s1}=8$

24

9

12

${\sigma}_{s2}=5$

21

11

12

${\sigma}_{s3}=3$

12

12

12

LB

${\sigma}_{s}=8$

18

15

15

CLB

${\sigma}_{s}=8$

24

5

6

For each image category and its related signal size, the parameters of the LG
channels are determined: the size of the channels, ${a}_{u}$; the number of 2D LG channels, ${P}_{2\mathrm{D}}$; and the number of 3D LG channels, ${P}_{3\mathrm{D}}$. The parameters of 2D and 3D LG channels are selected in the
experiments with ssCHO and vCHO models, respectively. The models are
investigated in the space of five families of LG channels defined by the value
of the channel spread parameter, ${a}_{u}=\{7,12,18,24,32\}$. For each family, the number of LG channels is varied in the
range of $P=1,\dots ,30$. The experiments are conducted with ${N}_{\text{tr}}=2000$ trainer pairs and ${N}_{\text{ts}}=1000$ tester pairs and for the second largest among four considered
values of signal magnitude ${a}_{s}$ given in Table 1. The
results of these experiments are illustrated in Fig. 3.

Table 3

MRMC Study Configurations^{
a
}

Background
Category

Number of Trainer Image Pairs (${N}_{\text{tr}}$)

Number of Readers (${N}_{\text{rd}}$)

WNB, CNB

${N}_{\text{tr}}=\{50,100,200,500,1000,2000\}$

${N}_{\text{rd}}=5$

${N}_{\text{tr}}=\{5000\}$

${N}_{\text{rd}}=2$

LB, CLB

${N}_{\text{tr}}=\{50,100,200,500,1000\}$

${N}_{\text{rd}}=5$

${N}_{\text{tr}}=\{2000\}$

${N}_{\text{rd}}=3$

The total number of each of WNB and CNB images is 11,000 image pairs, and the
total number of each of LB and CLB images is 7000 image pairs. For all study
configurations, the number of tester image pairs is fixed to ${N}_{\text{ts}}=1000$. No overlap exists between the trainer images and the tester
images.

Table 4

Terms of Eq. (17) for Three
Different Types of Model Observer Efficiency, η

Type of Efficiency

${\mathrm{SNR}}_{\text{curr}}$

${\mathrm{SNR}}_{\text{ref}}$

${\eta}_{\mathrm{CHO}}$

SNR of a given CHO

SNR of the IO

${\eta}_{{N}_{\text{tr}}}$

SNR of the CHO trained with ${N}_{\text{tr}}$ image pairs, ${N}_{\text{tr}}<5000$ (see Table 3)

SNR of the CHO trained with the maximum considered
number of trainer pairs, ${N}_{\text{tr}}=5000$

${\eta}_{\mathrm{ss},\mathrm{v}}$

SNR of the ssCHO

SNR of the vCHO

Table 5

Efficiency of CHO Models Applied on CNB Images with Different Spread of the
Signal: Efficiency of the CHO Model Relative to the IO Performance (${\mathit{\eta}}_{\mathrm{CHO}}$) and Efficiency of ssCHO Relative to the vCHO Performance (${\mathit{\eta}}_{\mathrm{ss},\mathrm{v}}$)^{
a
}

ssCHO

${\mathrm{msCHO}}_{a}$

${\mathrm{msCHO}}_{b}$

${\mathrm{msCHO}}_{c}$

vCHO

${\sigma}_{s}$

${a}_{s}$

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{ss},\mathrm{v}}$ (%)

8

0.25

69

85

86

91

$>100$

62

0.5

59

71

71

82

98

60

0.75

55

66

66

77

93

59

1

53

63

63

75

91

59

5

0.01

13

28

27

35

82

16

0.015

13

27

27

36

78

17

0.02

14

27

26

37

77

18

0.025

14

26

26

37

76

18

3

0.0025

12

36

35

46

88

13

0.0035

12

36

36

47

86

14

0.0045

12

36

35

47

85

14

0.0055

12

36

35

47

85

15

Three different values of signal spread parameter are considered: ${\sigma}_{s1}=8$, ${\sigma}_{s2}=5$, and ${\sigma}_{s3}=5$. For each ${\sigma}_{s}$, the exact same backgrounds are used and their lump spread
parameter is ${\sigma}_{b}=8$. For msCHO models, the efficiency for the ROI size of $R=11$ are given. The values of ${\eta}_{\mathrm{CHO}}$ and ${\eta}_{\mathrm{ss},\mathrm{v}}$ are calculated using Eq. (17) and as explained in Subsection 4C. The calculations are done for the MRMC configuration
with the number of trainer image pairs ${N}_{\text{tr}}=5000$.

