Composite Transformation Matrix Decomposition for Rasterized Data

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Solution Overview

Problem

Conventional systems face challenges in properly decomposing composite transformation matrices to match the transformation order of rendering devices, leading to inefficiencies in rendering rasterized data, particularly when rotation, scaling, and translation operations are performed in different orders.

Innovation Solution

A method and system that decompose composite transformation matrices using polar decomposition to derive rotation, scaling, and translation matrices, allowing these operations to be performed in any desired order, thereby aligning with the rendering device's operation sequence.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Device complexity

If conventional systems use composite transformation matrices to represent translation, scaling, and rotation operations, then the mathematical representation is compact and efficient, but the systems cannot properly decompose these matrices to match the fixed transformation order required by rendering devices

Engineering Contradiction:

Improvetransformation representationVSAvoidtransformation order compatibility

Core Design Contradiction:

Device complexityVSAdaptability or versatility

Solution Approach 1:

The patent segments the composite transformation matrix into individual transformation components (translation, scaling, rotation) in a specific order. By decomposing the composite matrix into sequential elementary transformations, the system can apply these operations in the order required by rendering devices while maintaining the mathematical efficiency of matrix representation.

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent applies reverse polar decomposition to invert the conventional transformation order. Instead of representing transformations in the order they are applied (translation, then scaling, then rotation), the system decomposes the composite matrix to extract operations in reverse order, allowing reordering to match rendering device requirements.

Inventive Principle:

Principle #13The other way round (Inversion)

2Adaptability or versatility

If the system maintains decoupling between algorithms and hardware specifics to enable abstract algorithm development, then algorithm portability and reusability improve, but the system loses direct control over transformation operation ordering

Engineering Contradiction:

Improvealgorithm portabilityVSAvoidtransformation operation order

Core Design Contradiction:

Adaptability or versatilityVSManufacturing precision

Solution Approach 1:

The patent introduces an intermediary decomposition process between the abstract algorithm layer and the hardware execution layer. The composite transformation matrix serves as an intermediary that encapsulates transformation information in a hardware-agnostic manner, while the reverse polar decomposition acts as a mediator that translates this abstract representation into the specific operation order required by rendering devices.

Inventive Principle:

Principle #24Intermediary (Mediator)

3Productivity

If conventional systems abandon the composite transformation matrix after rasterization, then the data flow simplifies for conventional printing applications, but transformation information is lost and cannot be applied to rasterized data

Engineering Contradiction:

Improvedata processing efficiencyVSAvoidtransformation information

Core Design Contradiction:

ProductivityVSLoss of information

Solution Approach 1:

The patent performs preliminary decomposition of the composite transformation matrix into individual transformation operations before the rasterization process completes. By extracting and preserving the transformation components in advance, the system ensures that transformation information remains available for application to rasterized data in subsequent processing stages, preventing information loss while maintaining processing efficiency.

Inventive Principle:

Principle #10Preliminary action

Data Source

PatentUS9659392B2Method and system for utilizing transformation matrices to process rasterized image data

Publication Date: 2017.05.23 XEROX CORP

AI summary

A method and system render rasterized data by receiving non-rasterized page description language data and a corresponding transformation matrix representing transformation operations to be performed. The non-rasterized page description language data is rasterizing to create rasterized data. The corresponding transformation matrix is decomposed into a plurality of individual transformation operation matrices and a discrete transformation operation value, from each corresponding individual transformation operation matrix, is generated for each transformation operation to be performed upon the rasterized data. The transformation operations are performed upon the rasterized data based upon the generated discrete transformation operation values.