^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

In [1] and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Móricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.

In a series of papers, Gelb and Tadmor published a means of edge detection from spectral data of a given function, [1-3], see also the review essay [

is known to be the real part of the complex power series

on the unit circle. Taking the imaginary part of this power series results in the conjugate trigonometric series

Note that often is called the conjugate series.

The coefficients and are computed from

For purposes of numerical analysis we are not concerned with trigonometric series but with the partial sums of them,

as with the partial sums of the conjugate series,

For sake of reference we note that the partial sums mentioned can be rewritten as convolutions

of with the Dirichlet kernels

The main result exploited by Gelb and Tadmor from the works of Fejér and Lukács cited above is summarized in Theorem II.8.13 of [

then

This property of the conjugate trigonomteric series is called the concentration property. Hence, the conjugate series is an indicator for a jump discontinuity which can be recovered by. The GelbTadmor theory exploits this property. As was shown in [

at, then

where denotes the Dirac distribution located at. As convergence is really slow Gelb and Tadmor developed so-called concentration factors so that finally they arrive at a concentrated conjugate partial sum

which directly leads to generalised conjugate kernels

with similar properties as the conjugated Dirichlet kernel. Given an admissible kernel (see Def. 1) they proved following theorem.

Theorem 1 (The Concentration Property) Let f(x) be a piecewise smooth function and denotes the set of its jump discontinuities. Consider the generalised conjugate partial sum

where is an admissible kernel. Then

With the help of this theorem Gelb and Tadmor were able to investigate the concentration factors in detail, in other words, they studied the conditions on the contration factor for being an admissible kernel. They analysed the case of being a continuous function and extended their results to the discrete case by constructing discrete concentration factors from the continuous ones. In this paper, the discrete factor is reformulated (Section 1.3.1) and a direct proof for the discrete concentration property is given using weaker preconditions (Section 1.3.2). Beforehand, some of the main classical results are reviewed (Section 1.2). Furthermore, an opportunity of generalising to the full 2-d case which was investigated by Móricz [

Our main motivation for studying these edge detectors is not in image processing but in steering the spectral viscosity filter in spectral difference and spectral Discontinuous Galerkin methods for the numerical solution of hyperbolic conservation laws, cp. [

Although Gelb and Tadmor took up only Theorem II.8.13 of [

Let us start with the oldest of our choice of papers, [

at a point of discontinuity. Fejér asks if there is a comparably simple limit process with which it would be possible to compute the rightand left-sided limits and itself and hence the height of the jump which is simply. In [

Theorem 2 (Fejér 1913) Let be a function obeying the Dirichlet conditions and a point of a jump discontinuity of. If g denotes the smallest (in fact: any) positive root of the transcendent equation

then the sequence

converges to, while the sequence

converges to. Hence the sequence

converges to.

In fact, this theorem can be generalised to the following form, [5, p.178].

Theorem 3 (Fejér 1913) Let and be any positive numbers and. Under the assumptions of Theorem 2 it follows that the sequence

converges to.

It was well known in Fejér’s days (and, in fact, proven by Fejér himself earlier) that the sequence of arithmetic means defined by

converges to under mild conditions on. Fejér argues that it might be possible to also extract the jump height at a discontinuity of the first kind from the sequence of arithmetic means. In fact, he was able to prove the following result, [5, p.179].

Theorem 4 (Fejér 1913) Let and be any positive numbers and. Then the sequence

converges to.

Fejér then turns to exploit the conjugate Fourier series for the computation of the jump height. Note that Fejér considers as conjugate series in contrast to (1.1) so that a minus sign has to be included if we compare to Gelb and Tadmor’s results. He notices that even for a function obeying the Dirichlet conditions the conjugate Fourier series need not be convergent (Fejér calls it “eigentlich divergent” meaning “intrinsically divergent”).

This can be seen from to which the conjugate series is. On the point is a point of a jump discontinuity of the first kind. However, the conjugate series evaluated at obviously gives the harmonic series.

However, Fejér was able to come up with universal convergence factors (therebye anticipating Gelb’s and Tadmor’s concentration factors) and to prove the following theorem, [5, p.183].

Theorem 5 (Fejér 1913) Under the Dirichlet conditions on it holds

where again is the smallest positive solution of (1.8). In points of smoothness of this limit gives zero.

Hence, if one already got hold of

as well as then

Fejér also investigated another method to compute the jump height [5, p.186f].

Theorem 6 (Fejér 1913) If f obeys the Dirichlet conditions and if is a function of bounded variation on, then

At points of continuity of this limit again gives zero.

While he showed in [

Theorem 7 (Fejér 1914) If a trigonometric series converges uniformly in, then the conjugate series converges almost everywhere, i.e. with the exception of a set of measure zero.

Fejér’s ideas were taken up by Ferenc (Franz) Lukács, who died prematurely in 1918, in his paper [

Theorem 8 (Lukács 1920) If f is Lebesgue-integrable on and -periodic and if the limit

exists, then3

Note that this theorem gives nothing but (1.5).