Table 6

Efficiency of Five CHO Models for Different Levels of the Signal ${\mathit{a}}_{\mathit{s}}$ while the Number of Trainer Images Increase: ${\eta}_{{N}_{{\mathrm{tr}|a}_{s}}}$^{
a
}

${a}_{s}$

0.25

0.5

0.75

1

0.25

0.5

0.75

1

0.25

0.5

0.75

1

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ssCHO

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for vCHO

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

0.55

1.02

1.48

1.94

0.70

1.31

1.92

2.53

${N}_{\text{tr}}=50$

41

74

84

88

55

82

87

88

100

56

81

89

92

61

84

90

93

200

69

91

95

97

73

92

96

97

500

89

96

98

99

90

97

98

99

1000

96

99

99

100

94

99

99

100

2000

99

100

100

100

99

100

100

100

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{a}$

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{b}$

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{c}$

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

0.61

1.12

1.62

2.11

0.61

1.12

1.62

2.11

0.63

1.20

1.75

2.30

${N}_{\text{tr}}=50$

13

51

69

75

23

58

72

78

0

0

3

5

100

34

71

81

85

37

70

82

87

11

26

36

42

200

44

83

91

93

54

86

93

95

16

43

60

68

500

78

94

96

97

81

94

97

98

41

70

80

85

1000

88

97

98

99

89

97

98

99

61

85

92

94

2000

97

99

99

100

97

99

99

100

85

95

97

98

For CNB images, the efficiency of CHO models ssCHO, ${\mathrm{msCHO}}_{a}$, ${\mathrm{msCHO}}_{b}$, ${\mathrm{msCHO}}_{c}$, and vCHO, trained with fewer image pairs relative to their
performance for the largest considered number of trainer images, ${\eta}_{{N}_{\text{tr}|{a}_{s}}}$, are calculated using Eq. (17) and as explained in Subsection 4C. For three msCHO models, the efficiency for the ROI size
of $R=11$ is given.

Table 7

Efficiency of msCHO Models for Different-Sized ROIs while the Number of
Trainer Images Increase: ${\eta}_{{N}_{\mathrm{tr}|R}}$^{
a
}

R

3

5

11

64

3

5

11

64

3

5

11

64

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{a}$

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{b}$

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{c}$

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

1.49

1.49

1.62

1.73

1.49

1.49

1.62

1.73

1.60

1.65

1.75

—

${N}_{\text{tr}}=50$

84

83

69

19

82

81

72

25

55

36

3

—

100

88

87

81

49

87

86

82

52

77

68

36

—

200

95

94

91

68

95

94

93

73

88

80

60

—

500

97

97

96

88

97

97

97

87

94

89

80

—

1000

99

99

98

94

99

99

98

94

98

96

92

—

2000

100

100

99

98

100

100

99

98

99

99

97

—

For CNB images, the efficiency of msCHO models ${\mathrm{msCHO}}_{a}$, ${\mathrm{msCHO}}_{b}$, and ${\mathrm{msCHO}}_{c}$, trained with fewer image pairs relative to their performance
for the largest considered number of trainer images, ${\eta}_{{N}_{\text{tr}|R}}$, are calculated using Eq. (17) and as explained in Subsection 4C. In particular, the efficiency for the signal magnitude
of ${a}_{s}=0.75$ for four different ROI sizes, $R=\{3,5,11,64\}$, is presented. Here, $R=64$ implies that the CHO is applied to all slices in the
image.

The following notation applies: M, number of voxels in the
image; ${M}_{\mathrm{FOV}}$, number of voxels in the FOV (LB, CLB); ${\sigma}_{s}$, spread parameter of the 3D Gaussian signal; ${a}_{s}$, magnitude of the 3D Gaussian signal; ${\sigma}_{b}$, standard deviation of the 3D Gaussian kernel (CNB) or spread
parameter of the 3D Gaussian lump (LB); ${a}_{b}$, peak intensity level in the background image; $\overline{K}$, mean number of lumps in the FOV (LB, CLB); ${L}_{x}$, ${L}_{y}$, and ${L}_{z}$, characteristic lengths of the asymmetrical lumps in the
x, y, and z directions,
respectively (CLB).

