## 3 Spectral Scattering Imaging
### Analysis of Scattering Images
#### 3.2.2 Correction of Scattering Profiles  
Scattering profiles obtained using the procedures described above may suffer from a certain degree of distortion due to the uneven instrumental responses and the surface curvature of the fruit [54; 55]. Hence, corrections are needed to address the distortion for accurately quantifying scattering profiles. The uneven instrumental responses refer to the uneven distribution of the pixel intensities when a flat homogeneous object is imaged under ideal uniform illumination. A number of factors, e.g., a varied viewing angle due to the finite fruit-camera distance, and the imperfection of imaging optics, can contribute to the uneven instrumental responses. Corrections for this problem can be made by imaging a flat homogenous reference sample like a Teflon disk. As shown in Figure 11a, a Teflon reference is placed on a motor-controlled stage that permits precise adjustment of the Teflon position; moving the reference back and forth causes intensity changes of the scattering image acquired, which are attributed to the uneven uniform instrumental response. Then, the response curve versus the displacement of the reference is extracted by recording the peak intensity values of the scattering image for each wavelength at each displacement position. With the response curve being known, correction is done by multiplying the raw scattering profile by the reciprocal of the normalized response curve. A more detailed description of the correction method can be found in [55]. The reference-based correction, however, has a major limitation in that it does not take into account the effect of the surface curvature of a fruit sample. The curved shape of the fruit can bring  
Figure 10: (**a**) Schematic of a multispectral scattering image and (**b**) the corresponding scattering profile obtained by radial averaging at a given wavelength. about two types of distortion, i.e., scattering distance distortion and intensity distortion [49]. For a given point \(m\) on the fruit surface, as shown in Figure 11b, the actual scattering distance for the point should be the arc length \(z\) instead of \(x\), and assuming the fruit is spherical with a radius \(r\), \(z\) can be calculated as follows:  
\[z=r\text{arctan}\frac{x}{\sqrt{r^{2}-x^{2}}} \tag{1}\]  
where \(r\) and \(x\) are the radius of the fruit and the horizontal distance of the point \(m\) from the light beam incident center. So, the distance distortion can be easily corrected according to Equation (1). The intensity distortion is mainly due to the varied viewing angle of the lens for different points on the fruit surface. From Figure 11b, the point closer to the incident center, has a larger viewing angle and thus a larger detected scattering intensity. To tackle this issue, a model-based method was proposed with an assumption that the angular distribution of scattering intensity on the fruit surface follows the Lambertian Cosine Law [49; 54; 56]. Then the detected scattering intensity, \(S_{d}\), for the point \(m\) can be expressed as follows:  
\[S_{d}=S_{total}f(x,r) \tag{2}\]  
where \(S_{total}=\pi I_{\text{max}}^{2}/4\) and \(I_{\text{max}}^{2}\) is the normal component of the scattering intensity at the point \(m_{z}\)\(f(x,r)\) is a scaling factor that is a function of \(x\) and \(r\). \(S_{d}\) can be also expressed as an integration of the scattering intensity over the viewing angle \(\alpha\), equal to the area of the fan-shape zone \(m_{1}\)-\(m\)-\(m_{2}\). Given the geometric relations shown in Figure 11b, \(f(x,r)\) can be derived as follows [54]:  
\[ \begin{array}{ll}f(x,r)&=\frac{1}{\pi}\bigg{[}\text{arctan} \bigg{(}\frac{x+k}{h+r-\sqrt{r^{2}-x^{2}}}\bigg{)}-\text{arctan}\bigg{(}\frac {x-k}{h+r-\sqrt{r^{2}-x^{2}}}\bigg{)}\bigg{]}\\ &+\frac{1}{2\pi}\sin 2\bigg{[}\text{arcsin}\bigg{(}\frac{x}{r}\bigg{)}+ \text{arctan}\bigg{(}\frac{x+k}{h+r-\sqrt{r^{2}-x^{2}}}\bigg{)}\bigg{]}\\ &-\frac{1}{2\pi}\sin 2\bigg{[}\text{arcsin}\bigg{(}\frac{x}{r}\bigg{)}+ \text{arctan}\bigg{(}\frac{x-k}{h+r-\sqrt{r^{2}-x^{2}}}\bigg{)}\bigg{]}\end{array}