In contrast to [

each. The grid points are given by

and applying the composite trapezoidal rule results in the formulae (for)

and

It is as obvious as important an observation that in the discrete case the data consists of jumps from grid point to grid point. The jumps of order are acceptable, but the jumps indicate a jump discontinuity in the underlying function. Hence, we indicate a jump discontinuity at a point by means of the grid cell in which the jump occurs. The jump is then characterized by

Unfortunetely, the convergence rate is very slow, which can be seen in

In the discrete case the continuous function is not sufficient to be a concentration factor since discrete data is pestered with jumps by the sheer nature of discrete data. Instead of using the continuous function alone one has to use a product of with the coefficient

, which leads to the simplest concentration factor

Note that in this paper we follow the notational conventions of Gelb and Tadmor [

and vice versa.

Example 3.1 We consider a test function taken from [

on. Note that exhibits exactly one discontinuity of the first kind at. We first compute the Fourier series of and the conjugate series without using a concentration kernel. In order to avoid interference from quadrature errors we always choose. In

with significantly better concentration rates as can be seen in

To prove the concentration property for factors in the discrete case in detail, we start with the definition of an admissible kernel taken directly from [

Definition 1 (Admissible Kernels) A conjugate kernel is called admissible if it satisfies the following four properties:

We call a bounded concentration factor admissible, if is an admissible kernel.

Inserting in the definition of an admissible kernel gives following equivalent conditions in terms of:

We will now prove the following theorem in detail (see [

Theorem 9 If the continuous concentration factor

satisfies the concentration property, then the generalised discrete conjugated Fourier partial sum with the factor

satisfies also the concentration property.

Proof 1 We have to investigate the generalised conjugate partial sum (1.6) and the discrete Fourier coefficients and. In using the -periodicity of and by summation by parts we get

Now the mid-term is moved into the first sum and telescoping of the last sum yields

Next we expand by and recognize the fact by periodicity of the function and

and we get

Finally

Note that denotes the midpoint of the cell which encloses the discontinuity at. In complete analogy we find

Turning to the discrete conjugate Fourier partial sum (1.6) and inserting the expressions for and as derived above, this yields for

By employing the formulae

and

it follows

Inserting with, we get

Since is admissible, we can now use the concentration property in the continuous case shown in [

thus.

Gelb and Tadmor proved in [

Theorem 10 Consider a discrete concentration function such that.

Then the factors are admissible and the concentration property is satisfied,

if the following conditions are met:

The main part of the proof is to show that the associated conjugated kernel is admissible. We will prove two useful lemmata beforehand.

Lemma 1 Assume that the concentration function satisfies

Then property (P2’) and hence (P2) hold.

Proof 2 Let. By continuity,

By summing such terms we get

Employing the series expansion of we arrive at

and by the series expansion of the logarithm it follows

An index transformation and expansion yields

Hence, the result is proven.

Lemma 2 Consider the conjugate kernel

with concentration function Then the following estimate holds:

Proof 3 Let. First, summation by parts yields

For the following calculation, we use

to get

Using and the telescoping sum of it follows

Once again summation by parts yields

We arrive at the following result

and hence

Now we use the identity

to estimate as follows:

With, and with the mean value theorem,

The required result is hence proven.

We can now show that our main Theorem 10 holds.

Proof 4 (Proof: of Theorem 10) It is sufficient to prove that is an admissible kernel. Then, by Theorem 1, it follows that satisfies the concentration property. Lemma 1 directly yields the required properties (P2) and (P2’), respectively. The remaining properties (P3’) and (P4’) follow form Lemma 2. We have for (P3’):

Using and for all leads to and so (P3’) is satisfied.

It remains to prove (P4’). We have

In the last step we used the fact that.

Hence we have shown (P2’)-(P4’) and thereby demonstrated that is an admissible kernel which finishes the proof.

To treat the 2D case we now consider the square and a - periodic function. The Fourier series associated with is then given as

where

As in the one-dimensional case there is a complex form given by

where

However, unlike in the one-dimensional case, a conjugate Fourier series in two dimensions can not be derived from a complex power series on a polydisc, cp. [

2) conjugation w.r.t. the second variable

3) conjugation w.r.t. both variables

cp. [

Much can be said about conjugate Fourier series in multiple space dimensions and the interested reader is referred to [12-17]. The two formal conjugate series’ for each of the single variables are of no interest to us but we can easily see from these that the complex form of the one-dimensional conjugate series is

with

We are only interested in the main result of [

Theorem 11 (Móricz 2001) Let, , and

If there exists a number such that

and if there is a constant such that

where, then

As is well known from finite difference calculus,

and so we may conclude that the partial sums of the conjugate Fourier series w.r.t. both variables gives rise to an indicator of the jump in the mixed derivative, namely. Some approaches of this fully 2D edge detection using generalised conjugated partial sums are covered in [

Summarizing this work we have first given a review of both classical as well as modern approaches to detect jump discontinuities using conjugated Fourier partial sums. The ideas proposed by Gelb and Tadmor [

Interesting questions arise when the 2-d case is considered. Since the expansion of the conjugated partial sum is not unique, there are three different approaches, in which two of them (the pseudo-2-d case) have been covered by Gelb and Tadmor [