Table 2

Parameters of the LG Channels^{
a
}

Background
Category

Signal Size

${a}_{u}$

${P}_{2\mathrm{D}}$

${P}_{3\mathrm{D}}$

WNB

${\sigma}_{s}=8$

12

3

4

CNB

${\sigma}_{s1}=8$

24

9

12

${\sigma}_{s2}=5$

21

11

12

${\sigma}_{s3}=3$

12

12

12

LB

${\sigma}_{s}=8$

18

15

15

CLB

${\sigma}_{s}=8$

24

5

6

For each image category and its related signal size, the parameters of the LG
channels are determined: the size of the channels, ${a}_{u}$; the number of 2D LG channels, ${P}_{2\mathrm{D}}$; and the number of 3D LG channels, ${P}_{3\mathrm{D}}$. The parameters of 2D and 3D LG channels are selected in the
experiments with ssCHO and vCHO models, respectively. The models are
investigated in the space of five families of LG channels defined by the value
of the channel spread parameter, ${a}_{u}=\{7,12,18,24,32\}$. For each family, the number of LG channels is varied in the
range of $P=1,\dots ,30$. The experiments are conducted with ${N}_{\text{tr}}=2000$ trainer pairs and ${N}_{\text{ts}}=1000$ tester pairs and for the second largest among four considered
values of signal magnitude ${a}_{s}$ given in Table 1. The
results of these experiments are illustrated in Fig. 3.

Table 3

MRMC Study Configurations^{
a
}

Background
Category

Number of Trainer Image Pairs (${N}_{\text{tr}}$)

Number of Readers (${N}_{\text{rd}}$)

WNB, CNB

${N}_{\text{tr}}=\{50,100,200,500,1000,2000\}$

${N}_{\text{rd}}=5$

${N}_{\text{tr}}=\{5000\}$

${N}_{\text{rd}}=2$

LB, CLB

${N}_{\text{tr}}=\{50,100,200,500,1000\}$

${N}_{\text{rd}}=5$

${N}_{\text{tr}}=\{2000\}$

${N}_{\text{rd}}=3$

The total number of each of WNB and CNB images is 11,000 image pairs, and the
total number of each of LB and CLB images is 7000 image pairs. For all study
configurations, the number of tester image pairs is fixed to ${N}_{\text{ts}}=1000$. No overlap exists between the trainer images and the tester
images.

Table 4

Terms of Eq. (17) for Three
Different Types of Model Observer Efficiency, η

Type of Efficiency

${\mathrm{SNR}}_{\text{curr}}$

${\mathrm{SNR}}_{\text{ref}}$

${\eta}_{\mathrm{CHO}}$

SNR of a given CHO

SNR of the IO

${\eta}_{{N}_{\text{tr}}}$

SNR of the CHO trained with ${N}_{\text{tr}}$ image pairs, ${N}_{\text{tr}}<5000$ (see Table 3)

SNR of the CHO trained with the maximum considered
number of trainer pairs, ${N}_{\text{tr}}=5000$

${\eta}_{\mathrm{ss},\mathrm{v}}$

SNR of the ssCHO

SNR of the vCHO

Table 5

Efficiency of CHO Models Applied on CNB Images with Different Spread of the
Signal: Efficiency of the CHO Model Relative to the IO Performance (${\mathit{\eta}}_{\mathrm{CHO}}$) and Efficiency of ssCHO Relative to the vCHO Performance (${\mathit{\eta}}_{\mathrm{ss},\mathrm{v}}$)^{
a
}

ssCHO

${\mathrm{msCHO}}_{a}$

${\mathrm{msCHO}}_{b}$

${\mathrm{msCHO}}_{c}$

vCHO

${\sigma}_{s}$

${a}_{s}$

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{CHO}}$ (%)

${\eta}_{\mathrm{ss},\mathrm{v}}$ (%)

