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**Contents:**

In cardio-vascular imaging, an essential pre-processing task is the enhancement of coronary arterial tree, commonly using gradient or other local operators. In Khan et al. An adaptive vessel detection scheme is proposed in Wu et al.

Filtering is an elementary operation in low level computer vision and a pre-processing stage in many biomedical image processing applications. Some edge-preserving filtering techniques for biomedical image smoothing have been proposed Rydell et al. At the end of this chapter some simulation results are given for biomedical image filtering using some of the proposed 2D filters, namely the directional narrow fan-filter with specified orientation and the zero-phase circular filter.

This section presents the types of analog and digital 1D recursive prototype filters which will be further used to derive the desired 2D filter characteristics. An analog IIR prototype filter of order N has a transfer function in variable s of the general form:. This general transfer function can be factorized into simpler rational functions of first and second order. Such a second-order rational function biquad can be written:. An elliptic approximation with very low ripple can be used for an almost maximally-flat low-order filter.

Next we consider two such low-pass LP prototypes with imposed specifications. In Fig.

This will be further used as a prototype for two-directional filters. A useful zero-phase prototype can be obtained from the general function 1 by preserving only the magnitude characteristics of the 1D filter; this prototype will be further used to design 2D zero-phase FIR filters of different types, specifically circular filters, with real-valued transfer functions. The most convenient for our purpose is the Chebyshev series expansion, because it yields an efficient approximation of a given function, which is uniform along the desired interval.

Therefore, prior to Chebyshev series calculation, we apply the change of variable:. Next let us consider a recursive digital filter of order N with the transfer function:.

2D Discrete Fourier Transform - Image Transforms - Digital Image Processing

For an odd order filter, H P z has at least one first-order factor:. The transfer function also contains second-order biquad functions, where in general the numerator and denominator polynomials have complex-conjugated roots:. We will further use the term template , common in the field of cellular neural networks, for the coefficient matrices of the numerator and denominator of a 2D transfer function H z 1 , z 2.

In this section a design method is proposed for 2D square-shaped diamond-type IIR filters. The design relies on an analog 1D maximally-flat low-pass prototype filter. To this filter a frequency transformation is applied, which yields a 2D filter with the desired square shape in the frequency plane. The proposed method combines the analytical approach with numerical approximations.

The standard diamond filter has the shape in the frequency plane as shown in Fig. In this chapter a more general case is approached, i. Next we refer to them as diamond-type filters, since they are more general than the diamond filter from Fig.

The diamond-type filter in Fig. Correspondingly, the diamond-type filter transfer function H D z 1 , z 2 results as a product of two partial transfer functions:. A more general filter belonging to this class is a rhomboidal filter, as shown in Fig. In this case the two oriented LP filters may have different bandwidths and their axes are no longer perpendicular to each other. In the complex plane s 1 , s 2 the above frequency transformation becomes:. The usual method to obtain a discrete filter from an analog prototype is the bilinear transform. This method is straightforward, still the resulted 2D filter will present linearity distortions in its shape, which increase towards the limits of the frequency plane as compared to the ideal frequency response.

This is mainly due to the so-called frequency warping effect of the bilinear transform, expressed by the continuous to discrete frequency mapping:. This error can be corrected by applying a pre-warping. In order to include the nonlinear mappings 21 into the frequency transformation, a rational approximation is needed. Substituting the nonlinear mappings 21 with approximate expression 22 into 18 we get the 1D to 2D mapping which includes the pre-warping along both frequency axes:.

In order to design the Three types of arithmetic operations can be performed on required 2D FIR filter, frequency specifications are entered real and complex 2D signals using these blocks, that is, in the required fields as shown in Fig. Related Papers. If you want to make it similar with amplitude vs time plot as you do with one dimensional signal, you can plot a intensity vs x vs y scattered dots with three axis as well. In this chapter we will mainly use analog prototypes, since the design turns out to be simpler and the 2D filters result of lower complexity. Pattern recognition The derived 2D filter could become unstable only if the numerical approximations introduce large errors. The transfer function in z is:.

