Approximation Theory is an art-research project in visual mathematics and data aesthetics. It consists of a series of prints and videos that visualize the mathematical idea of approximation.


Development of this project was partly funded by City University research grants 9610322 and 7004725.


This project is a collaboration between Felipe Cucker & Hector Rodriguez.

The methodology used in the work involves the choice of a set of fixed dictionaries or databases of images. Each dictionary has its own distinctive quality. For instance, some consist of linear or curved elements generated procedurally.

Figure 1. A few images from the rectilinear dictionary

Figure 2. A few images from the curvilinear dictionary

Other dictionaries might consist of more complicated images, such as frames from movies or Chinese characters.

Figure 3. A few images from the Chinese dictionary

Any other image can then be reconstructed as a weighted superposition of all or some of the images in the dictionary. This set of images is called a basis. For instance, this source image from the 1936 Summer Olympics can be reconstructed by superposing images from a basis of 3840 elements (each of them of the "linear" dictionary).

Figure 4. An image of a swimmer and a reconstruction of it

There exist mathematical procedures that identify how to superpose a given subset of images from the dictionary in a manner that most closely reconstructs or "approximates" the source image.

The character of the approximation depends on two kinds of factors: qualitative and quantitative.

The quantitative aspect has to do with the number of images that are used in the reconstruction; i.e., the size of the basis. The sequence of digital prints in Figure 5 shows several reconstructions of one source image using progressively larger bases from the same dictionary.

Figure 5. Reconstructions using 30, 60, 120, 360, 720, and 1920 images from the dictionary

The qualitative aspect has to do with the character of the images in the dictionary, for instance whether they are linear or curved. Figure 6 shows an image and three reconstructions obtained using bases of the same size from the three dictionaries shown in Figures 1-3. While the three results resemble the original, each has its own distinctive personality.

Figure 6. Three reconstructions with different dictionaries

This video illustrates the qualitative aspect of approximation theory. It shows reconstructions of short clips done with the three dictionaries above. In all three cases the size of the basis is moderate (250 images). This allows the viewer to focus on the reconstruction of the motion rather than on the accuracy of detail.

The following video encapsulates the fundamental character of the project in the form of a grid. Qualitative aspects of approximation theory are shown along the vertical axis: the same images are approximated using four qualitatively different dictionaries. The dictionaries consist of straight lines, curves, Chinese characters, frames from Godard's movie Alphaville. Quantitative aspects of approximation theory are shown along the horizontal axis. The same images are approximated using more and more elements from each of the four elements. As more elements are used, the reconstruction is closer to the original image but the distinctive character of the dictionary becomes less marked.

Our work uses source images drawn from the 1936 Summer Olympics, as shot by director Leni Riefenstahl. These images express a fascination with the human form, and one of our main concerns has to to do with the representation of the body in an algorithmic age. The images are also drawn from an age where the threat of fascism was very real, and manifested itself in the cult of the body. We are also confronting a growing sense of populist authoritarianism around the world. Our response to this threat is to reassert the value of rational analysis and the integration of art, science, and cultural critique. An additional reason to choose Riefenstahl's movie is that it naturally provides a diverse, yet thematically coherent, collection of short clips, rich in movement. The reconstruction of these clips with different dictionaries allows for a visual comparison of the dictionaries' characters.

This project can be exhibited as a set of prints that explore different aspects (both qualitative and quantitative) of mathematical approximation theory, together with videos that compare different reconstructions. The number of prints and videos can be adjusted to take into account venue and equipment constraints.

Preferred Setup

The exhibition comprises three sections or “chapters” that explore different aspects of the concept of approximation. Please contact the artists if you would like to exhibit the work on a smaller scale.


Section 1 – The concept of Mathematical Approximation

This section serves as an introduction to the visual concept of Approximation Theory - approximating a source image using a database, or a “dictionary”, of random, unrelated images. Viewer will walk into this area surrounded by 3 walls. The center wall displays a large print of the approximation of a photograph of an Olympic swimmer. The walls on the sides will show the entire dictionary of 3840 binary, rectilinear images upon which the final approximation is constructed. The brightness of all the elements is weighted according to their contribution to the approximation as well.

Figure 1. Rectilinear approximation of swimmer

Figure 2 & 3. Complete rectilinear dictionary, split into 2 walls

The image of the swimmer is to be in a wall opposite the entrance. The dictionary is to be located on two walls, each one on either side of the wall that displays the swimmer.

The recommended dimensions for the images on the walls are as follows:

Center Image of the swimmer: 1.4m wide x 1m high

Wall print of the rectilinear dictionary: 3.4m wide x 2m high x 2 walls

In a second room, the same image is to be shown, reconstructed from a different dictionary. The dictionary is also to be shown in the room.

