Without the optical anti-aliasing filter, a digital post-processing filter (not a sharpening filter this time) with optimized coefficients [0.06765983302, 0.8646803339, 0.06765983302] only reduces the mean-square error by 11 % compared to a reduction of 40 % from the combination of an optical anti-aliasing filter and a post-processing sharpening filter, so it is apparent that optical anti-aliasing is useful.
In Karel Fliegel, Modeling and Measurement of Image Sensor Characteristics, Radioengineering, vol. 13, no. 4, December 2004, he gives the optical transfer functions (OTFs) for different detector photosensitive area shapes. OTF is the Fourier transform of the spatial domain impulse response. The spatial domain impulse response, also called the point spread function (PSF), is a 2-d function that is constant-valued (assuming uniform sensitivity) inside the detector photosensitive area and zero outside it.
If there is an optical low-pass (blurring) filter in front of the sensor or implemented by shaking the sensor, another PSF describes it, and the total PSF is the convolution of the PSF of the sensor element and the PSF of the optical low-pass filter. The total OTF is the product of the two OTFs. Birefringenge-based optical filters (for example patent US 6937283 B1) create shifted copies of the image so their PSFs are sums of shifted and possibly scaled Dirac pulses. For example, a square arrangement of four Dirac pulses is equivalent to a cascade of a horizontal and a vertical comb filter.
An optical anti-aliasing filter also attenuates in-band frequencies that would not alias. Using a software sharpening filter the amplitudes of those frequencies can be restored. This also amplifies aliased frequencies, but because their attenuation in the combined OTF of the anti-aliasing filter and the detector wasis stronger, they retain some of that attenuation.
Here is an example analysis of a horizontal-vertical-separable total PSF that includes 1) a square photosensitive area, 2) a birefringent optical low-pass filter that has a PSF that consists of a square arrangement of Dirac pulses separated horizontally and vertically by 0.4284281154 times pixel width, and 3) a digital post-processing filter (operating on the pixel data), which can be separated to a cascade of a horizontal and a vertical filter, both with a PSF [-0.01536945896, 1.030738917, -0.01536945896]. A square grid of pixels with no gaps between pixels (100 % coverage of the imaging plane by photosensitive areas) is assumed. The analysis is done only along one of the separable dimensions for simplicity, but it would be better to do a 2-d analysis.
Figure 1. Red: combined one-dimensional PSF of the optical anti-aliasing filter, the square photosensitive area, and the digital sharpening filter. Turquoise: the ideal sensor PSF for a square grid (sinc function). Black: PSF of the square photosensitive area. Horizontal axis: displacement in units of one pixel width.
Figure 2. One-dimensional OTFs (Fourier transforms) of PSFs of Fig. 1, using the same color code. Horizontal axis: spatial frequency in units of radians per pixel width. There are some plotting artifacts that make parts of the red curve invisible.
Considering sinc (Fig. 1) as the ideal one-dimensional PSF, the mean square error for imaging spatial white noise is reduced by 40 % by using the optical anti-aliasing filter and the digital sharpening filter. The sum of squares error can be calculated as the integral over the square of the difference between sinc and the actual PSF, or by Parseval's theorem equivalently in the frequency domain using the differences of the OTFs (Fig. 2). I optimized the coefficients and the amount of separation in the optical anti-aliasing filter so that the mean square error is minimized.
In Karel Fliegel, Modeling and Measurement of Image Sensor Characteristics, Radioengineering, vol. 13, no. 4, December 2004, he gives the optical transfer functions (OTFs) for different detector photosensitive area shapes. OTF is the Fourier transform of the spatial domain impulse response. The spatial domain impulse response, also called the point spread function (PSF), is a 2-d function that is constant-valued (assuming uniform sensitivity) inside the detector photosensitive area and zero outside it.
If there is an optical low-pass (blurring) filter in front of the sensor or implemented by shaking the sensor, another PSF describes it, and the total PSF is the convolution of the PSF of the sensor element and the PSF of the optical low-pass filter. The total OTF is the product of the two OTFs. Birefringenge-based optical filters create shifted copies of the image so their PSFs are sums of shifted and possibly scaled Dirac pulses. For example, a square arrangement of four Dirac pulses is equivalent to a cascade of a horizontal and a vertical comb filter.
An optical anti-aliasing filter also attenuates in-band frequencies that would not alias. Using a software sharpening filter the amplitudes of those frequencies can be restored. This also amplifies aliased frequencies, but because their attenuation in the combined OTF of the anti-aliasing filter and the detector was stronger, they retain some of that attenuation.
In Karel Fliegel, Modeling and Measurement of Image Sensor Characteristics, Radioengineering, vol. 13, no. 4, December 2004, he gives the optical transfer functions (OTFs) for different detector photosensitive area shapes. OTF is the Fourier transform of the spatial domain impulse response. The spatial domain impulse response, also called the point spread function (PSF), is a 2-d function that is constant-valued (assuming uniform sensitivity) inside the detector photosensitive area and zero outside it.
If there is an optical low-pass (blurring) filter in front of the sensor or implemented by shaking the sensor, another PSF describes it, and the total PSF is the convolution of the PSF of the sensor element and the PSF of the optical low-pass filter. The total OTF is the product of the two OTFs. Birefringenge-based optical filters (for example patent US 6937283 B1) create shifted copies of the image so their PSFs are sums of shifted and possibly scaled Dirac pulses. For example, a square arrangement of four Dirac pulses is equivalent to a cascade of a horizontal and a vertical comb filter.
An optical anti-aliasing filter also attenuates in-band frequencies that would not alias. Using a software sharpening filter the amplitudes of those frequencies can be restored. This also amplifies aliased frequencies, but because their attenuation in the combined OTF of the anti-aliasing filter and the detector is stronger, they retain some of that attenuation.
Here is an example analysis of a horizontal-vertical-separable total PSF that includes 1) a square photosensitive area, 2) a birefringent optical low-pass filter that has a PSF that consists of a square arrangement of Dirac pulses separated horizontally and vertically by 0.4284281154 times pixel width, and 3) a digital post-processing filter (operating on the pixel data), which can be separated to a cascade of a horizontal and a vertical filter, both with a PSF [-0.01536945896, 1.030738917, -0.01536945896]. A square grid of pixels with no gaps between pixels (100 % coverage of the imaging plane by photosensitive areas) is assumed. The analysis is done only along one of the separable dimensions for simplicity, but it would be better to do a 2-d analysis.
Figure 1. Red: combined one-dimensional PSF of the optical anti-aliasing filter, the square photosensitive area, and the digital sharpening filter. Turquoise: the ideal sensor PSF for a square grid (sinc function). Black: PSF of the square photosensitive area. Horizontal axis: displacement in units of one pixel width.
Figure 2. One-dimensional OTFs (Fourier transforms) of PSFs of Fig. 1, using the same color code. Horizontal axis: spatial frequency in units of radians per pixel width. There are some plotting artifacts that make parts of the red curve invisible.
Considering sinc (Fig. 1) as the ideal one-dimensional PSF, the mean square error for imaging spatial white noise is reduced by 40 % by using the optical anti-aliasing filter and the digital sharpening filter. The sum of squares error can be calculated as the integral over the square of the difference between sinc and the actual PSF, or by Parseval's theorem equivalently in the frequency domain using the differences of the OTFs (Fig. 2). I optimized the coefficients and the amount of separation in the optical anti-aliasing filter so that the mean square error is minimized.