As can be noticed, the filter characteristic corrected by pre-warping has a good linearity, however it still twists towards the margins of the frequency plane. These marginal linearity distortions can be corrected using an additional LP filter. The resulted 2D square-shaped correction filter characteristic is shown in Fig.

The corrected version of the diamond-type filter from Fig. It can be easily noticed that the initial distortions have been eliminated. In this section an analytical design method in the frequency domain for 2D fan-type filters is proposed, starting from an 1D analog prototype filter, with a transfer function decomposed as a product of elementary functions. Since we envisage designing efficient 2D filters, of minimum order, recursive filters are used as prototypes, and the 2D fan-type filters will result recursive as well. A particular case is the two-quadrant fan filter, shown in Fig.

The 1D analog filter discussed in section 2 is used as prototype. Applying the same steps as in Section 3. The 2D transfer function for each biquad is complex.

The characteristics of a fan-type filter designed with this method and based on the prototype filter of order 4 given by 6 - 7 is shown in Fig. As with the diamond-type filter analyzed in the previous section, the fan-type filter characteristic features marginal linearity distortions which can be corrected using a LP filter, similar with the correction filter used in Section 3. Two corrected fan-type filters with specified parameters have the magnitudes and contour plots shown in Fig. The initial distortions have been eliminated.

With the same correction filter, we obtain the two-quadrant fan filter, shown in Fig.

In this section a design method based on spectral transformations is proposed for another class of 2D IIR filters, namely multi-directional filters. The design starts from an analog prototype with specified parameters. Applying an appropriate frequency transformation to the 1D transfer function, the desired 2D filter is directly obtained in a factorized form, like the filters designed in the previous sections. For two-directional filters, an example is given of extracting lines with two different orientations from a test image.

The spectral transformation used in the case of multi-directional filters is similar to the one presented in the previous section, derived for fan-type filters and given by 34 , In this section the design of two-directional and three-directional filters with specified orientation is detailed. The method can be easily generalized to arbitrary multi-directional filters.

A two-directional 2D filter is orientation-selective along two directions in the frequency plane. It is based on a selective resonant IIR prototype as given in section 2. The denominator matrix A 3 has complex elements. As with the previous types of filters, the marginal linearity distortions can be corrected using an additional LP square-shaped filter. The second two-directional filter in Fig.

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In order to design a three-directional filter like the one depicted in Fig. The prototype transfer function H P s in variable s will be in this case the sum of three elementary functions:. The numerator B P s of H P s from 38 has the general form:. We see that a 2 is real and a 0 , a 1 are generally complex.

Therefore the factorized prototype transfer function is:. As a general remark on the method, using an analog prototype instead of a digital one, as is currently done, simplifies the design in this case, as the frequency mapping results simpler and leads to a 2D filter of lower complexity. The designed filters result with complex coefficients, however such IIR filters can also be implemented Nikolova et al.

We approach here a particular class of 2D filters, namely filters whose frequency response is symmetric about the origin and has at the same time an angular periodicity. The contour plots of their frequency response, resulted as sections with planes parallel with the frequency plane, can be defined as closed curves which can be described in terms of a variable radius which is a periodic function of the current angle formed with one of the axes.

The elementary transfer functions 14 and 15 have the complex frequency responses:.

The proposed design method for these 2D filters is based on the frequency transformation:. For instance, the four-lobe filter with the contour plot given in Fig. The frequency transformation 48 can be also expressed as:. Using 55 instead of 53 , 54 will result in much more efficient 2D filters, which fully satisfy the imposed specifications. We will use here a Chebyshev low-pass second-order filter of the general form According to 46 we can write:.

The numerator results real because the imaginary part is cancelled. Substituting the expressions 55 into this complex frequency response we get the rational approximation:. We approach now the design of a particular filter type designed in polar coordinates, namely two-directional selective four-lobe filters along the two plane axes or with a specified orientation angle. Let us consider the radial function given by:.

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Using trigonometric identities, 59 becomes:. We get using 49 :.