2.2.0.8

The origin of what is now called a Fourier transform dates back to 1807 when Jean Baptiste Joseph Fourier defined the notion of representing a function as a trigonometric series. The discrete version of a Fourier transform (DFT) for a one-dimensional signal is defined as:

$$F(\mu) = \frac{1}{N} \sum\_{\mathbf{x}=0}^{N-1} f(\mathbf{x}) e^{-j2\pi \mathbf{u} \mathbf{x}/N} \tag{1}$$

for *u* = 0, ...*N* − 1, where *f*(*x*) is a real valued function, *N* represents the number of elements in the signal and *j* <sup>2</sup> <sup>=</sup> <sup>−</sup>1.

The effect of this transform is to capture the spatial relationships inherent in the signal f(x) and express these relationships as the sum of sinusoidal function (frequency components). Similarly, the discrete version of an inverse Fourier transform (IDFT) for a one-dimensional signal is defined as:

$$f(\mathbf{x}) = \sum\_{\mu=0}^{N-1} F(\mathbf{x}) e^{j2\pi \mathbf{u}\mathbf{x}/N} \tag{2}$$

for *x* = 0, ...*N* − 1, where *F*(*u*) is the Fourier transform of the real valued function *f*(*x*), *N* represents the number of elements in the signal and *j* <sup>2</sup> <sup>=</sup> <sup>−</sup>1.

The effect of this inverse Fourier transform is to take a signal in frequency domain back to the spatial domain.

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Prior to the invention of the digital computer, the Fourier series was employed as a purely analytic tool. However, since that time, the development of a class of computationally efficient algorithms, known as fast Fourier transforms (FFT), has meant the notion has become a useful computational tool (VanLoan, 1992). One of the most attractive computational properties of the FFT is its ability to process signals at higher resolution with a minimal increase in cost to complexity. Today, most of us benefit from fast Fourier transforms every day without even knowing it as these algorithms power a vast range of electronic technology such as digital cameras and cell phones.

#### 2.2.0.10

The relevance of a fast Fourier transform to this paper is its relationship to the notion of convolution. The convolution of two functions *f*(*x*) and *g*(*x*) is denoted by *f*(*x*) ∗ *g*(*x*) and 6 Will-be-set-by-IN-TECH

The importance of the SHIA addressing scheme is that it facilitates primitive image transformations of translation, rotation and scaling. One of these transformations that has proven to be of particular relevance to the Fourier transform is one that provides rotation and scaling. It is referred to as mapping *M*10 in the notation of SHIA. Fig. 2 (a) displays an image represented in a four level SHIA, size is 7<sup>4</sup> = 2401. Fig. 2 (b) represents the effect of applying

The critical observation to make in regard to the effect of *M*10 is that it produces multiple 'near' copies at reduced resolution of the input image. This transform will play a critical role

The origin of what is now called a Fourier transform dates back to 1807 when Jean Baptiste Joseph Fourier defined the notion of representing a function as a trigonometric series. The discrete version of a Fourier transform (DFT) for a one-dimensional signal is defined as:

> *N*−1 ∑ *x*=0

for *u* = 0, ...*N* − 1, where *f*(*x*) is a real valued function, *N* represents the number of elements

The effect of this transform is to capture the spatial relationships inherent in the signal f(x) and express these relationships as the sum of sinusoidal function (frequency components). Similarly, the discrete version of an inverse Fourier transform (IDFT) for a one-dimensional

for *x* = 0, ...*N* − 1, where *F*(*u*) is the Fourier transform of the real valued function *f*(*x*), *N*

The effect of this inverse Fourier transform is to take a signal in frequency domain back to the

Prior to the invention of the digital computer, the Fourier series was employed as a purely analytic tool. However, since that time, the development of a class of computationally efficient algorithms, known as fast Fourier transforms (FFT), has meant the notion has become a useful computational tool (VanLoan, 1992). One of the most attractive computational properties of the FFT is its ability to process signals at higher resolution with a minimal increase in cost to complexity. Today, most of us benefit from fast Fourier transforms every day without even knowing it as these algorithms power a vast range of electronic technology such as digital

The relevance of a fast Fourier transform to this paper is its relationship to the notion of convolution. The convolution of two functions *f*(*x*) and *g*(*x*) is denoted by *f*(*x*) ∗ *g*(*x*) and

<sup>2</sup> <sup>=</sup> <sup>−</sup>1.

