The target of filter was, on the one hand, to smooth the image edges and on the other , to eliminate in general the pixelation, that occurs with the interpolation filter applied to the normal NEGA zoom, this filter is much faster in execution and useful for certain increase levels, but in a very high increase, because is based on the filter of the nearest neighbor filter, its efficiency is very poor and insufficient, as can be seen in the image Fig.1.

For that reason, it was thought to study a new filter based on SINC.

The filter, according to the NIST manual, of the University of Cambridge on mathematical functions** ^{(1) }**is based on the mathematical function, called Sine Cardinal (Sinc), its mathematical interpretation has two forms.

**(1)***NIST handbook of mathematical functions. Edited by Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark. With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. xvi+951 pp. ISBN: 978-0-521-14063-8*

For the **NOT normalized function** and x ≠ 0. See Fig.2

To filter our signal type we have the **normalized function** for x ≠ 0. See Fig.3

For both cases if x = 0 then sinc (0) = 1

**Normalization, is the form of the function that we will use for our the filter,** this normalization, makes the definite that the function integral, for the real numbers is equal to 1 (in the non-normalized does not matter) the zeros of the normalized function itself , are integer values of x.

This normalized function, is the Fourier Transform of the rectangular function without scale.

The zero crossings of the sine wave, in the non-standard function are different multiples of zero, and integers in the normalized zero crossings SINC function are nonzero integers.

The SYNC function has the peculiarity, that in a signal processing it eliminates all the components of a certain cut-off frequency, without affecting the low frequencies, with a linear phase of response. What makes it an “ideal” filter and belonging to electronic filters called brick wall.

Its response to an impulse is a SYNC function in the time domain. See Fig.4

However, the response to the frequency of a SINC filter is a rectangular function. See Fig.5.

In the real application, each sample of the signal is assigned a scaled and translated copy of the Lanczos Kernel, which is nothing more than a SYNC function, with a window by the central lobe of a second time, the sum of these translated and scaled nuclei are Evaluate below.

Interpolation, is multivariable and is the best method of interpolation compared to other filters, such as cubic interpolation, linear, Mitchel, near neighbor point, etc …

This kernel is defined as the normalized SYNC function:

The parameter a is a positive integer, usually applied from 2 to 10, the Lanczos nucleus will have 2a-1 lobes a positive in the center and a-1 alternating positive and negative lobes on each side of the central. An example for values 2 and 3 can be seen in Fig.7.

The interpolation is defined for a one-dimensional signal as:

For integer values of i given samples S_{i} If for the value of S(x) interpolated in a true argument of x and arbitrary by the discrete convolution of the samples with the Lanczos kernel.

Where a is the filter size parameter and is the soil function. The limits of this sum are such that the nucleus is zero outside them.

While the parameter a is a positive integer, the nucleus is continuous everywhere, derivative is defined continuous everywhere even when x = ±a where both SINC functions go to zero. So the function S (x) belonging to the reconstructed signal will also be continuous, with continuous derivative.

The kernel will be zero in each integer argument x, except for x = 0 where it has a value of 1. The reconstructed signal interpolates, therefore, exactly the given samples and we will have S(x) = s_{i} yes for each integer argument x = i.

Jim Blinn (James F. Blinn)senior physicist and American computer scientist, expert in computer graphics at the NASA Jet Propulsion Laboratory (JPL) and the New York Institute of Technology. He affirmed that the Lanczos kernel with a = 3“keeps the frequencies low and rejects the high frequencies better than any filter (realizable) that we have seen so far”

__MacWinLinCPS study for filter adaptation in NEGA Acpc__

The problem of applying a > 1 to the lanczos kernel, is that, the interpolated signal may be negative even if the samples are positive, this may cause the interpolated signals, to be wider than the range encompassed by the discrete sample values, giving rise to to ring artifacts just before and after abrupt changes, in sample values this can lead to clipping artifacts. These effects are reduced if a = 2, but the disadvantage of lowering both the value, is that it loses detail in the interpolation in certain sudden frequency changes, and if we lower a = 1 the filter has almost no effect.

With a = 3 the results are acceptable, and do not generate too many ring or trim artifacts, but now we will see why it is not enough.

The image on the left represents a = 3 (Fig. 8) and the one on the right a = 8 (Fig.9) artifacts are more importants than in a = 3, but also the interior details of the letter are more Highlighted.

Due to this it has been chosen to touch up the filter to approximate the value in certain cases of a great increase, with the value a = 8, to correct the graphic noise caused by the artifacts generated by the filter, it is recommended to apply the focus filter, at any level that is acceptable according to the image, this filter was redesigned to adapt to this, applied a convolution in the samples, which causes a new high-pass filtering at certain frequencies, the application of these two filters together, makes the effects of a bandpass filter, limiting mostly high frequencies.

This is the result, after applying the **focus filter** to the previous images, with the kernel filter lanczos a = 3 and a = 8 respectively. The Lanczos kernel filter has a value of a = 8, which preserves the interior letter details, but the artifacts surrounding the letter in the zones greater abrupt frequency change of the samples.

The focus filter and Zoom HQ together on the NEGA Apcp , are the one that best optimizes the antialiasing and optimum interpolation for digital image enlargement,in addition to being tested and the one that obtains the best quality in enlargement, in any image type and especially, in written texts and documents, being ideal for forensic handwriting analysis expertise and documents examiners experts. NEGA is the only software in the market that applies the zoom, filters and tools only in the image, without distorting its origin, nor the indications that we have included as text arrows or markers, the activation of tools and filters follows an internally preset method , so in NEGA the activation order is not important, and the image is without any distortion when the filters are deactivated and zoomed in, however, if you use Photoshop, Corel or Gimp, and change the order of activation or deactivation Filters, zoom, etc. you will not get the same results.