Tomography reconstruction by entropy maximization with smoothing filtering

Figures & data

Figure 1. Image space f and the ith ray crossing the object.

Figure 2. Reconstructions with different parameters of β and the matrix M calculated with Equation (4). Notes: (a) β = 0, (b) β = 50, (c) β = 10², (d) β = 10³, (e) β = 10⁴ and (f) β = 10⁶.

Figure 3. Reconstruction with different parameters of β and the M matrix calculated with Equation (5). Notes: (a) β = 0, (b) β = 50, (c) β = 10², (d) β = 10³, (e) β = 10⁴ and (f) β = 10⁶.

Figure 4. Pixel-to-pixel error between the reconstructed and the original image, using Equations (4) (solid) and (5) (dashed).

Figure 5. Error between projections of the reconstruction image and the projection data, using Equations (4) (solid) and (5) (dashed).

Figure 6. Reconstructions with different β using Equation (4). Notes: (a) β = 0, (b) β = 10³, (c) β = 10⁴ and (d) β = 10⁶.

Figure 7. Reconstructions with different β using Equation (4). Notes: (a) β = 0, (b) β = 10³, (c) β = 10⁴, and (d) β = 10⁶.

Figure 8. Pixel-to-pixel standard deviation between the reconstructed and the original image, using Equations (4) (solid) and (5) (dashed).

Figure 9. Error between projections of the reconstruction image and the noiseless projection data, using Equations (4) (solid) and (5) (dashed).

Figure 10. Pixel-to pixel error between the reconstructed image and the original image data, using Equations (4) (solid) and (5) (dashed), and projections data with 2% random uniform noise.

Figure 11. Error between projections of the reconstruction image and the noisy projection data, using Equations (4) (solid) and (5) (dashed).

Figure 12. Reconstructions with 2% random-noise projections with Equation (4). Notes: (a) β = 0, (b) β = 10³, (c) β = 10⁴ and (d) β = 10⁵.

Figure 13. Reconstructions with 2% random-noise projections with Equation (5). Notes: (a) β = 0, (b) β = 10³, (c) β = 10⁴ and (d) β = 10⁵.

Figure 14. Increment of the pixel-to-pixel error relative to the noiseless value as the noise level of the projections increase.

Figure 15. Reconstructions with 2% random-noise projections and median filter. Notes: (a–c) planar MEM β = 0 without and with non-linear median filter applied (one pass) and (two pass), (d–e) using Equation (4) and optimum β without and with one pass median filter, (f–g) using Equation (5) and optimum β without and with one pass median filter.

Figure 16. Reconstructions with noiseless projections. Notes: (a–c) planar MEM β = 0 and non-linear median filter applied one pass and two pass, (d–e) using Equation (4) and optimum β without and with one pass median filter, (f–g) using Equation (5) and optimum β without and with one pass median filter.

Table 1. Pixel-to-pixel and projection errors obtained with different reconstructions of Figure 6.

Method	Noiseless projections		Projections with 2% noise
	σ	ε	σ	ε
β = 0	1186	4934	1677	12,864
β = 0 and one median filtering	926	15,199	1154	28,905
β = 0 and two median filtering	903	16,356	1095	32,056
Optimum β (minimum ε) with Equation (4)	917	4213	1267	9027
Optimum β (minimum ε) with Equation (4) and one median filtering	825	10,082	1015	16,858
Optimum β (minimum ε) with Equation (5)	947	4944	1305	10,922
Optimum β (minimum ε) with Equation (5) and two median filtering	869	10,476	1078	18,341