A Fractal Approach Towards Visual Analogy

We present a preliminary computational model of visual analogy that uses fractal image representations that rely only on the grayscale pixel values of input images, and are mathematical abstractions quite rigorously grounded in the theory of fractal image compression. We have applied this model of visual analogy to problems from the Raven’s Progressive Matrices intelligence test, and we describe in detail the fractal solution strategy as well as some preliminary results. Finally, we discuss the implications of using these fractal representations for memory recall and analogical reasoning. 1 Cognition, Computation, and Creativity We may study creative behavior at many levels of aggregation and abstraction ranging from environmental and genetic to neural and cognitive to social and cultural. Our work on computational creativity focuses on the cognitive level. Although at present there is little agreement in the cognitive sciences about the proper characterization of creativity, nevertheless there is broad and deep consensus that analogy is a core cognitive process of creative behavior, e.g. [1]–[8]. Indeed, some cognitive scientists have argued that analogy is a core process not only of creativity, but all cognition, including perception [6][9]. Thus, understanding the computational processes of analogy seems critical to understanding and developing computational models of creativity. We may classify computational theories of analogy into two broad categories. In one category are theories that propose general-purpose mechanisms for mapping and transfer of relations from one problem to another, e.g. production systems [10], structure mapping [11][12], schema induction [13], and constraint satisfaction [14]. These mechanism theories typically make few commitments about the contents of knowledge; indeed, their mechanisms are general-purpose because they are agnostic towards knowledge contents. In the other category are computational theories that describe contents of knowledge that drive analogies in creative tasks such as story understanding [15], diagram understanding [16][17], scientific problem solving [18][19], innovative design [20][21][22], and calligraphy [23]. These content theories of analogy, too, describe computational processes, but their processes are driven by the contents of knowledge (and corresponding vocabularies for capturing the knowledge contents). The extant content and mechanism computational theories of analogy nevertheless are united in their use of propositional representations: From the perspective of the content theories of analogy, the mechanism theories offer potential substrates for computational implementation. We present a preliminary computational theory of visual analogy that is fundamentally different from both content and mechanism theories because it uses fractal representations instead of propositional ones. Visual analogy is a topic of longstanding interest in computational creativity because of its central role in many creative tasks such as intelligence tests [24]. In particular, in this paper, we focus on geometric analogy problems that are a type of visual analogy problems: geometric analogy problems focus solely on shapes, sizes and locations of the shapes, and spatial relations among the shapes. Fractal image representations rely only on the grayscale pixel values of input images, and are mathematical abstractions quite rigorously grounded in the theory of fractal image compression [25]. Below, first we describe how fractal representations can be used for computing features and similarity among distinct images, and how these operations can be combined to form a technique for creating visual analogies using purely pictorial inputs. Second, we present initial experimental results from using this method on geometric analogy problems that occur on the standardized intelligence test called the Raven’s Progressive Matrices test [26]. Third, we discuss how this fractal model of visual analogy may be considered to be a mechanism for creativity. 2 Fractal Image Representation Consider the general form of an analogy problem as: