4 Contrast enhancement

Because some features are hardly detectable by eye in an image, we often transform it before display. Histogram equalization is one the most well-known methods for contrast enhancement. Such an approach is generally useful for images with a poor intensity distribution. Since edges play a fundamental role in image understanding, a way to enhance the contrast is to enhance the edges. For example, we can add to the original image its Laplacian ( $I'= I + \gamma \Delta I$, where $\gamma$ is a parameter). Only features at the finest scale are enhanced (linearly). For a high $\gamma$ value, only the high frequencies are visible.

Since the curvelet transform is well-adapted to represent images containing edges, it is a good candidate for edge enhancement. Curvelet coefficients can be modified in order to enhance edges in an image. The idea is to not modify curvelet coefficients which are either at the noise level, in order to not amplify the noise, or larger than a given threshold. Largest coefficients corresponds to strong edges which do not need to be amplified. Therefore, only curvelets coefficients with an absolute value in $[T_{{\rm min}},T_{{\rm max}}]$ are modified, where $T_{{\rm min}}$ and $T_{{\rm max}}$ must be fixed. We define the following function $y_{\rm c}$which modifies the values of the curvelet coefficients:

 

$$y_{\rm c}(x) = 1 \mbox{ if } x < T_{{\rm min}}$$

$$y_{\rm c}(x) = \frac{x-T_{{\rm min}}}{T_{{\rm min}}}\left(\frac{T... ...ight)^p + \frac{2T_{{\rm min}}-x}{T_{{\rm min}}} \mbox{ if } x < 2T_{{\rm min}}$$

$$y_{\rm c}(x) = \left(\frac{T_{{\rm max}}}{x}\right)^p \mbox{ if } 2T_{{\rm min}} \le x < T_{{\rm max}}$$

$$y_{\rm c}(x) = 1 \mbox{ if }x \ge T_{{\rm max}}.$$ (10)

Figure 8: Enhanced coefficients versus original coefficients. Parameters are $T_{{\rm max}} =$ 30, c=5 and p=0.5.

p determines the degree of non-linearity. $T_{{\rm min}}$ is derived from the noise level, $T_{{\rm min}} = c \sigma$. A c value larger than 3 guaranties that the noise will not be amplified. The $T_{{\rm max}}$ parameter can be defined either from the noise standard deviation ( $T_{{\rm max}} = K_{\rm m} \sigma$) or from the maximum curvelet coefficient $M_{\rm c}$ of the relative band ( $T_{{\rm max}} = l M_{\rm c}$, with l < 1). The first choice allows the user to define the coefficients to amplify as a function of their signal to noise ratio, while the second one gives an easy and general way to fix the $T_{{\rm max}}$ parameter independently of the range of the pixel values. Figure 8 shows the curve representing the enhanced coefficients versus the original coefficients.

The curvelet enhancement method consists of the following steps:

1.

Estimate the noise standard deviation $\sigma$ in the input image I.

2.

Calculate the curvelet transform of the input image. We get a set of bands w_j, each band w_j contains N_j coefficients and corresponds to a given resolution level.

3.

Calculate the noise standard deviation $\sigma_j$ for each band j of the curvelet transform (see Starck et al. 2002 more details on this step).

4.

For each band j do

• Calculate the maximum M_j of the band.
• Multiply each curvelet coefficient w_j,k by $y_{\rm c}(\mid w_{j,k} \mid )$.

5.

Reconstruct the enhanced image from the modified curvelet coefficients.

Example: Saturn image

Figure 9: Top, Saturn image and its histogram equalization. Bottom, Saturn image enhancement the Laplacian method and by the curvelet transform.

Figure 9 shows respectively a part of the Saturn image, the histogram equalized image, the Laplacian enhanced image and the curvelet multiscale edge enhanced image (parameters were p=0.5, c=3, and l=0.5). The curvelet multiscale edge enhanced image shows clearly better the rings and edges of Saturn.