Figure 4. Curvilinear approximation of swimmer

Figure 5 & 6. Complete curvilinear dictionary, split into 2 walls

Section 2 – Qualitative

This area will consist of a set of 4 prints, aligned as a 2-by-2 grid, showing an image of a diver in the air juxtaposed with 3 different reconstructions using the different dictionaries (rectilinear, curvilinear and traditional Chinese daxie numerals) of the same size.

This set of images illustrates how properties of the dictionaries affect the visual quality of the final reconstruction. For instance, the geometric properties of each of the dictionaries can be clearly recognized in their respective reconstruction.

Figure 7. 2-by-2 qualitative set of diving images

The recommended dimensions the image set are as follows:

Dimensions for each of the 4 prints: 32cm wide x 40 cm high

In addition, the room should contain a monitor (dimensions variable) that displays several clips illustrating different sets of pre-rendered moving images from the film Olympia (1938). The video clips are on .mov format and can be played from a VLC media player.

The following is an image that illustrates one of the clips.

Section 3 - Quantitative

This section will present a series of images approximated from a photograph of Jesse Owens in the 1936 Olympics, showcasing a quantitative progression in correspondence to the original source. The series contains 9 prints; all of them are approximated using the elements from the same set of rectilinear dictionary, but an increasing size of basis, so that the elements used in one approximation is also a subset of its subsequent approximations.

It is recommended that the series should be displayed as a single row to enhance a sense of ascendance of accuracy when the size of the basis is increasing along the series. Another option would be to arrange the 9 prints into a 3-by-3 grid, with the size of the basis increasing from the top-left corner to the bottom-right.

Figure 8. 3-by-3 setup of the Owens series

The recommended dimensions the image set are as follows:

Dimensions for each of the 9 prints: 75cm wide x 50 cm high

Mathematical framework

The idea of approximation underpins many aspects of digital technology, including machine learning, computer vision, and digital image and sound processing. In many of these applications, it is vital to replace a complicated function (for instance, one that is expensive to compute or tedious to describe) by a simpler one. Digital data compression, for instance, often involves the replacement of image or audio data by a shorter description. Approximation theory aims to find the best approximation of the desired input.

The context of approximation theory is the following: there is a target function f which we want to approximate with another function g to be selected from within a set S of approximant functions. The main problem is that of estimating how good this g can be, that is, how small can the error of replacing f by g be. This problem presupposes a way of measuring this error (usually a distance in a space of functions where both f and g live). In addition, it depends on both the target function f and the set of approximant functions S. This dependence has both quantitative and qualitative aspects.

On the quantitative side there are parameters, such as the dimension of the linear space spanned by S (typically, when S spans a finite dimensional linear space) or the radius of S (typically, when S is a closed ball in a linear space, possibly of infinite dimension).

Our project is an experimental exploration of the quantitative and qualitative aspects of approximation.

Intuitively, a digital image can be viewed as a position or a point in a space of possible images. More precisely, an image of N (grayscale or "black-and-white") pixels is identified with a vector of N real numbers in the interval [0, 1], with 0 being black and 1 being white. Our target function (for instance a photograph that we wish to reconstruct) is a point in this N-dimensional vector space. A set S of approximant functions is a subset of the unit cube in this space which we will assume it is closed under weighted sums. In particular, S is a set of N-pixel images (the "dictionary").

Any target image is to be approximately reconstructed as a weighted sum (linear combination) of the elements of a basis (a selection of images from the dictionary). Intuitively, we can think of this procedure as a superposition or mix of brightened or darkened versions of the various images in the basis. By changing the intensities of the images in the basis in appropriate ways, the resulting combination will resemble approximately the target image. The "appropriate ways" of mixing the images in the basis are determined automatically by mathematically projecting the target image onto the subspace spanned by the images in the basis. The measure of error implicitly used is Euclidean distance.

The quantitative side mainly involves the number of images to be used in any approximation, i.e., the size of the basis. We might choose to employ only a relatively small subset of images in the dictionary. As the number of images employed decreases, the fidelity of the resulting reconstruction also decreases, and vice versa.

The qualitative side involves the character of the images in the dictionary at hand. Our project explores how the use of different kinds of images (for instance geometrically abstract frames consisting of lines or checkerboards, photographs of different locations in the world or still frames from classical films) affects the visual quality of the reconstruction.

We are currently exploring how the use of advanced mathematical tehcniques, such as "L1-minimization", may affect the visual character of the result.