2 Data and methods

We use Johnson B-band CCD images of a group of spiral galaxies obtained at the prime focus of the 2.5 m Isaac Newton Telescope, at the Observatorio del Roque de los Muchachos, La Palma. The images were reduced using standard routines, with a seeing-limited angular resolution close to 1 arcsec. The observations and data reduction are described in detail in Paper II. In their original form (before deprojection) the orientation is north-right, east-top. In this paper all images are deprojected. The positive $\widehat{x}$-axes corresponds to the PAs listed in Table 2.

2.1 Fourier transform method

A complete and detailed description of this method can be found in Puerari & Dottori (1992). We give a brief review here of the procedure, and of the notation and formulae used.

We consider the decomposition of a given distribution of intensities of a set of continuous two-dimensional coplanar points into a superposition of m-armed logarithmic spirals. Each point, associated with a pixel of the image, with polar coordinates ($r_j,\theta_j$) in the galaxian plane, is weighted by the intensity of the corresponding pixel. An m-armed logarithmic spiral is expressed by $r=r_0 \exp(-(m/p)\theta)$, where m is the number of arms, i.e. the angular periodicity number, and p is related to the pitch angle i of the spiral via the relation $\tan i = -m/p$. The distribution of the points that follow the former conditions can be written in terms of $\delta$-functions, and its Fourier transform is then expressed by  

$$A(m,p)=\int_{-\infty}^{+\infty}\int_{-\pi}^{+\pi} \sum_{i=1}... ...rm d}u{\rm d}\theta/ \sum_{i=1}^{I} \sum_{j=1}^{J} \omega_{ij},$$ (1)

where $\omega_{ij}$ is the weight of the point (ij), r_ij and $\theta_{ij}$ its deprojected radial and angular coordinates and $u_{ij}\equiv \ln r_{ij}$. Eq. (1) has the simpler form:

$$\begin{eqnarray} A(m,p)\!=\!\left(\!\sum_{i=1}^{I}\! \sum_{j=1}^{J} \omega_{ij}\... ... \right] \right)\!/ \sum_{i=1}^{I} \!\sum_{j=1}^{J}\! \omega_{ij}.\end{eqnarray}$$ (2)

The complex surface brightness of each m component is the sum of the inverse Fourier transforms of the complex function |A(m,p)|:

 

$$S(u,\theta)=\frac{1}{{e}^{2u}} \cdot \frac{\sum_{i=1}^{I} \s... ..._{-\infty}^{+\infty} G_m(p) A(m,p){e}^{i(pu+m\theta)}{\rm d}p.$$ (3)

G_m(p) is a high frequency filter used to emphasize the main component, i.e. the spiral with $\tan i = -m/p^m_{_{\rm max}}$:

$$\begin{eqnarray} G_m(p) = \exp \left( -\frac{1}{2}((p-p^m_{_{\rm max}})/25)^2\right)\end{eqnarray}$$ (4)

where $p^m_{_{\rm max}}$ is the value of p^m for which the amplitude of |A(m,p)| is a maximum. This filter is also used to smooth the Fourier coefficient spectra at the ends of the interval, so |A(m,p)| goes to zero for large values of |p|.

The righthand side of Eq. (3) can be separated by variables in the form:

$$\begin{eqnarray} S(r,\theta) \equiv S(u,\theta) = \sum_m S_m(u)\cdot e^{im\theta},\end{eqnarray}$$ (5)

where

$$\begin{eqnarray} S_m(u) \!=\! \frac{1}{e^{2u}}\frac{\sum_{i=1}^{I}\! \sum_{j=1}^... ...4\pi} \!\!\int^{\infty}_{-\infty} \! G_m(p) A(m,p)e^{ipu}{\rm d}p.\end{eqnarray}$$ (6)

Finally, to construct the different m-images, denoted as S_m (S_m(r) and S_m do not vary so much when are calculated without G(p) filter, only the arm edges are slightly noisier), we take the real part of the $S_m(r)e^{i\theta m}$, giving all negative pixels value zero.

The relative importance of the different spiral modes can be estimated from

$$\begin{eqnarray} f_m=\frac{S_m}{S_1+S_2+S_3+S_4+S_5},\end{eqnarray}$$ (7)

with

$$\begin{eqnarray} S_m=\int^{r_2}_{r_1} S_m(r)\,{\rm d}r, \nonumber\end{eqnarray}$$

where we have changed S₆ to S₁ from the original formula of Considère & Athanassoula (1988), because in our decomposition the latter is simply noise, whereas the former keeps information, and the lower limit in the integration from 0 to r₁, because our values usually diverge at $r\sim0$, and we wish to avoid the influence of oval bulges, which favour S₂ components.

2.2 Symmetric and asymmetric images

This method has been developed and applied by EEM and consists in decomposing a given image into images of different degrees of symmetry through rotations and subtractions, thereby obtaining the bi-, tri- and four-symmetric images S2, S3 and S4, respectively, together with the asymmetric image A2. This method maintains the non-linear spiral arm form, without introducing spurious inter-arm components. The authors also claim that this technique shows the m=1,2,3,4,... sub-components simultaneously over the whole image, which prevents confusion with field stars or star formation zones.

According to EEM, not only do galaxies apparently showing a high degree of symmetry, such as NGC 157 and NGC 4321 in this sample, have an important tri-symmetric component that is not evident in the images, but even galaxies with lower symmetry, such as NGC 6764 and NGC 7723, also possess this tri-symmetry. The 3-S structure extends from the 3:1 ILR to the 3:1 OLR, and, correspondingly, the 4-S structure from the 4:1 ILR to the 4:1 OLR.