8

0.25

69

85

86

91

$>100$

62

0.5

59

71

71

82

98

60

0.75

55

66

66

77

93

59

1

53

63

63

75

91

59

5

0.01

13

28

27

35

82

16

0.015

13

27

27

36

78

17

0.02

14

27

26

37

77

18

0.025

14

26

26

37

76

18

3

0.0025

12

36

35

46

88

13

0.0035

12

36

36

47

86

14

0.0045

12

36

35

47

85

14

0.0055

12

36

35

47

85

15

Three different values of signal spread parameter are considered: ${\sigma}_{s1}=8$, ${\sigma}_{s2}=5$, and ${\sigma}_{s3}=5$. For each ${\sigma}_{s}$, the exact same backgrounds are used and their lump spread
parameter is ${\sigma}_{b}=8$. For msCHO models, the efficiency for the ROI size of $R=11$ are given. The values of ${\eta}_{\mathrm{CHO}}$ and ${\eta}_{\mathrm{ss},\mathrm{v}}$ are calculated using Eq. (17) and as explained in Subsection 4C. The calculations are done for the MRMC configuration
with the number of trainer image pairs ${N}_{\text{tr}}=5000$.

Table 6

Efficiency of Five CHO Models for Different Levels of the Signal ${\mathit{a}}_{\mathit{s}}$ while the Number of Trainer Images Increase: ${\eta}_{{N}_{{\mathrm{tr}|a}_{s}}}$^{
a
}

${a}_{s}$

0.25

0.5

0.75

1

0.25

0.5

0.75

1

0.25

0.5

0.75

1

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ssCHO

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for vCHO

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

0.55

1.02

1.48

1.94

0.70

1.31

1.92

2.53

${N}_{\text{tr}}=50$

41

74

84

88

55

82

87

88

100

56

81

89

92

61

84

90

93

200

69

91

95

97

73

92

96

97

500

89

96

98

99

90

97

98

99

1000

96

99

99

100

94

99

99

100

2000

99

100

100

100

99

100

100

100

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{a}$

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{b}$

${\eta}_{{N}_{\text{tr}|{a}_{s}}}$ (%) for ${\mathrm{msCHO}}_{c}$

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

0.61

1.12

1.62

2.11

0.61

1.12

1.62

2.11

0.63

1.20

1.75

2.30

${N}_{\text{tr}}=50$

13

51

69

75

23

58

72

78

0

0

3

5

100

34

71

81

85

37

70

82

87

11

26

36

42

200

44

83

91

93

54

86

93

95

16

43

60

68

500

78

94

96

97

81

94

97

98

41

70

80

85

1000

88

97

98

99

89

97

98

99

61

85

92

94

2000

97

99

99

100

97

99

99

100

85

95

97

98

For CNB images, the efficiency of CHO models ssCHO, ${\mathrm{msCHO}}_{a}$, ${\mathrm{msCHO}}_{b}$, ${\mathrm{msCHO}}_{c}$, and vCHO, trained with fewer image pairs relative to their
performance for the largest considered number of trainer images, ${\eta}_{{N}_{\text{tr}|{a}_{s}}}$, are calculated using Eq. (17) and as explained in Subsection 4C. For three msCHO models, the efficiency for the ROI size
of $R=11$ is given.

Table 7

Efficiency of msCHO Models for Different-Sized ROIs while the Number of
Trainer Images Increase: ${\eta}_{{N}_{\mathrm{tr}|R}}$^{
a
}

R

3

5

11

64

3

5

11

64

3

5

11

64

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{a}$

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{b}$

${\eta}_{{N}_{\text{tr}|R}}$ (%) for ${\mathrm{msCHO}}_{c}$

${\mathrm{SNR}}_{{N}_{\text{tr}}=5000}$

1.49

1.49

1.62

1.73

1.49

1.49

1.62

1.73

1.60

1.65

1.75

—

${N}_{\text{tr}}=50$

84

83

69

19

82

81

72

25

55

36

3

—

100

88

87

81

49

87

86

82

52

77

68

36

—

200

95

94

91

68

95

94

93

73

88

80

60

—

500

97

97

96

88

97

97

97

87

94

89

80

—

1000

99

99

98

94

99

99

98

94

98

96

92

—

2000

100

100

99

98

100

100

99

98

99

99

97

—

For CNB images, the efficiency of msCHO models ${\mathrm{msCHO}}_{a}$, ${\mathrm{msCHO}}_{b}$, and ${\mathrm{msCHO}}_{c}$, trained with fewer image pairs relative to their performance
for the largest considered number of trainer images, ${\eta}_{{N}_{\text{tr}|R}}$, are calculated using Eq. (17) and as explained in Subsection 4C. In particular, the efficiency for the signal magnitude
of ${a}_{s}=0.75$ for four different ROI sizes, $R=\{3,5,11,64\}$, is presented. Here, $R=64$ implies that the CHO is applied to all slices in the
image.