*N*−1 ∑ *u*=0

*f*(*x*)*e*

<sup>−</sup>*j*2*πux*/*<sup>N</sup>* (1)

*F*(*x*)*ej*2*πux*/*<sup>N</sup>* (2)

*<sup>F</sup>*(*u*) = <sup>1</sup>

*f*(*x*) =

*N*

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*M*102 = *M*100 to this image.

in the proposed FFT.

in the signal and *j*

signal is defined as:

spatial domain.

cameras and cell phones.

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2.2.0.10

<sup>2</sup> <sup>=</sup> <sup>−</sup>1.

represents the number of elements in the signal and *j*

(a) Four Level SHIA

Fig. 2. Displays (a) an image of a duck represented on a four-level SHIA; (b) the result of applying SHIA transform M10 twice to the image displayed in (a). There are four observable effects: 1) multiple near copies of the input image (a), 2) each copy is rotated by the same angle, 3) each copy is scaled by the same amount, 4) applying *M*10 twice to the image displayed in (b) results in the image displayed in (a).

its discrete definition is

$$f(\mathbf{x}) \* \mathbf{g}(\mathbf{x}) = \sum\_{a \in \mathcal{g}} f(\mathbf{x}) \mathbf{g}(\mathbf{x} - a) \tag{3}$$

(a) Intermediate step (b) Fourier transform

<sup>189</sup> Cortical Specification of a Fast Fourier

Transform Supports a Convolution Model of Visual Perception

(c) Intermediate step (d) Inverse Fourier transform

Fig. 3. Displays the results of applying the special case of GOSH-FFT, that is PaSH-FFT, to image of Fig. 2 (a), with n=4 and m=2. The four sub figures display intermediate results of PaSH-FFT: (a) on completion of first iteration of PaSH-FFT to Fig. 2; (b) Fourier transform on completion of second iteration; (c) on completion of first iteration of inverse PaSH-FFT; (d)

through before being output as a convolved value. With the cortical convolution conundrum thus fully formulated, in this section we will establish a specification of a sufficient sequence of steps to address the issue. This specification will unfold in three steps. First, we will discuss how the SHIA transform *M*10 manifests in cortical architecture. We will then employ this manifestation to demonstrate how neural circuitry accommodates PaSH-FFT. Lastly, we will

A critical component of the fast Fourier transform, PaSH-FFT, is the transform *M*10. Consequently, it is an imperative of our argument that the redistribution properties of *M*10 be accounted for in the neural circuitry of the visual system. To this end we now argue that the required effects of *M*10 are accounted for by the long-range properties of patchy connections

show how the cortical manifestation of PaSH-FFT supports long-range convolution.

Inverse PaSH-FFT on completion of second iteration.

**4.1 Cortical manifestation of M10**

A well known result to researchers in the field of signal processing is the Convolution Theorem, which relates convolution in the spatial domain to convolution in the frequency domain. For two functions, *f*(*x*) and *g*(*x*), let *F*(*x*) and *G*(*x*) represent the Fourier transform of *f*(*x*) and *g*(*x*) respectively. The Convolution Theorem states that,

$$f(\mathbf{x}) \* g(\mathbf{x}) \rightleftharpoons F(\mathbf{x})G(\mathbf{x}) \tag{4}$$

In other words, the convolution of two functions in the spatial domain can be achieved by the multiplication of the functions in the frequency domain.
