Analysis

The Leaf scanner was characterized using a step tablet consisting of several sections of known optical density. The step tablet was placed in the scanner and scanned as 4"x5" portrait B&W positive image. The image produced was analyzed using the NIH Image software. This process was then repeated three more times to test for reliability. Results are shown below.

The average of the four scans was then modeled as the function:

where pv is the pixel value of the image and OD is the optical density of the step in the step tablet. Results are shown below.

This data was then transformed to produce a linear graph of optical density as a function of pixel value. This conversion was accomplished with the formula:

where pv is the original pixel value and cpv is the converted pixel value. The transformed data was then fitted to a line. A plot of the optical density vs. converted pixel value and the corresponding line fit are shown below.

As a check, this line fit was used to calculate the optical density from the pixel measurement data. Results are displayed in the table below.

Pixel Value	Converted Pixel Value	Calculated Optical Density	Optical Density	Deviation
13.51	0.04	0.04	0.04	0.00
34.89	0.14	0.14	0.14	0.00
56.21	0.24	0.24	0.24	0.00
76.97	0.34	0.34	0.34	0.00
96.33	0.45	0.45	0.44	0.01
113.20	0.54	0.54	0.54	0.00
129.19	0.63	0.64	0.63	0.01
146.65	0.75	0.75	0.74	0.01
160.68	0.84	0.84	0.84	0.00
174.00	0.94	0.94	0.94	0.00
186.33	1.04	1.04	1.04	0.00
198.04	1.14	1.14	1.14	0.00
208.70	1.24	1.25	1.24	0.01
217.13	1.33	1.34	1.35	-0.01
225.26	1.43	1.44	1.44	0.00
232.95	1.54	1.54	1.54	0.00
239.59	1.64	1.65	1.65	0.00
245.27	1.76	1.77	1.75	0.02
249.87	1.88	1.89	1.86	0.03
253.52	2.03	2.04	1.96	0.08

Thus the above curve was chosen as the model for the Leaf scanner.

Sensitivity to White Space in Scan

The step tablet was once again scanned, this time with varying degrees of white space surrounding the prescan image of the tablet. Results are shown below.

Scan1 had very little white space, Scan2 had some white space, and Scan3 had a large amount of white space in the image. As shown, there is little difference between the scans; the average standard deviation between the scans was 0.09 pixel values. (Standard deviations for each step of optical density were computed and then averaged.) Also, the magnitude of the deviations had no correlation with the value of optical density.

Sensitivity to Scan Resolution

The optical density strip was scanned once again, this time with the scan resolution varying from 500 to 5000 dpi. As shown, there is almost no difference in pixel value between scans of different resolution.

Alternative Fits for the Leaf Scanner

Although the above power model is very accurate, it is somewhat hard to explain physically. Moreover, the shape of the pixel value -- optical density curve suggests that perhaps a logarithmic or exponential fit might provide both an accurate and an explainable model for this scanner. Therefore, this section is included to demonstrate that neither the logarithmic nor the exponential curve can model the leaf scanner as accurately as the one presented in the previous section.

 Logarithmic Fit

First, the Leaf scanner was modeled as a logarithmic function. As shown below, the logarithmic fit fails to model the scanner accurately.

Exponential Fit

The Leaf was also modeled as a function of an exponential. Again, this model is also somewhat inaccurate.

Quantitative Measurement of Model Accuracy

To further illustrate the inaccuracy of these models, the following table displays the deviations of the three models from the actual data. All deviations are in pixel values.

Optical Density	Actual Pixel Value	Deviation of Logarithmic Fit	Deviation of Exponential Fit	Deviation of Power Fit
0.04	13.51	0.00	-0.26	-0.54
0.14	34.89	1.25	2.52	0.39
0.24	56.21	5.79	3.42	0.27
0.34	76.97	7.32	3.08	-0.39
0.44	96.33	7.37	2.50	-0.74
0.54	113.20	7.70	2.89	0.30
0.63	129.19	5.72	1.25	-0.50
0.74	146.65	3.85	-0.09	-0.60
0.84	160.68	2.81	-0.70	0.02
0.94	174.00	1.54	-1.69	0.26
1.04	186.33	0.46	-2.68	0.42
1.14	198.04	-0.69	-3.96	0.13
1.24	208.70	-1.38	-5.04	-0.18
1.35	217.13	0.55	-3.81	1.55
1.44	225.26	0.48	-4.68	0.78
1.54	232.95	1.35	-4.92	0.27
1.65	239.59	3.69	-4.06	0.33
1.75	245.27	5.82	-3.49	-0.35
1.86	249.87	9.46	-1.80	-0.63
1.96	253.52	13.01	-0.22	-1.41

As shown above, the power fit is the most accurate of the three models. Therefore, the power model was chosen as the model for the Leaf scanner.

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