Younes Benkarroum, Gabor T. Herman, and Stuart W. Rowland, "Blob parameter selection for image representation," J. Opt. Soc. Am. A 32, 1898-1915 (2015)

A technique for optimizing parameters for image representation using blob basis functions is presented and demonstrated. The exact choice of the basis functions significantly influences the quality of the image representation. It has been previously established that using spherically symmetric volume elements (blobs) as basis functions, instead of the more traditional voxels, yields superior representations of real objects, provided that the parameters that occur in the definition of the family of blobs are appropriately tuned. The technique presented in this paper makes use of an extra degree of freedom, which has been previously ignored, in the blob parameter space. The efficacy of the resulting parameters is illustrated.

R. Simon and N. Mukunda J. Opt. Soc. Am. A 17(12) 2440-2463 (2000)

References

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First Nine Roots (Zero Crossings) of ${J}_{7/2}$ with ${j}_{r}$ Denoting the $r$th Root

${j}_{r}$

Root

${j}_{1}$

6.987932

${j}_{2}$

10.417119

${j}_{3}$

13.698023

${j}_{4}$

16.923621

${j}_{5}$

20.121806

${j}_{6}$

23.304247

${j}_{7}$

26.476764

${j}_{8}$

29.642605

${j}_{9}$

32.803732

Table 2.

Desirable Blob Parameters for Various Pairs of Roots of ${J}_{7/2}$

${j}_{{r}_{1}}$, ${j}_{{r}_{2}}$

$a$

$\alpha $

${j}_{1}$, ${j}_{2}$

$1.738875\beta $

3.294537

${j}_{1}$, ${j}_{3}$

$2.651778\beta $

9.485434

${j}_{1}$, ${j}_{4}$

$3.469269\beta $

13.738507

${j}_{1}$, ${j}_{5}$

$4.247117\beta $

17.527826

${j}_{1}$, ${j}_{6}$

$5.003932\beta $

21.105107

${j}_{1}$, ${j}_{7}$

$5.748062\beta $

24.563319

${j}_{1}$, ${j}_{8}$

$6.483890\beta $

27.946764

${j}_{1}$, ${j}_{9}$

$7.213964\beta $

31.279745

Table 3.

Values ${q}_{3}/{q}_{0}$, ${q}_{4}/{q}_{0}$, and ${q}_{5}/{q}_{0}$ of ${\widehat{w}}_{2,a,\alpha}(R)/{\widehat{w}}_{2,a,\alpha}(0)$ at Distances ${R}_{3}=\sqrt{6}/2\beta $, ${R}_{4}=\sqrt{10}/2\beta $, and ${R}_{5}=2/\beta $, Respectively, from the Origin^{a}

${j}_{{r}_{1}}$, ${j}_{{r}_{2}}$

${q}_{3}/{q}_{0}$

${q}_{4}/{q}_{0}$

${q}_{5}/{q}_{0}$

${j}_{1}$, ${j}_{2}$

$1.378761\times {10}^{-3}$

$-\mathrm{1.1.28555}\times {10}^{-3}$

$2.429460\times {10}^{-5}$

${j}_{1}$, ${j}_{3}$

$2.030869\times {10}^{-5}$

$-1.095569\times {10}^{-5}$

$6.241931\times {10}^{-6}$

${j}_{1}$, ${j}_{4}$

$-1.759383\times {10}^{-7}$

$-1.873667\times {10}^{-7}$

$1.125839\times {10}^{-7}$

${j}_{1}$, ${j}_{5}$

$-1.011915\times {10}^{-8}$

$-3.034945\times {10}^{-9}$

$2.024441\times {10}^{-9}$

${j}_{1}$, ${j}_{6}$

$1.885230\times {10}^{-10}$

$-2.960094\times {10}^{-12}$

$3.525051\times {10}^{-11}$

${j}_{1}$, ${j}_{7}$

$8.038446\times {10}^{-12}$

$2.974920\times {10}^{-12}$

$4.761025\times {10}^{-13}$

${j}_{1}$, ${j}_{8}$

$-3.048963\times {10}^{-13}$

$1.672062\times {10}^{-13}$

$-2.001146\times {10}^{-15}$

${j}_{1}$, ${j}_{9}$

$-5.836311\times {10}^{-15}$

$5.456460\times {10}^{-15}$

$-5.990216\times {10}^{-16}$

The $a$ and $\alpha $ are as in Table 2 with $\beta =1/\sqrt{2}$.

Table 4.

Values of the Error ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ for the $({r}_{1},{r}_{2})$ Pairs $(1\le {r}_{1}<{r}_{2}\le 9)$ with $\beta =1/\sqrt{2}$ and $\theta =4$

${r}_{1}\backslash {r}_{2}$

2

3

4

5

6

7

8

9

1

0.47367

0.24554

0.23447

0.26334

0.29724

0.32804

0.35421

0.37599

2

0.23470

0.26398

0.29721

0.32778

0.35393

0.37574

3

0.31407

0.30922

0.33244

0.35601

0.37675

4

0.36309

0.36702

0.38176

5

0.42496

0.40043

Table 5.

The Pairs $({r}_{1},{r}_{2})$ that Minimize the Error ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ for Various Values of $\theta $ with $\beta =1/\sqrt{2}$, Together with ${E}_{\beta ,\theta ,\mathrm{STD}}$ (Which is Greater Than the Optimal ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ in All Cases)

$\theta $

$({r}_{1},{r}_{2})$

${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$

${E}_{\beta ,\theta ,\mathrm{STD}}$

¼

(1,2)

0.10354

0.16390

1

(1,3)

0.17054

0.17817

2

(1,3)

0.19554

0.19719

4

(1,4)

0.23447

0.23524

8

(2,4)

0.25586

0.31133

Table 6.

