To achieve rotation, scale, and translation invariance in Fourier analysis, one can use a technique called scale-invariant feature transform (SIFT). SIFT is a widely used method in computer vision for detecting and describing local features in images.

Here are the steps to achieve rotation, scale, and translation invariance using SIFT in Fourier analysis:

1. Apply the Fourier transform to the image.
2. Detect key points in the Fourier transformed image using the SIFT algorithm. The SIFT algorithm will identify the features in the Fourier transformed image that are robust to changes in scale, rotation, and translation.
3. Compute the descriptors for each key point using SIFT. Descriptors are a set of local features that describe the characteristics of the image around the key point.
4. Match the descriptors of the Fourier transformed image with the descriptors of a reference image to determine the similarity between the two images.
5. Use the matched key points to estimate the transformation parameters (scale, rotation, and translation) between the Fourier transformed image and the reference image.
6. Apply the estimated transformation parameters to the original image to achieve rotation, scale, and translation invariance.

By using SIFT to detect and describe features in the Fourier transformed image, the resulting descriptors are invariant to changes in scale, rotation, and translation. This allows for the matching of features between images with different orientations and scales, and also enables the estimation of the transformation parameters necessary to achieve invariance.