Relative Errors Measured in the Central Slab of the Abdomen and Thorax Reconstructions

Voxels

Standard Blobs $\alpha =10.4$

Standard Blobs $\alpha =10.444256$

Recommended Blobs

Abdomen

0.221928

0.066382

0.066458

0.063463

Thorax

0.220511

0.079533

0.079568

0.080059

Tables (6)

Table 1.

First Nine Roots (Zero Crossings) of ${J}_{7/2}$ with ${j}_{r}$ Denoting the $r$th Root

${j}_{r}$

Root

${j}_{1}$

6.987932

${j}_{2}$

10.417119

${j}_{3}$

13.698023

${j}_{4}$

16.923621

${j}_{5}$

20.121806

${j}_{6}$

23.304247

${j}_{7}$

26.476764

${j}_{8}$

29.642605

${j}_{9}$

32.803732

Table 2.

Desirable Blob Parameters for Various Pairs of Roots of ${J}_{7/2}$

${j}_{{r}_{1}}$, ${j}_{{r}_{2}}$

$a$

$\alpha $

${j}_{1}$, ${j}_{2}$

$1.738875\beta $

3.294537

${j}_{1}$, ${j}_{3}$

$2.651778\beta $

9.485434

${j}_{1}$, ${j}_{4}$

$3.469269\beta $

13.738507

${j}_{1}$, ${j}_{5}$

$4.247117\beta $

17.527826

${j}_{1}$, ${j}_{6}$

$5.003932\beta $

21.105107

${j}_{1}$, ${j}_{7}$

$5.748062\beta $

24.563319

${j}_{1}$, ${j}_{8}$

$6.483890\beta $

27.946764

${j}_{1}$, ${j}_{9}$

$7.213964\beta $

31.279745

Table 3.

Values ${q}_{3}/{q}_{0}$, ${q}_{4}/{q}_{0}$, and ${q}_{5}/{q}_{0}$ of ${\widehat{w}}_{2,a,\alpha}(R)/{\widehat{w}}_{2,a,\alpha}(0)$ at Distances ${R}_{3}=\sqrt{6}/2\beta $, ${R}_{4}=\sqrt{10}/2\beta $, and ${R}_{5}=2/\beta $, Respectively, from the Origin^{a}

${j}_{{r}_{1}}$, ${j}_{{r}_{2}}$

${q}_{3}/{q}_{0}$

${q}_{4}/{q}_{0}$

${q}_{5}/{q}_{0}$

${j}_{1}$, ${j}_{2}$

$1.378761\times {10}^{-3}$

$-\mathrm{1.1.28555}\times {10}^{-3}$

$2.429460\times {10}^{-5}$

${j}_{1}$, ${j}_{3}$

$2.030869\times {10}^{-5}$

$-1.095569\times {10}^{-5}$

$6.241931\times {10}^{-6}$

${j}_{1}$, ${j}_{4}$

$-1.759383\times {10}^{-7}$

$-1.873667\times {10}^{-7}$

$1.125839\times {10}^{-7}$

${j}_{1}$, ${j}_{5}$

$-1.011915\times {10}^{-8}$

$-3.034945\times {10}^{-9}$

$2.024441\times {10}^{-9}$

${j}_{1}$, ${j}_{6}$

$1.885230\times {10}^{-10}$

$-2.960094\times {10}^{-12}$

$3.525051\times {10}^{-11}$

${j}_{1}$, ${j}_{7}$

$8.038446\times {10}^{-12}$

$2.974920\times {10}^{-12}$

$4.761025\times {10}^{-13}$

${j}_{1}$, ${j}_{8}$

$-3.048963\times {10}^{-13}$

$1.672062\times {10}^{-13}$

$-2.001146\times {10}^{-15}$

${j}_{1}$, ${j}_{9}$

$-5.836311\times {10}^{-15}$

$5.456460\times {10}^{-15}$

$-5.990216\times {10}^{-16}$

The $a$ and $\alpha $ are as in Table 2 with $\beta =1/\sqrt{2}$.

Table 4.

Values of the Error ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ for the $({r}_{1},{r}_{2})$ Pairs $(1\le {r}_{1}<{r}_{2}\le 9)$ with $\beta =1/\sqrt{2}$ and $\theta =4$

${r}_{1}\backslash {r}_{2}$

2

3

4

5

6

7

8

9

1

0.47367

0.24554

0.23447

0.26334

0.29724

0.32804

0.35421

0.37599

2

0.23470

0.26398

0.29721

0.32778

0.35393

0.37574

3

0.31407

0.30922

0.33244

0.35601

0.37675

4

0.36309

0.36702

0.38176

5

0.42496

0.40043

Table 5.

The Pairs $({r}_{1},{r}_{2})$ that Minimize the Error ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ for Various Values of $\theta $ with $\beta =1/\sqrt{2}$, Together with ${E}_{\beta ,\theta ,\mathrm{STD}}$ (Which is Greater Than the Optimal ${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$ in All Cases)

$\theta $

$({r}_{1},{r}_{2})$

${E}_{\beta ,\theta ,{r}_{1},{r}_{2}}$

${E}_{\beta ,\theta ,\mathrm{STD}}$

¼

(1,2)

0.10354

0.16390

1

(1,3)

0.17054

0.17817

2

(1,3)

0.19554

0.19719

4

(1,4)

0.23447

0.23524

8

(2,4)

0.25586

0.31133

Table 6.

Relative Errors Measured in the Central Slab of the Abdomen and Thorax Reconstructions